Rs Aggarwal for Class 6 Math Chapter 1 - Number System

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Rs Aggarwal Solutions for Class 6 Math Chapter 1 Number System are provided here with simple step-by-step explanations. These solutions for Number System are extremely popular among Class 6 students for Math Number System Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal Book of Class 6 Math Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal Solutions. All Rs Aggarwal Solutions for class Class 6 Math are prepared by experts and are 100% accurate.

Page No 82:

Question 1:

Write the fraction representing the shaded portion:
(i) Figure
(ii) Figure
(iii) Figure
(iv) Figure
(v) Figure
(vi) Figure

Answer:

(i) The shaded portion is 3 parts of the whole figure.
     $∴$ $\frac{3}{4}$
(ii) The shaded portion is 1 parts of the whole figure.
     $∴$ $\frac{1}{4}$
(iii) The shaded portion is 2 parts of the whole figure.
     $∴$ $\frac{2}{3}$
(iv) The shaded portion is 3 parts of the whole figure.
     $∴$ $\frac{3}{10}$
(v)The shaded portion is 4 parts of the whole figure.
     $∴$ $\frac{4}{9}$
(vi) The shaded portion is 3 parts of the whole figure.
   $∴$ $\frac{3}{8}$

Page No 82:

Question 2:

Shade $\frac{4}{9}$ of the given figure.
Figure

Answer:

Page No 82:

Question 3:

In the given figure, if we say that the shaded region is $\frac{1}{4}$ , then identify the error in it.
Figure

Answer:

The given rectangle is not divided into four equal parts.

Thus, the shaded region is not equal to $\frac{1}{4}$ of the whole.

Page No 82:

Question 4:

Write a fraction for each of the following:
(i) three-fourths
(ii) four-sevenths
(iii) two-fifths
(iv) three-tenths
(v) one-eighth
(vi) five-sixths
(vii) eight-ninths
(viii) seven-twelfths

Answer:

(i) $\frac{3}{4}$ (ii) $\frac{4}{7}$ (iii) $\frac{2}{5}$ (iv) $\frac{3}{10}$ (v) $\frac{1}{8}$
(vi) $\frac{5}{6}$ (vii) $\frac{8}{9}$ (viii) $\frac{7}{12}$

Page No 83:

Question 5:

Write down the numerator and the denominator of each of the fractions given below:
(i) $\frac{4}{9}$
(ii) $\frac{6}{11}$
(iii) $\frac{8}{15}$
(iv) $\frac{12}{17}$
(v) $\frac{5}{1}$

Answer:

Numerator Denominator
(i) 4       9
(ii) 6 11
(iii) 8    15
(iv) 12 17
(v) 5    1

Page No 83:

Question 6:

Write down the fraction in which
(i) numerator = 3, denominator = 8
(ii) numerator = 5, denominator = 12
(iii) numerator = 7, denominator = 16
(iv) numerator = 8, denominator = 15

Answer:

(i) $\frac{3}{8}$ (ii) $\frac{5}{12}$ (iii) $\frac{7}{16}$ (iv) $\frac{8}{15}$

Page No 83:

Question 7:

Write down the fractional number for each of the following:
(i) $\frac{2}{3}$
(ii) $\frac{4}{9}$
(iii) $\frac{2}{5}$
(iv) $\frac{7}{10}$
(v) $\frac{1}{3}$
(vi) $\frac{3}{4}$
(vii) $\frac{3}{8}$
(viii) $\frac{9}{14}$
(ix) $\frac{5}{11}$
(x) $\frac{6}{15}$

Answer:

(i) two-thirds
(ii) four $-$ ninths
(iii) two $-$ fifths
(iv) seven $-$ tenths
(v) one $-$ thirds
(vi) three $-$ fourths
(vii) three $-$ eighths
(viii) nine $-$ fourteenths
(ix) five $-$ elevenths
(x) six $-$ fifteenths

Page No 83:

Question 8:

What fraction of an hour is 24 minutes?

Answer:

We know: 1 hour = 60 minutes
∴ The required fraction = $\frac{24}{60} = \frac{2}{5}$

Page No 83:

Question 9:

How many natural numbers are there from 2 to 10? What fraction of them are prime numbers?

Answer:

There are total 9 natural numbers from 2 to 10. They are 2, 3, 4, 5, 6, 7, 8, 9, 10.
Out of these natural numbers, 2, 3, 5, 7 are the prime numbers.
∴ The required fraction = $\frac{4}{9}$ .

Page No 83:

Question 10:

Determine:
(i) $\frac{2}{3}$ of 15 pens
(ii) $\frac{2}{3}$ of 27 balls
(iii) $\frac{2}{3}$ of 36 balloons

Answer:

(i) $\frac{2}{3}$ of 15 pens = $(\frac{2}{3_{1}} \times \frac{15^{5}}{1}) = 10 pens$
(ii) $\frac{2}{3}$ of 27 balls = $(\frac{2}{3_{1}} \times \frac{27^{9}}{1}) = 18 balls$
(iii) $\frac{2}{3}$ of 36 balloons = $(\frac{2}{3_{1}} \times \frac{36^{12}}{1}) = 24 balloons$

Page No 83:

Question 11:

Determine:
(i) $\frac{3}{4}$ of 16 cups
(ii) $\frac{3}{4}$ of 28 rackets
(iii) $\frac{3}{4}$ of 32 books

Answer:

(i) $\frac{3}{4}$ of 16 cups = $(\frac{3}{4_{1}} \times \frac{16^{4}}{1}) = 12 cups$
(ii) $\frac{3}{4}$ of 28 rackets = $(\frac{3}{4_{1}} \times \frac{28^{7}}{1}) = 21 rackets$
(iii) $\frac{3}{4}$ of 32 books = $(\frac{3}{4_{1}} \times \frac{32^{8}}{1}) = 24 books$

Page No 83:

Question 12:

Neelam has 25 pencils. She gives $\frac{4}{5}$ of them to Meena. How many pencils does Meena get? How many pencils are left with Neelam?

Answer:

Neelam gives $\frac{4}{5}$ of 25 pencils to Meena.
$(\frac{4}{5_{1}} \times \frac{25^{5}}{1}) = 20 Pencils$
Thus, Meena gets 20 pencils.
∴ Number of pencils left with Neelam = 25 $-$ 20 = 5 pencils
Thus, 5 pencils are left with Neelam.

Page No 83:

Question 13:

Represent each of the following fractions on the number line:
(i) $\frac{3}{8}$
(ii) $\frac{5}{9}$
(iii) $\frac{4}{7}$
(iv) $\frac{2}{5}$
(v) $\frac{1}{4}$

Answer:

Draw a 0 to 1 on a number line. Label point 1 as A and mark the starting point as 0.

(i) Divide the number line from 0 to 1 into 8 equal parts and take out 3 parts from it to reach point P.

(ii) Divide the number line from 0 to 1 into 9 equal parts and take out 5 parts from it to reach point P

.

(iii) Divide the number line from 0 to 1 into 7 equal parts and take out 4 parts from it to reach point P.

(Iv) Divide the number line from 0 to 1 into 5 equal parts and take out 2 parts from it to reach point P.

(v) Divide the number line from 0 to 1 into 4 equal parts and take out 1 part from it to reach point P.

Page No 85:

Question 1:

Which of the following are proper fractions?
$\frac{1}{2}, \frac{3}{5}, \frac{10}{7} \frac{7}{4}, 2, \frac{15}{8}, \frac{16}{16}, \frac{10}{11}, \frac{23}{10}$

Answer:

$\frac{1}{2}, \frac{3}{5}, \frac{10}{11}$

Page No 85:

Question 2:

Which of the following are improper fractions?
$\frac{3}{2}, \frac{5}{6}, \frac{9}{4}, \frac{8}{8}, 3, \frac{27}{16}, \frac{23}{31}, \frac{19}{18}, \frac{10}{13}, \frac{26}{26}$

Answer:

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction. Hence, $\frac{3}{2}, \frac{9}{4}, \frac{8}{8}, \frac{27}{16}, \frac{19}{18} and \frac{26}{26}$ are improper fractions.

Page No 85:

Question 3:

Write six improper fractions with denominator 5.

Answer:

Clearly, $\frac{6}{5}, \frac{7}{5}, \frac{8}{5}, \frac{9}{5}, \frac{11}{5} and \frac{12}{5}$ are improper fractions, each with 5 as the denominator.

Page No 85:

Question 4:

Write six improper fractions with numerator 13.

Answer:

Clearly, $\frac{13}{2}, \frac{13}{3}, \frac{13}{4}, \frac{13}{5}, \frac{13}{6}, \frac{13}{7}$ are improper fractions, each with 13 as the numerator.

Page No 85:

Question 5:

Convert each of the following into an improper fraction:
(i) $5 \frac{5}{7}$
(ii) $9 \frac{3}{8}$
(iii) $6 \frac{3}{10}$
(iv) $3 \frac{5}{11}$
(v) $10 \frac{9}{14}$
(vi) $12 \frac{7}{15}$
(vii) $8 \frac{8}{13}$
(viii) $51 \frac{2}{3}$

Answer:

We have:
(i) $5 \frac{5}{7} = \frac{(5 \times 7) + 5}{7} = \frac{40}{7}$

(ii) $9 \frac{3}{8} = \frac{(9 \times 8) + 3}{8} = \frac{75}{8}$

(iii) $6 \frac{3}{10} = \frac{(6 \times 10) + 3}{10} = \frac{63}{10}$

(iv) $3 \frac{5}{11} = \frac{(3 \times 11) + 5}{11} = \frac{38}{11}$

(v) $10 \frac{9}{14} = \frac{(10 \times 14) + 9}{14} = \frac{149}{14}$

(vi) $12 \frac{7}{15} = \frac{(12 \times 15) + 7}{15} = \frac{187}{15}$

(vii) $8 \frac{8}{13} = \frac{(8 \times 13) + 8}{13} = \frac{112}{13}$

(viii) $51 \frac{2}{3} = \frac{(51 \times 3) + 2}{3} = \frac{155}{3}$

Page No 85:

Question 6:

Convert each of the following into a mixed fraction:
(i) $\frac{17}{5}$
(ii) $\frac{62}{7}$
(iii) $\frac{101}{8}$
(iv) $\frac{95}{13}$
(v) $\frac{81}{11}$
(vi) $\frac{87}{16}$
(vii) $\frac{103}{12}$
(viii) $\frac{117}{20}$

Answer:

(i) On dividing 17 by 5, we get:
Quotient = 3
    Remainder = 2
∴ $\frac{17}{5} = 3 + \frac{2}{5} = 3 \frac{2}{5}$

(ii) On dividing 62 by 7, we get:
Quotient = 8
    Remainder = 6
∴ $\frac{62}{7} = 8 + \frac{6}{7} = 8 \frac{6}{7}$

(iii) On dividing 101 by 8, we get:
Quotient = 12
    Remainder = 5
∴ $\frac{101}{8} = 12 + \frac{5}{8} = 12 \frac{5}{8}$

(iv) On dividing 95 by 13, we get:
Quotient = 7
    Remainder = 4
∴ $\frac{95}{13} = 7 + \frac{4}{13} = 7 \frac{4}{13}$

(v) On dividing 81 by 11, we get:
Quotient = 7
    Remainder = 4
∴ $\frac{81}{11} = 7 + \frac{4}{11} = 7 \frac{4}{11}$

(vi) On dividing 87 by 16, we get:
Quotient = 5
    Remainder = 7
∴ $\frac{87}{16} = 5 + \frac{7}{16} = 5 \frac{7}{16}$

(vii) On dividing 103 by 12, we get:
Quotient = 8
    Remainder = 7
∴ $\frac{103}{12} = 8 + \frac{7}{12} = 8 \frac{7}{12}$

(viii) On dividing 117 by 20, we get:
Quotient = 5
    Remainder = 17
∴ $\frac{117}{20} = 5 + \frac{17}{20} = 5 \frac{17}{20}$

Page No 85:

Question 7:

Fill up the blanks with '>', '<' or '=':
(i) $\frac{1}{2} 1$
(ii) $\frac{3}{4} 1$
(iii) $1 \frac{6}{7}$
(iv) $\frac{6}{6} 1$
(v) $\frac{3016}{3016} 1$
(vi) $\frac{11}{5} 1$

Answer:

An improper fraction is greater than 1. Hence, it is always greater than a proper fraction, which is less than 1.
(i) $\frac{1}{2} < 1$

(ii) $\frac{3}{4} < 1$

(iii) $1 > \frac{6}{7}$

(iv) $\frac{6}{6} = 1$

(v) $\frac{3016}{3016} = 1$

(vi) $\frac{11}{5} > 1$

Page No 86:

Question 8:

Draw number lines and locate the following points:
(i) $\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \frac{4}{4}$
(ii) $\frac{1}{8}, \frac{2}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}$
(iii) $\frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{8}{5}$

Answer:

(i) Draw a number line. Mark 0 as the starting point and 1 as the ending point.
Then, divide 0 to 1 in four equal parts, where each part is equal to 1/4.
Show the consecutive parts as 1/4, 1/2, 3/4 and at 1 show 4/4 = 1.

(ii) Draw 0 to 1 on a number line. Divide the segment into 8 equal parts, each part corresponds to 1/8. Show the consecutive parts as 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8 and 8/8. Highlight the required ones only.

(iii) Draw 0 to 2 on a number line. Divide the segment between 0 and 1 into 5 equal parts, where each part is equal to 1/5.
Show 2/5, 3/5, 4/5 and 8/5 3 parts away from 1 towards 2. (1 < 8/5 < 2)

Page No 89:

Question 1:

Write five fractions equivalent to each of the following:
(i) $\frac{2}{3}$
(ii) $\frac{4}{5}$
(iii) $\frac{5}{8}$
(iv) $\frac{7}{10}$
(v) $\frac{3}{7}$
(vi) $\frac{6}{11}$
(vii) $\frac{7}{9}$
(viii) $\frac{5}{12}$

Answer:

(i) $\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{2 \times 3}{3 \times 3} = \frac{2 \times 4}{3 \times 4} = \frac{2 \times 5}{3 \times 5} = \frac{2 \times 6}{3 \times 6}$

∴ $\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12} = \frac{10}{15} = \frac{12}{18}$

Hence, the five fractions equivalent to $\frac{2}{3}$ are $\frac{4}{6}, \frac{6}{9}, \frac{8}{12}, \frac{10}{15} and \frac{12}{18}$ .

(ii) $\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{4 \times 3}{5 \times 3} = \frac{4 \times 4}{5 \times 4} = \frac{4 \times 5}{5 \times 5} = \frac{4 \times 6}{5 \times 6}$

∴ $\frac{4}{5} = \frac{8}{10} = \frac{12}{15} = \frac{16}{20} = \frac{20}{25} = \frac{24}{30}$

Hence, the five fractions equivalent to $\frac{4}{5}$ are $\frac{8}{10}, \frac{12}{15}, \frac{16}{20}, \frac{20}{25} and \frac{24}{30}$ .

(iii) $\frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{5 \times 3}{8 \times 3} = \frac{5 \times 4}{8 \times 4} = \frac{5 \times 5}{8 \times 5} = \frac{5 \times 6}{8 \times 6}$

∴ $\frac{5}{8} = \frac{10}{16} = \frac{15}{24} = \frac{20}{32} = \frac{25}{40} = \frac{30}{48}$

Hence, the five fractions equivalent to $\frac{5}{8}$ are $\frac{10}{16}, \frac{15}{24}, \frac{20}{32}, \frac{25}{40} and \frac{30}{48}$ .

(iv) $\frac{7}{10} = \frac{7 \times 2}{10 \times 2} = \frac{7 \times 3}{10 \times 3} = \frac{7 \times 4}{10 \times 4} = \frac{7 \times 5}{10 \times 5} = \frac{7 \times 6}{10 \times 6}$

∴ $\frac{7}{10} = \frac{14}{20} = \frac{21}{30} = \frac{28}{40} = \frac{35}{50} = \frac{42}{60}$

Hence, the five fractions equivalent to $\frac{7}{10}$ are $\frac{14}{20}, \frac{21}{30}, \frac{28}{40}, \frac{35}{50} and \frac{42}{60}$ .

(v) $\frac{3}{7} = \frac{3 \times 2}{7 \times 2} = \frac{3 \times 3}{7 \times 3} = \frac{3 \times 4}{7 \times 4} = \frac{3 \times 5}{7 \times 5} = \frac{3 \times 6}{7 \times 6}$

∴ $\frac{3}{7} = \frac{6}{14} = \frac{9}{21} = \frac{12}{28} = \frac{15}{35} = \frac{18}{42}$

Hence, the five fractions equivalent to $\frac{3}{7}$ are $\frac{6}{14}, \frac{9}{21}, \frac{12}{28}, \frac{15}{35} and \frac{18}{42}$ .

(vi) $\frac{6}{11} = \frac{6 \times 2}{11 \times 2} = \frac{6 \times 3}{11 \times 3} = \frac{6 \times 4}{11 \times 4} = \frac{6 \times 5}{11 \times 5} = \frac{6 \times 6}{11 \times 6}$

∴ $\frac{6}{11} = \frac{12}{22} = \frac{18}{33} = \frac{24}{44} = \frac{30}{55} = \frac{36}{66}$

Hence, the five fractions equivalent to $\frac{6}{11}$ are $\frac{12}{22}, \frac{18}{33}, \frac{24}{44}, \frac{30}{55} and \frac{36}{66}$ .

(vii) $\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{7 \times 3}{9 \times 3} = \frac{7 \times 4}{9 \times 4} = \frac{7 \times 5}{9 \times 5} = \frac{7 \times 6}{9 \times 6}$

∴ $\frac{7}{9} = \frac{14}{18} = \frac{21}{27} = \frac{28}{36} = \frac{35}{45} = \frac{42}{54}$

Hence, the five fractions equivalent to $\frac{7}{9}$ are $\frac{14}{18}, \frac{21}{27}, \frac{28}{36}, \frac{35}{45} and \frac{42}{54}$ .

(viii) $\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{5 \times 3}{12 \times 3} = \frac{5 \times 4}{12 \times 4} = \frac{5 \times 5}{12 \times 5} = \frac{5 \times 6}{12 \times 6}$

∴ $\frac{5}{12} = \frac{10}{24} = \frac{15}{36} = \frac{20}{48} = \frac{25}{60} = \frac{30}{72}$

Hence, the five fractions equivalent to $\frac{5}{12}$ are $\frac{10}{24}, \frac{15}{36}, \frac{20}{48}, \frac{25}{60} and \frac{30}{72}$ .

Page No 89:

Question 2:

Which of the following are the pairs of equivalent fractions?
(i) $\frac{5}{6} and \frac{20}{24}$
(ii) $\frac{3}{8} and \frac{15}{40}$
(iii) $\frac{4}{7} and \frac{16}{21}$
(iv) $\frac{2}{9} and \frac{14}{63}$
(v) $\frac{1}{3} and \frac{9}{24}$
(vi) $\frac{2}{3} and \frac{33}{22}$

Answer:

The pairs of equivalent fractions are as follows:
(i) $\frac{5}{6} and \frac{20}{24} (\frac{20}{24} = \frac{5 \times 4}{6 \times 4})$
(ii) $\frac{3}{8} and \frac{15}{40} (\frac{15}{40} = \frac{3 \times 5}{8 \times 5})$
(iv) $\frac{2}{9} and \frac{14}{63} (\frac{14}{63} = \frac{2 \times 7}{9 \times 7})$

Page No 89:

Question 3:

Find the equivalent fraction of $\frac{3}{5}$ having
(i) denominator 30
(ii) numerator 24

Answer:

(i) Let $\frac{3}{5} = \frac{}{30}$
Clearly, 30 = 5 $\times$ 6
So, we multiply the numerator by 6.

∴ $\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$
Hence, the required fraction is $\frac{18}{30}$ .
(ii) Let $\frac{3}{5} = \frac{24}{}$
Clearly, 24 = 3 $\times$ 8
So, we multiply the denominator by 8.

∴ $\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}$
Hence, the required fraction is $\frac{24}{40}$ .

Page No 89:

Question 4:

Find the equivalent fraction of $\frac{5}{9}$ having
(i) denominator 54
(ii) numerator 35

Answer:

(i) Let $\frac{5}{9} = \frac{}{54}$
Clearly, 54 = 9 $\times$ 6
So, we multiply the numerator by 6.
∴ $\frac{5}{9} = \frac{5 \times 6}{9 \times 6} = \frac{30}{54}$
Hence, the required fraction is $\frac{30}{54}$ .
(ii) Let $\frac{5}{9} = \frac{35}{}$
Clearly, 35 = 5 $\times$ 7
So, we multiply the denominator by 7.
∴ $\frac{5}{9} = \frac{5 \times 7}{9 \times 7} = \frac{35}{63}$
Hence, the required fraction is $\frac{35}{63}$ .

Page No 89:

Question 5:

Find the equivalent fraction of $\frac{6}{11}$ having
(i) denominator 77
(ii) numerator 60

Answer:

(i) Let $\frac{6}{11} = \frac{}{77}$
Clearly, 77 = 11 $\times$ 7
So, we multiply the numerator by 7.

∴ $\frac{6}{11} = \frac{6 \times 7}{11 \times 7} = \frac{42}{77}$
Hence, the required fraction is $\frac{42}{77}$ .
(ii) Let $\frac{6}{11} = \frac{60}{}$
Clearly, 60 = 6 $\times$ 10
So, we multiply the denominator by 10.

∴ $\frac{6}{11} = \frac{6 \times 10}{11 \times 10} = \frac{60}{110}$
Hence, the required fraction is $\frac{60}{110}$ .

Page No 89:

Question 6:

Find the equivalent fraction of $\frac{24}{30}$ having numerator 4.

Answer:

Let $\frac{24}{30} = \frac{4}{}$
Clearly, 4 = 24 $\div$ 6
So, we divide the denominator by 6.
∴ $\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}$
Hence, the required fraction is $\frac{4}{5}$ .

Page No 89:

Question 7:

Find the equivalent fraction of $\frac{36}{48}$ with
(i) numerator 9
(ii) denominator 4

Answer:

(i) Let $\frac{36}{48} = \frac{9}{}$
Clearly, 9 = 36 $\div$ 4
So, we divide the denominator by 4.
∴ $\frac{36}{48} = \frac{36 \div 4}{48 \div 4} = \frac{9}{12}$
Hence, the required fraction is $\frac{9}{12}$ .
(ii) Let $\frac{36}{48} = \frac{}{4}$
Clearly, 4 = 48 $\div$ 12
So, we divide the numerator by 12.
∴ $\frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4}$
Hence, the required fraction is $\frac{3}{4}$ .

Page No 89:

Question 8:

Find the equivalent fraction of $\frac{56}{70}$ with
(i) numerator 4
(ii) denominator 10

Answer:

(i) Let $\frac{56}{70} = \frac{4}{}$
Clearly, 4 = 56 $\div$ 14
So, we divide the denominator by 14.
∴ $\frac{56}{70} = \frac{56 \div 14}{70 \div 14} = \frac{4}{5}$
Hence, the required fraction is $\frac{4}{5}$ .
(ii) Let $\frac{56}{70} = \frac{}{10}$
Clearly, 10 = 70 $\div$ 7
So, we divide the numerator by 7.
∴ $\frac{56}{70} = \frac{56 \times 7}{70 \times 7} = \frac{8}{10}$
Hence, the required fraction is $\frac{8}{10}$ .

Page No 89:

Question 9:

Reduce each of the following fractions into its simplest form:
(i) $\frac{9}{15}$
(ii) $\frac{48}{60}$
(iii) $\frac{84}{98}$
(iv) $\frac{150}{60}$
(v) $\frac{72}{90}$

Answer:

(i) Here, numerator = 9 and denominator = 15
Factors of 9 are 1, 3 and 9.
Factors of 15 are 1, 3, 5 and 15.
Common factors of 9 and 15 are 1 and 3.
H.C.F. of 9 and 15 is 3.
∴ $\frac{9}{15} = \frac{9 \div 3}{15 \div 3} = \frac{3}{5}$
Hence, the simplest form of $\frac{9}{15} is \frac{3}{5}$ .

(ii) Here, numerator = 48 and denominator = 60
Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Common factors of 48 and 60 are 1, 2, 3, 4, 6 and 12.
H.C.F. of 48 and 60 is 12.
∴ $\frac{48}{60} = \frac{48 \div 12}{60 \div 12} = \frac{4}{5}$
Hence, the simplest form of $\frac{48}{60} is \frac{4}{5}$ .

(iii) Here, numerator = 84 and denominator = 98
Factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 42 and 84.
Factors of 98 are 1, 2, 7, 14, 49 and 98.
Common factors of 84 and 98 are 1, 2, 7 and 14.
H.C.F. of 84 and 98 is 14.
∴ $\frac{84}{98} = \frac{84 \div 14}{98 \div 14} = \frac{6}{7}$
Hence, the simplest form of $\frac{84}{98} is \frac{6}{7}$ .

(iv) Here, numerator = 150 and denominator = 60
Factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 75 and 150.
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Common factors of 150 and 60 are 1, 2, 3, 5, 6, 10, 15 and 30.
H.C.F. of 150 and 60 is 30.
∴ $\frac{150}{60} = \frac{150 \div 30}{60 \div 30} = \frac{5}{2}$
Hence, the simplest form of $\frac{150}{60} is \frac{5}{2}$ .

(v) Here, numerator = 72 and denominator = 90
Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90.
Common factors of 72 and 90 are 1, 2, 3, 6, 9 and 18.
H.C.F. of 72 and 90 is 18.
∴ $\frac{72}{90} = \frac{72 \div 18}{90 \div 18} = \frac{4}{5}$
Hence, the simplest form of $\frac{72}{90} is \frac{4}{5}$ .

Page No 89:

Question 10:

Show that each of the following fractions is in the simplest form:
(i) $\frac{8}{11}$
(ii) $\frac{9}{14}$
(iii) $\frac{25}{36}$
(iv) $\frac{8}{15}$
(v) $\frac{21}{10}$

Answer:

(i) Here, numerator = 8 and denominator = 11
Factors of 8 are 1, 2, 4 and 8.
Factors of 11 are 1 and 11.

Common factor of 8 and 11 is 1.
Thus, H.C.F. of 8 and 11 is 1.
Hence,

\frac{8}{11}

is the simplest form.

(ii) Here, numerator = 9 and denominator = 14
Factors of 9 are 1, 3 and 9.
Factors of 14 are 1, 2, 7 and 14.

Common factor of 9 and 14 is 1.
Thus, H.C.F. of 9 and 14 is 1.
Hence,

\frac{9}{14}

is the simplest form.

(iii) Here, numerator = 25 and denominator = 36
Factors of 25 are 1, 5 and 25.
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Common factor of 25 and 36 is 1.
Thus, H.C.F. of 25 and 36 is 1.
Hence,

\frac{25}{36}

is the simplest form.

(iv) Here, numerator = 8 and denominator = 15
Factors of 8 are 1, 2, 4 and 8.
Factors of 15 are 1, 3, 5 and 15.

Common factor of 8 and 15 is 1.
Thus, H.C.F. of 8 and 15 is 1.
Hence,

\frac{8}{15}

is the simplest form.
(v) Here, numerator = 21 and denominator = 10
Factors of 21 are 1, 3, 7 and 21.
Factors of 10 are 1, 2, 5 and 10.

Common factor of 21 and 10 is 1.
Thus, H.C.F. of 21 and 10 is 1.
Hence,

\frac{21}{10}

is the simplest form.

Page No 90:

Question 11:

Replace by the correct number in each of the following:
(i) $\frac{2}{7} = \frac{8}{}$
(ii) $\frac{3}{5} = \frac{}{35}$
(iii) $\frac{5}{8} = \frac{20}{}$
(iv) $\frac{45}{60} = \frac{9}{}$
(v) $\frac{40}{56} = \frac{}{7}$
(vi) $\frac{42}{54} = \frac{7}{}$

Answer:

(i) 28 $(\frac{2}{7} = \frac{2 \times 4}{7 \times 4} = \frac{8}{28})$
(ii) 21 $(\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35})$
(iii) 32 $(\frac{5}{8} = \frac{5 \times 4}{8 \times 4} = \frac{20}{32})$
(iv) 12 $(\frac{45}{60} = \frac{45 \div 5}{60 \div 5} = \frac{9}{12})$
(v) 5 $(\frac{40}{56} = \frac{40 \div 8}{56 \div 8} = \frac{5}{7})$
(vi) 9 $(\frac{42}{54} = \frac{42 \div 6}{54 \div 6} = \frac{7}{9})$

Page No 93:

Question 1:

Define like and unlike fractions and give five examples of each.

Answer:

Like fractions:
Fractions having the same denominator are called like fractions.
Examples: $\frac{3}{11}, \frac{5}{11}, \frac{7}{11}, \frac{9}{11}, \frac{10}{11}$

Unlike fractions:
Fractions having different denominators are called unlike fractions.
Examples: $\frac{3}{4}, \frac{4}{5}, \frac{6}{7}, \frac{9}{11}, \frac{2}{13}$

Page No 93:

Question 2:

Convert $\frac{3}{5}, \frac{7}{10}, \frac{8}{15} and \frac{11}{30}$ into like fractions.

Answer:

The given fractions are $\frac{3}{5}, \frac{7}{10}, \frac{8}{15} and \frac{11}{30} .$

L.C.M. of 5, 10, 15 and 30 = (5 $\times$ 2 $\times$ 3) = 30
So, we convert the given fractions into equivalent fractions with 30 as the denominator.
(But, one of the fractions already has 30 as its denominator. So, there is no need to convert it into an equivalent fraction.)
Thus, we have:
$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}; \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30}; \frac{8}{15} = \frac{8 \times 2}{15 \times 2} = \frac{16}{30}$

Hence, the required like fractions are $\frac{18}{30}, \frac{21}{30}, \frac{16}{30} and \frac{11}{30} .$

Page No 93:

Question 3:

Convert $\frac{1}{4}, \frac{5}{8}, \frac{7}{12} and \frac{13}{24}$ into like fractions.

Answer:

The given fractions are $\frac{1}{4}, \frac{5}{8}, \frac{7}{12} and \frac{13}{24} .$
L.C.M. of 4, 8, 12 and 24 = (4 $\times$ 2 $\times$ 3) = 24
So, we convert the given fractions into equivalent fractions with 24 as the denominator.
(But one of the fractions already has 24 as the denominator. So, there is no need to convert it into an equivalent fraction.)
Thus, we have:
$\frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24}; \frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}; \frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24}$

Hence, the required like fractions are $\frac{6}{24}, \frac{15}{24}, \frac{14}{24} and \frac{13}{24} .$

Page No 93:

Question 4:

Fill in the place holders with the correct symbol > or <:
(i) $\frac{8}{9} \frac{5}{9}$
(ii) $\frac{9}{10} \frac{7}{10}$
(iii) $\frac{3}{7} \frac{6}{7}$
(iv) $\frac{11}{15} \frac{8}{15}$
(v) $\frac{6}{11} \frac{5}{11}$
(vi) $\frac{11}{20} \frac{17}{20}$

Answer:

Between two fractions with the same denominator, the one with the greater numerator is the greater of the two.

(i) >
(ii) >
(iii) <
(iv) >
(v) >
(vi) <

Page No 93:

Question 5:

Fill in the place holders with the correct symbol > or <:
(i) $\frac{3}{4} \frac{3}{5}$
(ii) $\frac{7}{8} \frac{7}{10}$
(iii) $\frac{4}{11} \frac{4}{9}$
(iv) $\frac{8}{11} \frac{8}{13}$
(v) $\frac{5}{12} \frac{5}{8}$
(vi) $\frac{11}{14} \frac{11}{15}$

Answer:

Between two fractions with the same numerator, the one with the smaller denominator is the greater of the two.

(i) >
(ii) >
(iii)<
(iv) >
(v) <
(vi) >

Page No 93:

Question 6:

Compare the fractions given below:
$\frac{4}{5}, \frac{5}{7}$

Answer:

$\frac{4}{5}, \frac{5}{7}$
By cross multiplying:
5 $\times$ 5 = 25 and 4 $\times$ 7 = 28
Clearly, 28 > 25
$∴$ $\frac{4}{5} > \frac{5}{7}$

Page No 93:

Question 7:

Compare the fractions given below:
$\frac{3}{8}, \frac{5}{6}$

Answer:

$\frac{3}{8}, \frac{5}{6}$
By cross multiplying:
3 $\times$ 6 = 18 and 5 $\times$ 8 = 40
Clearly, 18 < 40
$∴$ $\frac{3}{8} < \frac{5}{6}$

Page No 93:

Question 8:

Compare the fractions given below:
$\frac{7}{11}, \frac{6}{7}$

Answer:

$\frac{7}{11}, \frac{6}{7}$

By cross multiplying:
7 $\times$ 7 = 49 and 11 $\times$ 6 = 66
Clearly, 49 < 66
$∴$ $\frac{7}{11} < \frac{6}{7}$

Page No 93:

Question 9:

Compare the fractions given below:
$\frac{5}{6}, \frac{9}{11}$

Answer:

$\frac{7}{11}, \frac{6}{7}$
By cross multiplying:
5 $\times$ 11 = 55 and 9 $\times$ 6 = 54
Clearly, 55 > 54
$∴$ $\frac{5}{6} > \frac{9}{11}$

Page No 93:

Question 10:

Compare the fractions given below:
$\frac{2}{3}, \frac{4}{9}$

Answer:

$\frac{7}{11}, \frac{6}{7}$
By cross multiplying:
2 $\times$ 9 = 18 and 4 $\times$ 3 = 12
Clearly, 18 > 12
$∴$ $\frac{2}{3} > \frac{4}{9}$

Page No 93:

Question 11:

Compare the fractions given below:
$\frac{6}{13}, \frac{3}{4}$

Answer:

$\frac{6}{13}, \frac{3}{4}$
By cross multiplying:
6 $\times$ 4 = 24 and 13 $\times$ 3 = 39
Clearly, 24 < 39
$∴$ $\frac{6}{13} < \frac{3}{4}$

Page No 93:

Question 12:

Compare the fractions given below:
$\frac{3}{4}, \frac{5}{6}$

Answer:

$\frac{6}{13}, \frac{3}{4}$
By cross multiplying:
3 $\times$ 6 = 18 and 4 $\times$ 5 = 20
Clearly, 18 < 20
$∴$ $\frac{3}{4} < \frac{5}{6}$

Page No 93:

Question 13:

Compare the fractions given below:
$\frac{5}{8}, \frac{7}{12}$

Answer:

$\frac{5}{8}, \frac{7}{12}$
By cross multiplying:
5 $\times$ 12 = 60 and 8 $\times$ 7 = 56
Clearly, 60 > 56
$∴$ $\frac{5}{8} > \frac{7}{12}$

Page No 93:

Question 14:

Compare the fractions given below:
$\frac{4}{9}, \frac{5}{6}$

Answer:

L.C.M. of 9 and 6 = (3 $\times$ 3 $\times$ 2) = 18
Now, we convert $\frac{4}{9} and \frac{5}{6}$ into equivalent fractions having 18 as the denominator.
∴ $\frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18} and \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$ 4949

Clearly, $\frac{8}{18} < \frac{15}{18}$
$∴$ $\frac{4}{9} < \frac{5}{6}$

Page No 93:

Question 15:

Compare the fractions given below:
$\frac{4}{5}, \frac{7}{10}$

Answer:

L.C.M. of 5 and 10 = (5 $\times$ 2) = 10
Now, we convert $\frac{4}{5}$ into an equivalent fraction having 10 as the denominator as the other fraction has already 10 as its denominator.
∴ $\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$ 4949

Clearly, $\frac{8}{10} > \frac{7}{10}$
$∴$ $\frac{4}{5} > \frac{7}{10}$

Page No 93:

Question 16:

Compare the fractions given below:
$\frac{7}{8}, \frac{9}{10}$

Answer:

L.C.M. of 8 and 10 = (2 $\times$ 5 $\times$ 2 $\times$ 2) = 40
Now, we convert $\frac{7}{8} and \frac{9}{10}$ into equivalent fractions having 40 as the denominator.
∴ $\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40} and \frac{9}{10} = \frac{9 \times 4}{10 \times 4} = \frac{36}{40}$ 4949

Clearly, $\frac{35}{40} < \frac{36}{40}$
$∴$ $\frac{7}{8} < \frac{9}{10}$

Page No 93:

Question 17:

Compare the fractions given below:
$\frac{11}{12}, \frac{13}{15}$

Answer:

L.C.M. of 12 and 15 = (2 $\times$ 2 $\times$ 3 $\times$ 5) = 60
Now, we convert $\frac{11}{12} and \frac{13}{15}$ into equivalent fractions having 60 as the denominator.
∴ $\frac{11}{12} = \frac{11 \times 5}{12 \times 5} = \frac{55}{60} and \frac{13}{15} = \frac{13 \times 4}{15 \times 4} = \frac{52}{60}$ 4949

Clearly, $\frac{55}{60} > \frac{52}{60}$
$∴$ $\frac{11}{12} > \frac{13}{15}$

Page No 93:

Question 18:

Arrange the following fractions in ascending order:
$\frac{1}{2}, \frac{3}{4}, \frac{5}{6} and \frac{7}{8}$

Answer:

The given fractions are $\frac{1}{2}, \frac{3}{4}, \frac{5}{6} and \frac{7}{8}$ .
L.C.M. of 2, 4, 6 and 8 = (2 $\times$ 2 $\times$ 2 $\times$ 3) = 24
We convert each of the given fractions into an equivalent fraction with denominator 24.
Now, we have:
$\frac{1}{2} = \frac{1 \times 12}{2 \times 12} = \frac{12}{24}; \frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24} \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}; \frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}$

Clearly, $\frac{12}{24} < \frac{18}{24} < \frac{20}{24} < \frac{21}{24}$

∴ $\frac{1}{2} < \frac{3}{4} < \frac{5}{6} < \frac{7}{8}$
Hence, the given fractions can be arranged in the ascending order as follows:
$\frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}$

Page No 93:

Question 19:

Arrange the following fractions in ascending order:
$\frac{2}{3}, \frac{5}{6}, \frac{7}{9} and \frac{11}{18}$

Answer:

The given fractions are $\frac{2}{3}, \frac{5}{6}, \frac{7}{9} and \frac{11}{18} .$

L.C.M. of 3, 6, 9 and 18 = (3 $\times$ 2 $\times$ 3) = 18
So, we convert each of the fractions whose denominator is not equal to 18 into an equivalent fraction with denominator 18.
Now, we have:
$\frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}; \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}; \frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}$
Clearly, $\frac{11}{18} < \frac{12}{18} < \frac{14}{18} < \frac{15}{18}$
∴ $\frac{11}{18} < \frac{2}{3} < \frac{7}{9} < \frac{5}{6}$

Hence, the given fractions can be arranged in the ascending order as follows:
$\frac{11}{18}, \frac{2}{3}, \frac{7}{9}, \frac{5}{6}$

Page No 93:

Question 20:

Arrange the following fractions in ascending order:
$\frac{2}{5}, \frac{7}{10}, \frac{11}{15} and \frac{17}{30}$

Answer:

The given fractions are $\frac{2}{5}, \frac{7}{10}, \frac{11}{15} and \frac{17}{30} .$
L.C.M. of 5, 10, 15 and 30 = (2 $\times$ 5 $\times$ 3) = 30

So, we convert each of the fractions whose denominator is not equal to 30 into an equivalent fraction with denominator 30.
Now, we have:
$\frac{2}{5} = \frac{2 \times 6}{5 \times 6} = \frac{12}{30}; \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30}; \frac{11}{15} = \frac{11 \times 2}{15 \times 2} = \frac{22}{30}$
Clearly, $\frac{12}{30} < \frac{17}{30} < \frac{21}{30} < \frac{22}{30}$
∴ $\frac{2}{5} < \frac{17}{30} < \frac{7}{10} < \frac{11}{15}$

Hence, the given fractions can be arranged in the ascending order as follows:
$\frac{2}{5}, \frac{17}{30}, \frac{7}{10}, \frac{11}{15}$

Page No 93:

Question 21:

Arrange the following fractions in ascending order:
$\frac{3}{4}, \frac{7}{8}, \frac{11}{16} and \frac{23}{32}$

Answer:

The given fractions are $\frac{3}{4}, \frac{7}{8}, \frac{11}{16} and \frac{23}{32} .$
L.C.M. of 4, 8, 16 and 32 = (2 ⨯ 2 ⨯ 2 ⨯ 2 ⨯ 2) = 32

So, we convert each of the fractions whose denominator is not equal to 32 into an equivalent fraction with denominator 32.
Now, we have:
$\frac{3}{4} = \frac{3 \times 8}{4 \times 8} = \frac{24}{32}; \frac{7}{8} = \frac{7 \times 4}{8 \times 4} = \frac{28}{32}; \frac{11}{16} = \frac{11 \times 2}{16 \times 2} = \frac{22}{32}$
Clearly, $\frac{22}{32} < \frac{23}{32} < \frac{24}{32} < \frac{28}{32}$
∴ $\frac{11}{16} < \frac{23}{32} < \frac{3}{4} < \frac{7}{8}$

Hence, the given fractions can be arranged in the ascending order as follows:
$\frac{11}{16}, \frac{23}{32}, \frac{3}{4}, \frac{7}{8}$

Page No 93:

Question 22:

Arrange the following fractions in descending order:
$\frac{3}{4}, \frac{5}{8}, \frac{11}{12} and \frac{17}{24}$

Answer:

The given fractions are $\frac{3}{4}, \frac{5}{8}, \frac{11}{12} and \frac{17}{24} .$
L.C.M. of 4, 8, 12 and 24 = (2 ⨯ 2 ⨯ 2 ⨯ 3) = 24

So, we convert each of the fractions whose denominator is not equal to 24 into an equivalent fraction with denominator 24.
Thus, we have;
$\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}; \frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}; \frac{11}{12} = \frac{11 \times 2}{12 \times 2} = \frac{22}{24}$
Clearly, $\frac{22}{24} > \frac{18}{24} > \frac{17}{24} > \frac{15}{24}$

∴ $\frac{11}{12} > \frac{3}{4} > \frac{17}{24} > \frac{5}{8}$

Hence, the given fractions can be arranged in the descending order as follows:
$\frac{11}{12}, \frac{3}{4}, \frac{17}{24}, \frac{5}{8}$

Page No 93:

Question 23:

Arrange the following fractions in descending order:
$\frac{7}{9}, \frac{5}{12}, \frac{11}{18} and \frac{17}{36}$

Answer:

The given fractions are $\frac{7}{9}, \frac{5}{12}, \frac{11}{18} and \frac{17}{36} .$
L.C.M. of 9, 12, 18 and 36 = (3 ⨯ 3 ⨯ 2 ⨯ 2) = 36

We convert each of the fractions whose denominator is not equal to 36 into an equivalent fraction with denominator 36.
Thus, we have:
$\frac{7}{9} = \frac{7 \times 4}{9 \times 4} = \frac{28}{36}; \frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}; \frac{11}{18} = \frac{11 \times 2}{18 \times 2} = \frac{22}{36}$
Clearly, $\frac{28}{36} > \frac{22}{36} > \frac{17}{36} > \frac{15}{36}$

∴ $\frac{7}{9} > \frac{11}{18} > \frac{17}{36} > \frac{5}{12}$

Hence, the given fractions can be arranged in the descending order as follows:
$\frac{7}{9}, \frac{11}{18}, \frac{17}{36}, \frac{5}{12}$

Page No 93:

Question 24:

Arrange the following fractions in descending order:
$\frac{2}{3}, \frac{3}{5}, \frac{7}{10} and \frac{8}{15}$

Answer:

The given fractions are $\frac{2}{3}, \frac{3}{5}, \frac{7}{10} and \frac{8}{15} .$
L.C.M. of 3, 5,10 and 15 = (2 ⨯ 3 ⨯ 5) = 30

So, we convert each of the fractions into an equivalent fraction with denominator 30.
Thus, we have:
$\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}; \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}; \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30}; \frac{8}{15} = \frac{8 \times 2}{15 \times 2} = \frac{16}{30}$
Clearly, $\frac{21}{30} > \frac{20}{30} > \frac{18}{30} > \frac{16}{30}$
∴ $\frac{7}{10} > \frac{2}{3} > \frac{3}{5} > \frac{8}{15}$

Hence, the given fractions can be arranged in the descending order as follows:
$\frac{7}{10}, \frac{2}{3}, \frac{3}{5}, \frac{8}{15}$

Page No 93:

Question 25:

Arrange the following fractions in descending order:
$\frac{5}{7}, \frac{9}{14}, \frac{17}{21} and \frac{31}{42}$

Answer:

The given fractions are $\frac{5}{7}, \frac{9}{14}, \frac{17}{21} and \frac{31}{42} .$
L.C.M. of 7, 14, 21 and 42 = (2 ⨯ 3 ⨯ 7) = 42

We convert each one of the fractions whose denominator is not equal to 42 into an equivalent fraction with denominator 42.
Thus, we have:
$\frac{5}{7} = \frac{5 \times 6}{7 \times 6} = \frac{30}{42}; \frac{9}{14} = \frac{9 \times 3}{14 \times 3} = \frac{27}{42}; \frac{17}{21} = \frac{17 \times 2}{21 \times 2} = \frac{34}{42}$
Clearly, $\frac{34}{42} > \frac{31}{42} > \frac{30}{42} > \frac{27}{42}$
∴ $\frac{17}{21} > \frac{31}{42} > \frac{5}{7} > \frac{9}{14}$
Hence, the given fractions can be arranged in the descending order as follows:
$\frac{17}{21}, \frac{31}{42}, \frac{5}{7}, \frac{9}{14}$

Page No 93:

Question 26:

Arrange the following fractions in descending order:
$\frac{1}{12}, \frac{1}{23}, \frac{1}{7}, \frac{1}{9}, \frac{1}{17}, \frac{1}{50}$

Answer:

The given fractions are $\frac{1}{12}, \frac{1}{23}, \frac{1}{7}, \frac{1}{9}, \frac{1}{17} and \frac{1}{50} .$
As the fractions have the same numerator, we can follow the rule for the comparison of such fractions.
This rule states that when two fractions have the same numerator, the fraction having the smaller denominator is the greater one.
Clearly, $\frac{1}{7} > \frac{1}{9} > \frac{1}{12} > \frac{1}{17} > \frac{1}{23} > \frac{1}{50}$
Hence, the given fractions can be arranged in the descending order as follows:
$\frac{1}{7}, \frac{1}{9}, \frac{1}{12}, \frac{1}{17}, \frac{1}{23}, \frac{1}{50}$

Page No 93:

Question 27:

Arrange the following fractions in descending order:
$\frac{3}{7}, \frac{3}{11}, \frac{3}{5}, \frac{3}{13}, \frac{3}{4}, \frac{3}{17}$

Answer:

The given fractions are $\frac{3}{7}, \frac{3}{11}, \frac{3}{5}, \frac{3}{13}, \frac{3}{4} and \frac{3}{17} .$
As the fractions have the same numerator, so we can follow the rule for the comparison of such fractions.
This rule states that when two fractions have the same numerator, the fraction having the smaller denominator is the greater one.

Clearly, $\frac{3}{4} > \frac{3}{5} > \frac{3}{7} > \frac{3}{11} > \frac{3}{13} > \frac{3}{17}$
Hence, the given fractions can be arranged in the descending order as follows:
$\frac{3}{4}, \frac{3}{5}, \frac{3}{7}, \frac{3}{11}, \frac{3}{13}, \frac{3}{17}$

Page No 94:

Question 28:

Lalita read 30 pages of a book containing 100 pages while Sarita read $\frac{2}{5}$ of the book. Who read more?

Answer:

Lalita read 30 pages of a book having 100 pages.
Sarita read $\frac{2}{5}$ of the same book.
$\frac{2}{5}$ of 100 pages = $\frac{2}{5} \times 100 = \frac{200}{5} = 40 pages$
Hence, Sarita read more pages than Lalita as 40 is greater than 30.

Page No 94:

Question 29:

Rafiq exercised for $\frac{2}{3}$ hour, while Rohit exercise for $\frac{3}{4}$ hour. Who exercised for a longer time?

Answer:

To know who exercised for a longer time, we have to compare $\frac{2}{3} hour with \frac{3}{4} hour$ .
On cross multiplying:
4 $\times$ 2 = 8 and 3 $\times$ 3 = 9
Clearly, 8 < 9
$∴$ $\frac{2}{3} hour < \frac{3}{4} hour$
Hence, Rohit exercised for a longer time.

Page No 94:

Question 30:

In a school 20 students out of 25 passed in VI A, while 24 out of 30 passed in VI B. Which section gave better result?

Answer:

Fraction of students who passed in VI A = $\frac{20}{25} = \frac{20 \div 5}{25 \div 5} = \frac{4}{5}$

Fraction of students who passed in VI B = $\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}$
In both the sections, the fraction of students who passed is the same, so both the sections have the same result.

Page No 96:

Question 1:

Find the sum:
$\frac{5}{8} + \frac{1}{8}$

Answer:

The given fractions are like fractions.
We know:
Sum of like fractions = $\frac{Sum of the numerators}{C ommon d enominator}$
Thus, we have:
$\frac{5}{8} + \frac{1}{8} = \frac{(5 + 1)}{8} = \frac{6^{3}}{8_{4}} = \frac{3}{4}$

Page No 96:

Question 2:

Find the sum:
$\frac{4}{9} + \frac{8}{9}$

Answer:

The given fractions are like fractions.
We know:
Sum of like fractions = $\frac{Sum of the numeratos}{Common d enominator}$
Thus, we have:
$\frac{4}{9} + \frac{8}{9} = \frac{(4 + 8)}{9} = \frac{12^{4}}{9_{3}} = \frac{4}{3} = 1 \frac{1}{3}$

Page No 96:

Question 3:

Find the sum:
$1 \frac{3}{5} + 2 \frac{4}{5}$

Answer:

The given fractions are like fractions.
We know:
Sum of like fractions = $\frac{Sum of the numerators}{Common d enominator}$
Thus, we have:
$1 \frac{3}{5} + 2 \frac{4}{5} = \frac{8}{5} + \frac{14}{5} = \frac{(8 + 14)}{5} = \frac{22}{5} = 4 \frac{2}{5}$

Page No 96:

Question 4:

Find the sum:
$\frac{2}{9} + \frac{5}{6}$

Answer:

L.C.M. of 9 and 6 = (2 $\times$ 3 $\times$ 3) = 18

Now, we have:

$\frac{2}{9} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}; \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} ∴ \frac{2}{9} + \frac{5}{6} = \frac{4}{18} + \frac{15}{18} = \frac{(4 + 15)}{18} = \frac{19}{18} = 1 \frac{1}{18}$

Page No 96:

Question 5:

Find the sum:
$\frac{7}{12} + \frac{9}{16}$

Answer:

L.C.M. of 12 and 16 = (2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 3) = 48

Now, we have:

$\frac{7}{12} = \frac{7 \times 4}{12 \times 4} = \frac{28}{48}; \frac{9}{16} = \frac{9 \times 3}{16 \times 3} = \frac{27}{48} ∴ \frac{7}{12} + \frac{9}{16} = \frac{28}{48} + \frac{27}{48} = \frac{(28 + 27)}{48} = \frac{55}{48} = 1 \frac{7}{48}$

Page No 96:

Question 6:

Find the sum:
$\frac{4}{15} + \frac{17}{20}$

Answer:

L.C.M. of 15 and 20 = (3 $\times$ 5 $\times$ 2 $\times$ 2) = 60

$∴ \frac{4}{15} + \frac{17}{20} = \frac{(16 + 51)}{60} \{[60 \div 15 = 4, 4 \times 4 = 16] and [60 \div 20 = 3, 17 \times 3 = 51]\} = \frac{67}{60} = 1 \frac{7}{60}$

Page No 96:

Question 7:

Find the sum:
$2 \frac{3}{4} + 5 \frac{5}{6}$

Answer:

We have:

$2 \frac{3}{4} + 5 \frac{5}{6} = \frac{11}{4} + \frac{35}{6} L . C . M . of 4 and 6 = (2 \times 2 \times 3) = 12 = \frac{(66 + 140)}{24} \{[24 \div 4 = 6, 6 \times 11 = 66] and [24 \div 6 = 4, 4 \times 35 = 140]\} = \frac{206^{103}}{24_{12}} = \frac{103}{12} = 8 \frac{7}{12}$
234+556

Page No 96:

Question 8:

Find the sum:
$3 \frac{1}{8} + 1 \frac{5}{12}$

Answer:

We have:

$3 \frac{1}{8} + 1 \frac{5}{12} = \frac{25}{8} + \frac{17}{12} L . C . M . of 8 and 12 = (2 \times 2 \times 2 \times 3) = 24 = \frac{(75 + 34)}{24} \{[24 \div 8 = 3, 3 \times 25 = 75] and [24 \div 12 = 2, 2 \times 17 = 34]\} = \frac{109^{}}{24} = 4 \frac{13}{24}$
234+556

Page No 96:

Question 9:

Find the sum:
$2 \frac{7}{10} + 3 \frac{8}{15}$

Answer:

We have:

$2 \frac{7}{10} + 3 \frac{8}{15} = \frac{27}{10} + \frac{53}{15} L . C . M . of 10 and 15 = (2 \times 3 \times 5) = 30 = \frac{(81 + 106)}{30} \{[30 \div 10 = 3, 3 \times 27 = 81] and [30 \div 15 = 2, 2 \times 53 = 106]\} = \frac{187^{}}{30} = 6 \frac{7}{30}$
234+556

Page No 96:

Question 10:

Find the sum:
$3 \frac{2}{3} + 1 \frac{5}{6} + 2$

Answer:

We have:

$3 \frac{2}{3} + 1 \frac{5}{6} + 2 = \frac{11}{3} + \frac{11}{6} + \frac{2}{1} L . C . M . of 3 and 6 = (2 \times 3) = 6 = \frac{(22 + 11 + 12)}{6} \{[6 \div 3 = 2, 2 \times 11 = 22], [6 \div 6 = 1, 1 \times 11 = 11] and [6 \div 1 = 6, 6 \times 2 = 12]\} = \frac{45^{15}}{6_{2}} = \frac{15}{2} = 7 \frac{1}{2}$
234+556

Page No 96:

Question 11:

Find the sum:
$3 + 1 \frac{4}{15} + 1 \frac{3}{20}$

Answer:

We have:

$3 + 1 \frac{4}{15} + 1 \frac{3}{20} = \frac{3}{1} + \frac{19}{15} + \frac{23}{20} L . C . M . of 15 and 20 = (2 \times 2 \times 3 \times 5) = 60 = \frac{(180 + 76 + 69)}{60} \{[60 \div 1 = 60, 60 \times 3 = 180], [60 \div 15 = 4, 4 \times 19 = 76] and [60 \div 20 = 3, 3 \times 23 = 69]\} = \frac{325^{65}}{60_{12}} = \frac{65}{12} = 5 \frac{5}{12}$
234+556

Page No 96:

Question 12:

Find the sum:
$3 \frac{1}{3} + 4 \frac{1}{4} + 6 \frac{1}{6}$

Answer:

We have:

$3 \frac{1}{3} + 4 \frac{1}{4} + 6 \frac{1}{6} = \frac{10}{3} + \frac{17}{4} + \frac{37}{6} L . C . M . of 3, 4 and 6 = (2 \times 2 \times 3) = 12 = \frac{(40 + 51 + 74)}{12} \{[12 \div 3 = 4, 4 \times 10 = 40], [12 \div 4 = 3, 3 \times 17 = 51] and [12 \div 6 = 2, 2 \times 37 = 74]\} = \frac{165^{55}}{12_{4}} = \frac{55}{4} = 13 \frac{3}{4}$
234+556

Page No 96:

Question 13:

Find the sum:
$\frac{2}{3} + 3 \frac{1}{6} + 4 \frac{2}{9} + 2 \frac{5}{18}$

Answer:

We have:

$\frac{2}{3} + 3 \frac{1}{6} + 4 \frac{2}{9} + 2 \frac{5}{18} = \frac{2}{3} + \frac{19}{6} + \frac{38}{9} + \frac{41}{18} L . C . M . of 3, 6 and 9 = (2 \times 3 \times 3) = 18 = \frac{(12 + 57 + 76 + 41)}{18} \{[18 \div 3 = 6, 6 \times 2 = 12], [18 \div 6 = 3, 3 \times 19 = 57], [18 \div 9 = 2, 2 \times 38 = 76] and [18 \div 18 = 1, 1 \times 41 = 41]\} = \frac{186^{31}}{18_{3}} = \frac{31}{3} = 10 \frac{1}{3}$
234+556

Page No 96:

Question 14:

Find the sum:
$2 \frac{1}{3} + 1 \frac{1}{4} + 2 \frac{5}{6} + 3 \frac{7}{12}$

Answer:

We have:

$2 \frac{1}{3} + 1 \frac{1}{4} + 2 \frac{5}{6} + 3 \frac{7}{12} = \frac{7}{3} + \frac{5}{4} + \frac{17}{6} + \frac{43}{12} L . C . M . of 3, 4, 6 and 12 = (2 \times 2 \times 3) = 12 = \frac{(28 + 15 + 34 + 43)}{12} \{[12 \div 3 = 4, 4 \times 7 = 28], [12 \div 4 = 3, 3 \times 5 = 15], [12 \div 6 = 2, 2 \times 17 = 34] and [12 \div 12 = 1, 1 \times 43 = 43]\} = \frac{120^{10}}{12_{1}} = 10$
234+556

Page No 96:

Question 15:

Find the sum:
$2 + \frac{3}{4} + 1 \frac{5}{8} + 3 \frac{7}{16}$

Answer:

We have:

$2 + \frac{3}{4} + 1 \frac{5}{8} + 3 \frac{7}{16} = \frac{2}{1} + \frac{3}{4} + \frac{13}{8} + \frac{55}{16} L . C . M . of 4, 8, and 16 = (2 \times 2 \times 2 \times 2) = 16 = \frac{(32 + 12 + 26 + 55)}{16} \{[16 \div 1 = 16, 16 \times 2 = 32], [16 \div 4 = 4, 4 \times 3 = 12], [16 \div 8 = 2, 2 \times 13 = 26] and [16 \div 16 = 1, 1 \times 55 = 55]\} = \frac{125}{16} = 7 \frac{13}{16}$
234+556

Page No 96:

Question 16:

Rohit bought a pencil for Rs $3 \frac{2}{5}$ and an eraser for Rs $2 \frac{7}{10}$ . What is the total cost of both the articles?

Answer:

Total cost of both articles = Cost of pencil + Cost of eraser
Thus, we have:
$Rs 3 \frac{2}{5} + Rs 2 \frac{7}{10} = \frac{17}{5} + \frac{27}{10} = \frac{(34 + 27)}{10} (L . C . M . of 5 and 10 = (5 \times 2) = 10) = \frac{61}{10} = Rs 6 \frac{1}{10}$
Hence, the total cost of both the articles is $Rs 6 \frac{1}{10}$ .

Page No 96:

Question 17:

Sohini bought $4 \frac{1}{2} m$ of cloth for her kurta and $2 \frac{2}{3} m$ of cloth for her pyjamas. Ho much cloth did she purchase in all?

Answer:

Total cloth purchased by Sohini = Cloth for kurta + Cloth for pyjamas
Thus, we have:
$(4 \frac{1}{2} + 2 \frac{2}{3}) m = (\frac{9}{2} + \frac{8}{3}) m (L . C . M . of 2 and 3 = (2 \times 3) = 6) = (\frac{(27 + 16)}{6}) m \{[6 \div 2 = 3, 3 \times 9 = 27] and [6 \div 3 = 2, 2 \times 8 = 16]\} = (\frac{43}{6}) m = 7 \frac{1}{6} m$
$∴$ Total length of cloth purchased = $7 \frac{1}{6} m$

Page No 96:

Question 18:

While coming back home from his school, Kishan covered $4 \frac{3}{4}$ km by rickshaw and $1 \frac{1}{2}$ km on foot. What is the distance of his house from the school?

Answer:

Distance from Kishan's house to school = Distance covered by him by rickshaw + Distance covered by him on foot
Thus, we have:
$(4 \frac{3}{4} + 1 \frac{1}{2}) km = (\frac{19}{4} + \frac{3}{2}) km = (\frac{(19 + 6)}{4}) km (L . C . M . of 2 and 4 = (2 \times 2) = 4) = (\frac{25}{4}) km = 6 \frac{1}{4} km$

Hence, the distance from Kishan's house to school is $6 \frac{1}{4} km$ .

Page No 96:

Question 19:

The weight of an empty gas cylinder is $16 \frac{4}{5}$ kg and it contains $14 \frac{2}{3}$ kg of gas. What is the weight of the cylinder filled with gas?

Answer:

Weight of the cylinder filled with gas = Weight of the empty cylinder + Weight of the gas inside the cylinder
Thus, we have:
$(16 \frac{4}{5} + 14 \frac{2}{3}) kg = (\frac{84}{5} + \frac{44}{3}) kg (L . C . M . of 5 and 3 = (3 \times 5) = 15) = (\frac{(252 + 220)}{15}) kg = (\frac{472}{15}) kg = 31 \frac{7}{15} kg$
Hence, the weight of the cylinder filled with gas is $31 \frac{7}{15} kg$ .

Page No 99:

Question 1:

Find the difference:
$\frac{5}{8} - \frac{1}{8}$

Answer:

Difference of like fractions = Difference of numerator $\div$ Common denominator
$\frac{5}{8} - \frac{1}{8} = \frac{(5 - 1)}{8} = \frac{4^{1}}{8_{2}} = \frac{1}{2}$

Page No 99:

Question 2:

Find the difference:
$\frac{7}{12} - \frac{5}{12}$

Answer:

Difference of like fractions = Difference of numerator $\div$ Common denominator
$\frac{7}{12} - \frac{5}{12} = \frac{(7 - 5)}{12} = \frac{2^{1}}{12_{6}} = \frac{1}{6}$

Page No 99:

Question 3:

Find the difference:
$4 \frac{3}{7} - 2 \frac{4}{7}$

Answer:

Difference of like fractions = Difference of numerator $\div$ Common denominator
$4 \frac{3}{7} - 2 \frac{4}{7} = \frac{31}{7} - \frac{18}{7} = \frac{(31 - 18)}{7} = \frac{13^{}}{7}$

Page No 99:

Question 4:

Find the difference:
$\frac{5}{6} - \frac{4}{9}$

Answer:

$\frac{5}{6} - \frac{4}{9}$

$3 6, 9 2 2, 3 3 1, 3 1, 1$
L.C.M. of 6 and 9 = (3 $\times$ 2 $\times$ 3) = 18
Now, we have:
$\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}; \frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18} ∴ \frac{5}{6} - \frac{4}{9} = \frac{15}{18} - \frac{8}{18} = \frac{(15 - 8)}{18} = \frac{7}{18}$

Page No 99:

Question 5:

Find the difference:
$\frac{1}{2} - \frac{3}{8}$

Answer:

$\frac{1}{2} - \frac{3}{8}$

L.C.M. of 2 and 8 = (2 $\times$ 2 $\times$ 2) = 8
Now, we have:
$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8} ∴ \frac{1}{2} - \frac{3}{8} = \frac{4}{8} - \frac{3}{8} = \frac{(4 - 3)}{8} = \frac{1}{8}$

Page No 99:

Question 6:

Find the difference:
$\frac{5}{8} - \frac{7}{12}$

Answer:

$\frac{5}{8} - \frac{7}{12}$

$2 8, 12 2 4, 6 2 2, 3 3 1, 3 1, 1$
L.C.M. of 8 and 12 = (2 $\times$ 2 $\times$ 2 $\times$ 3) = 24
Now, we have:
$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}; \frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24} ∴ \frac{5}{8} - \frac{7}{12} = \frac{15}{24} - \frac{14}{24} = \frac{(15 - 14)}{24} = \frac{1}{24}$

Page No 99:

Question 7:

Find the difference:
$2 \frac{7}{9} - 1 \frac{8}{15}$

Answer:

$2 \frac{7}{9} - 1 \frac{8}{15} = \frac{25}{9} - \frac{23}{15} 3 9, 15 3 3, 5 5 1, 5 1, 1 L . C . M . of 9 and 15 = (3 \times 3 \times 5) = 45 ∴ \frac{25}{9} - \frac{23}{15} = \frac{(125 - 69)}{45} = \frac{56}{45} = 1 \frac{11}{45} \{[45 \div 9 = 5, 5 \times 25 = 125] and [45 \div 15 = 3, 3 \times 23 = 69]\}$

Page No 99:

Question 8:

Find the difference:
$3 \frac{5}{8} - 2 \frac{5}{12}$

Answer:

$3 \frac{5}{8} - 2 \frac{5}{12} = \frac{29}{8} - \frac{29}{12} 2 8, 12 24, 6 2 2, 3 3 1, 3 1, 1 L . C . M . of 8 and 12 = (2 \times 2 \times 2 \times 3) = 24 ∴ \frac{29}{8} - \frac{29}{12} = \frac{(87 - 58)}{24} = \frac{29}{24} = 1 \frac{5}{24} \{[24 \div 8 = 3, 3 \times 29 = 87] and [24 \div 12 = 2, 2 \times 29 = 58]\}$

Page No 99:

Question 9:

Find the difference:
$2 \frac{3}{10} - 1 \frac{7}{15}$

Answer:

$2 \frac{3}{10} - 1 \frac{7}{15} = \frac{23}{10} - \frac{22}{15} 5 10, 15 2 2, 3 3 1, 3 1, 1 L . C . M . of 10 and 15 = (2 \times 3 \times 5) = 30 = \frac{(69 - 44)}{30} \{[30 \div 10 = 3, 3 \times 23 = 69] and [30 \div 15 = 2, 2 \times 22 = 44]\} = \frac{25^{5}}{30_{6}} = \frac{5}{6}$

Page No 99:

Question 10:

Find the difference:
$6 \frac{2}{3} - 3 \frac{3}{4}$

Answer:

$6 \frac{2}{3} - 3 \frac{3}{4} = \frac{20}{3} - \frac{15}{4} L . C . M . of 3 and 4 = (2 \times 2 \times 3) = 12 = \frac{(80 - 45)}{12} \{[12 \div 3 = 4, 4 \times 20 = 80] and [12 \div 4 = 3, 3 \times 15 = 45]\} = \frac{35}{12} = 2 \frac{11}{12}$

Page No 99:

Question 11:

Find the difference:
$7 - 5 \frac{2}{3}$

Answer:

$7 - 5 \frac{2}{3} = \frac{7}{1} - \frac{17}{3} L . C . M . of 1 and 3 = 3 = \frac{(21 - 17)}{3} \{[3 \div 1 = 3, 3 \times 7 = 21] and [3 \div 3 = 1, 1 \times 17 = 17]\} = \frac{4}{3} = 1 \frac{1}{3}$

Page No 99:

Question 12:

Find the difference:
$10 - 6 \frac{3}{8}$

Answer:

$10 - 6 \frac{3}{8} = \frac{10}{1} - \frac{51}{8} L . C . M . of 1 and 8 = 8 = \frac{(80 - 51)}{8} \{[8 \div 1 = 8, 8 \times 10 = 80] and [8 \div 8 = 1, 1 \times 51 = 51]\} = \frac{29}{8} = 3 \frac{5}{8}$

Page No 99:

Question 13:

Simplify:
$\frac{5}{6} - \frac{4}{9} + \frac{2}{3}$

Answer:

We have:

$\frac{5}{6} - \frac{4}{9} + \frac{2}{3} L . C . M . of 3, 6 and 9 = (2 \times 3 \times 3) = 18 = \frac{(15 - 8 + 12)}{18} \{[18 \div 6 = 3, 3 \times 5 = 15], [18 \div 9 = 2, 2 \times 4 = 8] and [18 \div 3 = 6, 6 \times 2 = 12]\} = \frac{(27 - 8)}{18} = \frac{19}{18} = 1 \frac{1}{18}$
$3 3, 6, 9 3 1, 2, 3 2 1, 2, 1 1, 1, 1$

Page No 99:

Question 14:

Simplify:
$\frac{5}{8} + \frac{3}{4} - \frac{7}{12}$

Answer:

We have:
$\frac{5}{8} + \frac{3}{4} - \frac{7}{12} 2 4, 8, 12 2 2, 4, 6 2 1, 2, 3 3 1, 1, 3 1, 1, 1 L . C . M . of 4, 8 and 12 = (2 \times 2 \times 2 \times 3) = 24 = \frac{(15 + 18 - 14)}{24} \{[24 \div 8 = 3, 3 \times 5 = 15], [24 \div 4 = 6, 6 \times 3 = 18] and [24 \div 12 = 2, 2 \times 7 = 14]\} = \frac{(33 - 14)}{24} = \frac{19}{24}$
234+556

Page No 99:

Question 15:

Simplify:
$2 + \frac{11}{15} - \frac{5}{9}$

Answer:

We have: $\frac{2}{1} + \frac{11}{15} - \frac{5}{9} 3 1, 15, 9 3 1, 5, 3 5 1, 5, 1 1, 1, 1 L . C . M . of 15 and 9 = (3 \times 3 \times 5) = 45 = \frac{(90 + 33 - 25)}{45} \{[45 \div 1 = 45, 45 \times 2 = 90], [45 \div 15 = 3, 3 \times 11 = 33] and [45 \div 9 = 5, 5 \times 5 = 25]\} = \frac{(90 + 8)}{45} = \frac{98}{45} = 2 \frac{8}{45}$
234+556

Page No 99:

Question 16:

Simplify:
$5 \frac{3}{4} - 4 \frac{5}{12} + 3 \frac{1}{6}$

Answer:

We have:
$5 \frac{3}{4} - 4 \frac{5}{12} + 3 \frac{1}{6} = \frac{23}{4} - \frac{53}{12} + \frac{19}{6} L . C . M . of 4, 12 and 6 = (2 \times 2 \times 3) = 12 2 4, 12, 6 2 2, 6, 3 3 1, 2, 3 2 1, 2, 1 1, 1, 1 = \frac{(69 - 53 + 38)}{12} \{[12 \div 4 = 3, 3 \times 23 = 69], [12 \div 12 = 1, 1 \times 53 = 53] and [12 \div 6 = 2, 2 \times 19 = 38]\} = \frac{(107 - 53)}{12} = \frac{54}{12} = \frac{9}{2} = 4 \frac{1}{2}$
234+556

Page No 99:

Question 17:

Simplify:
$2 + 5 \frac{7}{10} - 3 \frac{14}{15}$

Answer:

We have: $2 + 5 \frac{7}{10} - 3 \frac{14}{15} = \frac{2}{1} + \frac{57}{10} - \frac{59}{15} 5 1, 10, 15 2 1, 2, 3 3 1, 1, 3 1, 1, 1 L . C . M . of 10 and 15 = (2 \times 5 \times 3) = 30 = \frac{(60 + 171 - 118)}{30} \{[30 \div 1 = 30, 30 \times 2 = 60], [30 \div 10 = 3, 3 \times 57 = 171] and [30 \div 15 = 2, 2 \times 59 = 118]\} = \frac{(231 - 118)}{30} = \frac{113}{30} = 3 \frac{23}{30}$

Page No 99:

Question 18:

Simplify:
$8 - 3 \frac{1}{2} - 2 \frac{1}{4}$

Answer:

We have:
$8 - 3 \frac{1}{2} - 2 \frac{1}{4} = \frac{8}{1} - \frac{7}{2} - \frac{9}{4} 2 1, 2, 4 2 1, 1, 2 1, 1, 1 L . C . M . of 1, 2 and 4 = (2 \times 2) = 4 = \frac{(32 - 14 - 9)}{4} \{[4 \div 1 = 4, 4 \times 8 = 32], [4 \div 2 = 2, 2 \times 7 = 14] and [4 \div 4 = 1, 1 \times 9 = 9]\} = \frac{(32 - 23)}{4} = \frac{9}{4} = 2 \frac{1}{4}$

Page No 99:

Question 19:

Simplify:
$8 \frac{5}{6} - 3 \frac{3}{8} + 2 \frac{7}{12}$

Answer:

We have:
$8 \frac{5}{6} - 3 \frac{3}{8} + 2 \frac{7}{12} = \frac{53}{6} - \frac{27}{8} + \frac{31}{12} 2 6, 8, 12 2 3, 4, 6 3 3, 2, 3 2 1, 2, 1 1, 1, 1 L . C . M . of 6, 8 and 12 = (2 \times 2 \times 2 \times 3) = 24 = \frac{(212 - 81 + 62)}{24} \{[24 \div 6 = 4, 4 \times 53 = 212], [24 \div 8 = 3, 3 \times 27 = 81] and [24 \div 12 = 2, 2 \times 31 = 62]\} = \frac{(274 - 81)}{24} = \frac{193}{24} = 8 \frac{1}{24}$

Page No 99:

Question 20:

Simplify:
$6 \frac{1}{6} - 5 \frac{1}{5} + 3 \frac{1}{3}$

Answer:

We have:
$6 \frac{1}{6} - 5 \frac{1}{5} + 3 \frac{1}{3} = \frac{37}{6} - \frac{26}{5} + \frac{10}{3} 2 6, 5, 3 3 3, 5, 3 5 1, 5, 1 1, 1, 1 L . C . M . of 6, 5 and 3 = (2 \times 5 \times 3) = 30 = \frac{(185 - 156 + 100)}{30} \{[30 \div 6 = 5, 5 \times 37 = 185], [30 \div 5 = 6, 6 \times 26 = 156], and [30 \div 3 = 10, 10 \times 10 = 100]\} = \frac{(285 - 156)}{30} = \frac{129^{43}}{30_{10}} = 4 \frac{3}{10}$

Page No 99:

Question 21:

Simplify:
$3 + 1 \frac{1}{5} + \frac{2}{3} - \frac{7}{15}$

Answer:

We have:
$3 + 1 \frac{1}{5} + \frac{2}{3} - \frac{7}{15} = \frac{3}{1} + \frac{6}{5} + \frac{2}{3} - \frac{7}{15} 5 5, 3, 15 3 1, 3, 3 1, 1, 1 L . C . M . of 1, 5, 3 and 15 = (5 \times 3) = 15 = \frac{(45 + 18 + 10 - 7)}{15} \{[15 \div 1 = 15, 15 \times 3 = 45], [15 \div 5 = 3, 3 \times 6 = 18], [15 \div 3 = 5, 5 \times 2 = 10] and [15 \div 15 = 1, 1 \times 7 = 7]\} = \frac{(73 - 7)}{15} = \frac{66^{22}}{15_{5}} = \frac{22}{5} = 4 \frac{2}{5}$

Page No 99:

Question 22:

What should be added to $9 \frac{2}{3}$ to get 19?

Answer:

Let x be added to $9 \frac{2}{3}$ to get 19.

$∴ 9 \frac{2}{3} + x = 19 Thus, we have : x = 19 - 9 \frac{2}{3} = \frac{19}{1} - \frac{29}{3} L . C . M . of 1 and 3 is 3 . = \frac{(57 - 29)}{3} \{[3 \div 1 = 3, 3 \times 19 = 57] and [3 \div 3 = 1, 1 \times 29 = 29]\} = \frac{28}{3} = 9 \frac{1}{3}$ 923

Page No 99:

Question 23:

What should be added to $6 \frac{7}{15}$ to get $8 \frac{1}{5}$ ?

Answer:

Let x be added to $6 \frac{7}{15}$ to get $8 \frac{1}{5}$ .
$∴ 6 \frac{7}{15} + x = 8 \frac{1}{5} Therefore, we have : x = 8 \frac{1}{5} - 6 \frac{7}{15} = \frac{41}{5} - \frac{97}{15} L . C . M . of 5 and 15 = (5 \times 3) = 15 = \frac{(123 - 97)}{15} \{[15 \div 5 = 3, 3 \times 41 = 123] and [15 \div 15 = 1, 1 \times 97 = 97]\} = \frac{26}{15} = 1 \frac{11}{15}$

Page No 99:

Question 24:

Subtract the sum of $3 \frac{5}{9}$ and $3 \frac{1}{3}$ from the sum of $5 \frac{5}{6}$ and $4 \frac{1}{9}$ .

Answer:

$(5 \frac{5}{6} + 4 \frac{1}{9}) - (3 \frac{5}{9} + 3 \frac{1}{3}) = (\frac{35}{6} + \frac{37}{9}) - (\frac{32}{9} + \frac{10}{3}) 2 6, 9, 3 3 3, 9, 3 3 1, 3, 1 1, 1, 1 L . C . M . of 3, 6, 9 = (2 \times 3 \times 3) = 18 = \frac{[105 + 74] - [64 + 60]}{18} \{[18 \div 6 = 3, 3 \times 35 = 105] and [18 \div 9 = 2, 2 \times 37 = 74]\} \{[18 \div 9 = 2, 2 \times 32 = 64] and [18 \div 3 = 6, 6 \times 10 = 60]\} = \frac{[179] - [124]}{18} = \frac{55}{18} = 3 \frac{1}{18}$

Page No 99:

Question 25:

Of $\frac{3}{4}$ and $\frac{5}{7}$ , which is greater and by how much?

Answer:

Let us compare $\frac{3}{4} and \frac{5}{7}$ .
3 $\times$ 7 = 21 and 4 $\times$ 5 = 20
Clearly, 21 > 20
∴ $\frac{3}{4} > \frac{5}{7}$
Required difference:
$= \frac{3}{4} - \frac{5}{7} L . C . M . of 4 and 7 = (2 \times 2 \times 7) = 28 = \frac{21 - 20}{28} \{[28 \div 4 = 7, 7 \times 3 = 21] and [28 \div 7 = 4, 4 \times 5 = 20]\} = \frac{1}{28}$
Hence, $\frac{3}{4} is greater than \frac{5}{7} by \frac{1}{28}$ .

Page No 99:

Question 26:

Mrs Soni bought $7 \frac{1}{2}$ litres of milk. Out of this milk, $5 \frac{3}{4}$ litres was consumed. How much milk is left with her?

Answer:

Amount of milk left with Mrs. Soni = Total amount of milk bought by her $-$ Amount of milk consumed
$∴$ Amount of milk left with Mrs. Soni $= 7 \frac{1}{2} - 5 \frac{3}{4} = \frac{15}{2} - \frac{23}{4} L . C . M . of 2 and 4 = (2 \times 2) = 4 = \frac{(30 - 23)}{4} \{[4 \div 2 = 2, 2 \times 15 = 30] and [4 \div 4 = 1, 1 \times 23 = 23]\} = \frac{7}{4} = 1 \frac{3}{4} litres$

$∴$ Milk left with Mrs. Soni = $1 \frac{3}{4} litres$

Page No 99:

Question 27:

A film show lasted for $3 \frac{1}{3}$ hours. Out of his time, $1 \frac{3}{4}$ hours was spent on advertisements. What was the actual duration of the film?

Answer:

Actual duration of the film = Total duration of the show $-$ Time spent on advertisements
$= (3 \frac{1}{3} - 1 \frac{3}{4}) hours = (\frac{10}{3} - \frac{7}{4}) hours L . C . M . of 3 and 4 = (2 \times 2 \times 3) = 12 = (\frac{40 - 21}{12}) hours \{[12 \div 3 = 4, 4 \times 10 = 40] and [12 \div 4 = 3, 3 \times 7 = 21]\} = (\frac{19}{12}) hours = 1 \frac{7}{12} hours$
Thus, the actual duration of the film was $1 \frac{7}{12} hours$ .

Page No 99:

Question 28:

In one day, a rickshaw puller earned Rs $137 \frac{1}{2}$ . Out of this money, he spent Rs $56 \frac{3}{4}$ on food. How much money is left with him?

Answer:

Money left with the rickshaw puller = Money earned by him in a day $-$ Money spent by him on food
$= Rs (137 \frac{1}{2} - 56 \frac{3}{4}) L . C . M . of 2 and 4 = (2 \times 2) = 4 = Rs (\frac{275}{2} - \frac{227}{4}) \{[4 \div 2 = 2, 2 \times 275 = 550] and [4 \div 4 = 1, 1 \times 227 = 227]\} = Rs (\frac{550 - 227}{4}) = Rs (\frac{323}{4}) = Rs 80 \frac{3}{4}$
Hence, Rs $80 \frac{3}{4}$ is left with the rickshaw puller.

Page No 99:

Question 29:

A piece of wire, $2 \frac{3}{4}$ metres long, broke into two pieces. One piece is $\frac{5}{8}$ metre long. How long is the other piece?

Answer:

The length of the other piece = (Length of the wire $-$ Length of one piece)
$= (2 \frac{3}{4} - \frac{5}{8}) m = (\frac{11}{4} - \frac{5}{8}) m L . C . M . of 4 and 8 = (2 \times 2 \times 2) = 8 = (\frac{22 - 5}{8}) m \{[8 \div 4 = 2, 2 \times 11 = 22] and [8 \div 8 = 1, 1 \times 5 = 5]\} = (\frac{17}{8}) m = 2 \frac{1}{8} m$
Hence, the other piece is $2 \frac{1}{8} m$ long.

Page No 99:

Question 1:

A fraction equivalent to $\frac{3}{5}$ is
(a) $\frac{3 + 2}{5 + 2}$
(b) $\frac{3 - 2}{5 - 2}$
(c) $\frac{3 \times 2}{5 \times 2}$
(d) none of these

Answer:

(c) $\frac{3 \times 2}{5 \times 2}$

Page No 99:

Question 2:

A fraction equivalent to $\frac{8}{12}$ is
(a) $\frac{8 + 4}{12 + 4}$
(b) $\frac{8 - 4}{12 - 4}$
(c) $\frac{8 \div 4}{12 \div 4}$
(d) none of these

Answer:

(c) $\frac{8 \div 4}{12 \div 4}$

Page No 100:

Question 3:

A fraction equivalent to $\frac{24}{36}$ is
(a) $\frac{3}{4}$
(b) $\frac{2}{3}$
(c) $\frac{8}{9}$
(d) none of these

Answer:

$(b) \frac{2}{3} Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 . Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 . Common factors of 24 and 36 are 1, 2, 3, 4, 6, 12 . H . C . F . = 12 Dividing both the numerator and the denominator by 12 : \frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}$

Page No 100:

Question 4:

If $\frac{3}{4}$ is equivalent to $\frac{x}{20}$ then the value of x is
(a) 15
(b) 18
(c)12
(d) none of these

Answer:

(a) 15

Explanation:
$(\frac{3}{4} = \frac{x}{20}) We have : 20 = 4 \times 5 So, we have to multiply the numerator by 5 . ∴ x = 3 \times 5 = 15$

Page No 100:

Question 5:

If $\frac{45}{60}$ is equivalent to $\frac{3}{x}$ then the value of x is
(a) 4
(b) 5
(c) 6
(d) none of these

Answer:

(a) 4

Explanation:
$(\frac{45}{60} = \frac{3}{x}) Now, 3 = 45 \div 15 So, we have to divide the denominator by 15 . ∴ x = 60 \div 15 = 4$

Page No 100:

Question 6:

Which of the following are like fractions?
(a) $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}$
(b) $\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$
(c) $\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}$
(d) none of these

Answer:

(c) $\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}$

(Fractions having the same denominator are called like fractions.)

Page No 100:

Question 7:

Which of the following is a proper fraction?
(a) $\frac{5}{3}$
(b) 5
(c) $1 \frac{2}{5}$
(d) none of these

Answer:

(d) none of these

In a proper fraction, the numerator is less than the denominator.

Page No 100:

Question 8:

Which of the following is a proper fractions?
(a) $\frac{7}{8}$
(b) $1 \frac{7}{8}$
(c) $\frac{8}{7}$
(d) none of these

Answer:

(a) $\frac{7}{8}$
In a proper fraction, the numerator is less than the denominator.

Page No 100:

Question 9:

Which of the following statements is correct?
(a) $\frac{3}{4} < \frac{3}{5}$
(b) $\frac{3}{4} > \frac{3}{5}$
(c) $\frac{3}{4}$ and $\frac{3}{5}$ cannot be compared

Answer:

(b) $\frac{3}{4} > \frac{3}{5}$
Between the two fractions with the same numerator, the one with the smaller denominator is the greater.

Page No 100:

Question 10:

The smallest of the fractions $\frac{3}{5}, \frac{2}{3}, \frac{5}{6}, \frac{7}{10}$ is
(a) $\frac{2}{3}$
(b) $\frac{7}{10}$
(c) $\frac{3}{5}$
(d) $\frac{5}{6}$

Answer:

(c) $\frac{3}{5}$

$2 5, 3, 6, 10 5 5, 3, 3, 5 3 1, 3, 3, 1 1, 1, 1, 1$

L.C.M. of 5, 3, 6 and 10 = (2 $\times$ 3 $\times$ 5) = 30
Thus, we have:
$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} ∴ The smallest fraction = \frac{18}{30} = \frac{3}{5}$

Page No 100:

Question 11:

The largest of the fractions $\frac{4}{5}, \frac{4}{7}, \frac{4}{9}, \frac{4}{11}$ is
(a) $\frac{4}{11}$
(b) $\frac{4}{5}$
(c) $\frac{4}{7}$
(d) $\frac{4}{9}$

Answer:

( b ) $\frac{4}{5}$
Among the given fractions with the same numerator, the one with the smallest denominator is the greatest.

Page No 100:

Question 12:

The smallest of the fractions $\frac{6}{11}, \frac{7}{11}, \frac{8}{11}, \frac{9}{11}$ is
(a) $\frac{6}{11}$
(b) $\frac{7}{11}$
(c) $\frac{8}{11}$
(d) $\frac{9}{11}$

Answer:

(a) $\frac{6}{11}$
Among like fractions, the fraction with the smallest numerator is the smallest.

Page No 100:

Question 13:

The smallest of the fractions $\frac{3}{4}, \frac{5}{6}, \frac{7}{12}, \frac{2}{3}$ is
(a) $\frac{2}{3}$
(b) $\frac{3}{4}$
(c) $\frac{5}{6}$
(d) $\frac{7}{12}$

Answer:

(d) $\frac{7}{12}$

Explanation:
$2 4, 6, 12, 3 2 2, 3, 6, 3 3 1, 3, 3, 3 1, 1, 1, 1$

L.C.M. of 4, 6, 12 and 3 = (2 $\times$ 2 $\times$ 3) = 12
Thus, we have:
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \frac{7}{12} Clearly, \frac{7}{12} is the smallest fraction .$

Page No 100:

Question 14:

$4 \frac{3}{5} = ?$
(a) $\frac{17}{5}$
(b) $\frac{23}{5}$
(c) $\frac{17}{3}$
(d) none of these

Answer:

(b) $\frac{23}{5}$

Page No 100:

Question 15:

$\frac{34}{7} = ?$
(a) $3 \frac{4}{7}$
(b) $7 \frac{3}{4}$
(c) $4 \frac{6}{7}$
(d) none of these

Answer:

(c) $4 \frac{6}{7}$
On dividing 34 by 7:
Quotient = 4
Remainder = 6
$\frac{34}{7} = 4 + \frac{6}{7} = 4 \frac{6}{7}$

Page No 101:

Question 16:

$\frac{5}{8} + \frac{1}{8} = ?$
(a) $\frac{3}{8}$
(b) $\frac{3}{4}$
(c) 6
(d) none of these

Answer:

(b) $\frac{3}{4}$

Explanation:

Addition of like fractions = Sum of the numerators / Common denominator
=

\frac{5}{8} + \frac{1}{8} = \frac{(5 + 1)}{8} = \frac{6^{3}}{8_{4}} = \frac{3}{4}

Page No 101:

Question 17:

$\frac{5}{8} - \frac{1}{8} = ?$
(a) $\frac{1}{4}$
(b) $\frac{1}{2}$
(c) $\frac{1}{16}$
(d) none of these

Answer:

(b) $\frac{1}{2}$
Explanation:
$\frac{5}{8} - \frac{1}{8} = \frac{(5 - 1)}{8} = \frac{4^{1}}{8_{2}} = \frac{1}{2}$

Page No 101:

Question 18:

$3 \frac{3}{4} - 2 \frac{1}{4} = ?$
(a) $1 \frac{1}{2}$
(b) $1 \frac{1}{4}$
(c) $\frac{1}{4}$
(d) none of these

Answer:

$(a) 1 \frac{1}{2} Explanation : 3 \frac{3}{4} - 2 \frac{1}{4} \Rightarrow \frac{15}{4} - \frac{9}{4} \Rightarrow \frac{(15 - 9)}{4} \Rightarrow \frac{6}{4} = \frac{3}{2} = 1 \frac{1}{2}$

Page No 101:

Question 19:

$\frac{5}{6} + \frac{2}{3} - \frac{4}{9} = ?$
(a) $1 \frac{1}{3}$
(b) $1 \frac{1}{6}$
(c) $1 \frac{1}{9}$
(d) $1 \frac{1}{18}$

Answer:

(d) $1 \frac{1}{18}$

Explanation:
$3 3, 6, 9 2 1, 2, 3 3 1, 1, 3 1, 1, 1$

$\frac{5}{6} + \frac{2}{3} - \frac{4}{9} (L . C . M . of 3, 6 and 9 = (2 \times 3 \times 3) = 18) = \frac{(15 + 12 - 8)}{18} \{[18 \div 6 = 3, 3 \times 5 = 15], [18 \div 3 = 6, 6 \times 2 = 12] and [18 \div 9 = 2, 2 \times 4 = 8]\} = \frac{(27 - 8)}{18} = \frac{19}{18} = 1 \frac{1}{18}$

Page No 101:

Question 20:

Which is greater: $3 \frac{1}{3} or \frac{33}{10}$ ?
(a) $3 \frac{1}{3}$
(b) $\frac{33}{10}$
(c) both are equal

Answer:

(a) $3 \frac{1}{3}$

Explanation:
Let us compare $3 \frac{1}{3} and \frac{33}{10} or \frac{10}{3} and \frac{33}{10}$ .
10 ⨯ 10 = 100 and 3 ⨯ 33 = 99
Clearly, 100 > 99
∴ $\frac{10}{3} > \frac{33}{10} or 3 \frac{1}{3} > \frac{33}{10}$

Page No 103:

Question 1:

Define a fraction. Give five examples of fractions.

Answer:

A fraction is defined as a number representing a part of a whole, where the whole may be a single object or a group of objects.

Examples: $\frac{5}{7}, \frac{8}{5}, \frac{2}{3}, \frac{4}{3}, \frac{4}{9}$

Page No 103:

Question 2:

What fraction of an hour is 35 minutes?

Answer:

An hour has 60 minutes.
$∴$ Fraction for 35 minutes = $\frac{35^{7}}{60_{12}} = \frac{7}{12}$
Hence, $\frac{7}{12}$ part of an hour is equal to 35 minutes.

Page No 103:

Question 3:

Find the equivalent fraction of 5/8 with denominator 56.

Answer:

56 = 8 ⨯ 7
So, we need to multiply the numerator by 7.

∴

\frac{5}{8} = \frac{5 \times 7}{8 \times 7} = \frac{35}{56}

Hence, the required fraction is

\frac{35}{56}

.

Page No 103:

Question 4:

Represent $2 \frac{3}{5}$ on the number line.

Answer:

Let OA = AB = BC = 1 unit
$∴$ OB = 2 units and OC = 3 units
Divide BC into 5 equal parts and take 3 parts out to reach point P.
Clearly, point P represents the number $2 \frac{3}{5}$ .

Page No 103:

Question 5:

Find the sum $2 \frac{4}{5} + 1 \frac{3}{10} + 3 \frac{1}{15}$ .

Answer:

We have:
$2 \frac{4}{5} + 1 \frac{3}{10} + 3 \frac{1}{15} = \frac{14}{5} + \frac{13}{10} + \frac{46}{15} 5 5, 10, 15 2 1, 2, 3 3 1, 1, 3 1, 1, 1 L . C . M . of 5, 10 and 15 = (5 \times 2 \times 3) = 30 = \frac{84 + 39 + 92}{30} \{[30 \div 5 = 6, 6 \times 14 = 84], [30 \div 10 = 3, 3 \times 13 = 39] and [30 \div 15 = 2, 2 \times 46 = 92]\} = \frac{215^{43}}{30_{6}} = \frac{43}{6} = 7 \frac{1}{6}$

Page No 103:

Question 6:

The cost of a pen is Rs $16 \frac{2}{3}$ and that of a pencil is Rs $4 \frac{1}{6}$ .
Which costs more and by how much?

Answer:

Cost of a pen = $Rs 16 \frac{2}{3} = Rs \frac{50}{3} = Rs \frac{50 \times 2}{3 \times 2} = Rs \frac{100}{6}$

Cost of a pencil = $Rs 4 \frac{1}{6} = Rs \frac{25}{6}$
$\frac{100}{6} > \frac{25}{6} ∴ Rs 16 \frac{2}{3} > Rs 4 \frac{1}{6}$
So, the cost of a pen is more than the cost of a pencil.
Difference between their costs:
$= Rs (\frac{50}{3} - \frac{25}{6}) = Rs (\frac{100 - 25}{6}) = Rs (\frac{75^{25}}{6_{2}}) = Rs (\frac{25}{2}) = Rs 12 \frac{1}{2}$
Hence, the cost of a pen is Rs $12 \frac{1}{2}$ more than the cost of a pencil.

Page No 103:

Question 7:

Of $\frac{3}{4}$ and $\frac{5}{7}$ , which is greater and by how much?

Answer:

Let us compare $\frac{3}{4} and \frac{5}{7}$ .
By cross multiplying:
3 ⨯ 7 = 21 and 4 ⨯ 5 = 20
Clearly, 21 > 20
∴ $\frac{3}{4} > \frac{5}{7}$
Their difference:
$\frac{3}{4} - \frac{5}{7} L . C . M . of 4 and 7 = (2 \times 2 \times 7) = 28 = \frac{21 - 20}{28} \{[28 \div 4 = 7, 7 \times 3 = 21] and [28 \div 7 = 4, 4 \times 5 = 20]\} = \frac{1}{28}$
Hence, $\frac{3}{4} is greater than \frac{5}{7} by \frac{1}{28} .$

Page No 103:

Question 8:

Convert the fractions $\frac{1}{2}, \frac{2}{3}, \frac{4}{9}$ and $\frac{5}{6}$ into like fractions.

Answer:

$The given fractions are \frac{1}{2}, \frac{2}{3}, \frac{4}{9}, \frac{5}{6} .$
L.C.M. of 2, 3, 9 and 6 = (2 ⨯ 3 ⨯ 3) = 18
Now, we have:
$\frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18} \frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18} \frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18} \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} Hence, \frac{9}{18}, \frac{12}{18}, \frac{8}{18} and \frac{15}{18} are like fractions .$

Page No 103:

Question 9:

Find the equivalent fraction of $\frac{3}{5}$ having denominator 30.

Answer:

$Let \frac{3}{5} = \frac{}{30}$

30 = 5 ⨯ 6
So, we have to multiply the numerator by 6 to get the equivalent fraction having denominator 30.

∴ $\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$

Thus, $\frac{18}{30} is the equivalent fraction of \frac{3}{5} .$

Page No 103:

Question 10:

Reduce $\frac{84}{98}$ to the simplest form.

Answer:

The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
The factors of 98 are 1, 2, 7, 14, 49, 98.
The common factors of 84 and 98 are 1, 2, 7, 14.
The H.C.F. of 84 and 98 is 14.
Dividing both the numerator and the denominator by the H.C.F.:
$\frac{84}{98} = \frac{84 \div 14}{98 \div 14} = \frac{6}{7}$

Page No 103:

Question 11:

$\frac{24}{11}$ is an example of
(a) a proper fraction
(b) an improper fraction
(c) a mixed fraction
(d) none of these

Answer:

(b) an improper fraction

In an improper fraction, the numerator is greater than the denominator.

Page No 103:

Question 12:

$\frac{3}{8}$ is an example of
(a) a proper fraction
(b) an improper fraction
(c) a mixed fraction
(d) none of these

Answer:

(a) proper fraction

In a proper fraction, the numerator is less than the denominator.

Page No 103:

Question 13:

$\frac{3}{8}$ and $\frac{5}{12}$ on comparison give
(a) $\frac{3}{8} > \frac{5}{12}$
(b) $\frac{3}{8} < \frac{5}{12}$
(c) $\frac{3}{8} = \frac{5}{12}$
(d) none of these

Answer:

(b) $\frac{3}{8} < \frac{5}{12}$

Considering $\frac{3}{8} and \frac{5}{12}$ :

$On cross multiplying, we get : 3 \times 12 = 36 and 8 \times 5 = 40 Clearly, 36 < 40 ∴ \frac{3}{8} < \frac{5}{12}$

Page No 103:

Question 14:

The largest of the fractions $\frac{2}{3}, \frac{5}{9}, \frac{1}{2}$ and $\frac{7}{12}$ is
(a) $\frac{2}{3}$
(b) $\frac{5}{9}$
(c) $\frac{7}{12}$
(d) $\frac{1}{2}$

Answer:

(a) $\frac{2}{3}$

Explanation:
L.C.M. of 3, 9, 2 and 12 = ( 2 ⨯ 2 ⨯ 3 ⨯ 3) = 36
Now, we have:
$\frac{2}{3} = \frac{2 \times 12}{3 \times 12} = \frac{24}{36} \frac{5}{9} = \frac{5 \times 4}{9 \times 4} = \frac{20}{36} \frac{1}{2} = \frac{1 \times 18}{2 \times 18} = \frac{18}{36} \frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36} Hence, \frac{24}{36} = \frac{2}{3} is the largest fraction .$

Page No 103:

Question 15:

$3 \frac{3}{4} - 1 \frac{1}{2} = ?$
(a) $2 \frac{1}{2}$
(b) $2 \frac{1}{4}$
(c) $1 \frac{1}{2}$
(d) $1 \frac{1}{4}$

Answer:

(b) $2 \frac{1}{4}$
Explanation:

$3 \frac{3}{4} - 1 \frac{1}{2} = \frac{15}{4} - \frac{3}{2} (L . C . M . of 2 and 4 = (2 \times 2) = 4) = \frac{15 - 6}{4} = \frac{9}{4} = 2 \frac{1}{4}$

Page No 103:

Question 16:

Which of the following are like fractions?
(a) $\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$
(b) $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}$
(c) $\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}$
(d) none of these

Answer:

(c) $\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}$
Like fractions have same the denominator.

Page No 104:

Question 17:

$? - \frac{8}{21} = \frac{8}{21}$
(a) 0
(b) 1
(c) $\frac{21}{8}$
(d) $\frac{16}{21}$

Answer:

(d) $\frac{16}{21}$

$? - \frac{8}{21} = \frac{8}{21} ? = \frac{8}{21} + \frac{8}{21} = \frac{16}{21}$

Page No 104:

Question 18:

Fill in the blanks:
(i) $9 \frac{2}{3} + . . . . . . = 19$
(ii) $6 \frac{1}{6} - ? = \frac{29}{30}$
(iii) $7 - 5 \frac{2}{3} = . . . . . .$
(iv) $\frac{72}{90}$ reduced to simples form is ......
(v) $\frac{42}{54} = \frac{7}{}$

Answer:

$(i) 9 \frac{2}{3} + . . . . . . = 19 . . . . . . = 19 - 9 \frac{2}{3} . . . . . . = \frac{19}{1} - \frac{29}{3} . . . . . . = \frac{57 - 29}{3} . . . . . . = \frac{28}{3} . . . . . = 9 \frac{1}{3}$

(ii)
$6 \frac{1}{6} - ? = \frac{29}{30} ? = 6 \frac{1}{6} - \frac{29}{30} ? = \frac{37}{6} - \frac{29}{30} ? = \frac{185 - 29}{30} ? = \frac{156^{26}}{30_{5}} ? = 5 \frac{1}{5}$

(iii)
$7 - 5 \frac{2}{3} = \frac{7}{1} - \frac{17}{3} = \frac{21 - 17}{3} = \frac{4}{3} = 1 \frac{1}{3}$

(iv)

$The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 . The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 . The common factors of 72 and 90 are 1, 2, 3, 6, 9, 18 . H . C . F . of 72 and 90 is 18 . \frac{72 \div 18}{90 \div 18} = \frac{4}{5}$

(v)

$\frac{42}{54} = \frac{7}{9} \Rightarrow \frac{42 \div 6}{54 \div 6} = \frac{7}{9}$

Page No 104:

Question 19:

Write 'T' for true and 'F' for false for each of the statements given below:
(a) $3 \frac{1}{3} > \frac{33}{10}$ .
(b) $8 - 1 \frac{5}{6} = 7 \frac{1}{6}$ .
(c) $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$ are like fractions.
(d) $\frac{3}{5}$ lies between 3 and 5.
(e) Among $\frac{1}{2}, \frac{1}{3}, \frac{3}{4}, \frac{4}{3}$ the largest fractions is $\frac{4}{3}$ .

Answer:

(a) T
(b) F    $(\frac{8}{1} - \frac{11}{6} = \frac{48 - 11}{6} = \frac{37}{6} = 6 \frac{1}{6})$
(c) F (Because like fractions have the same denominator.)
(d) F    (It lies between 0 and 1 as all proper fractions are less than 1.)
(e) T    (Because it is an improper fraction, while the others are proper fractions.)

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