NCERT Solutions for Class 6 Maths Chapter 10 - Ratio, Proportion And Unitary Method

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Question 1:

Question 2:

(i) 24:56 =     24  =     24 ​÷ 8     =    3     
                       56       56 â€‹÷ 8            7
As the H.C.F. of 3 and 7 is 1, the simplest form of 24:56 is 3:7.

(ii) 84 paise to Rs 3 = Rs 0.84 to R. 3 =  0.84  =  0.84​ ÷ 3    =  0.28   =   28    =  28 ​÷  4   =   7  
                                                                   3            3 â€‹÷ 3             1          100      100 â€‹÷ 4        25
As the H.C.F. of 7 and 25 is 1, the simplest form of  0.84:3 is 7:25.

(iii) 4 kg:750 g = 4000 g:750 g =   4000 ​÷ 250   =    16   
                                                            750 â€‹÷ 250            3
As the H.C.F. of 16 and 3 is 1, the simplest form of 4000:750 is 16:3.

(iv) 1.8 kg:6 kg  =   1.8   =   18   =  18 ​÷ 6  =     3  
                               6           60      60 â€‹÷ 6        10 
As the H.C.F. of 3 and 10 is 1, the simplest form of 1.8:6 is 3:1.

(v) 48 minutes to 1 hour = 48 minutes to 60 minutes = 48:60 =  48 ​÷ 12   =  4   
                                                                                                 60 â€‹÷ 12        5
As the H.C.F. of 4 and 5 is 1, the simplest form of 48:60 is 4:5.
 
(vi) 2.4 km to 900 m = 2400m:900m =    2400   =   24    =   24 ​÷ 3   =    8  
                                                               900          9           9 â€‹÷ 3          3
 As the H.C.F. of 8 and 3 is 1, the simplest form of 2400:900 is 8:3.

                    

Question 3:

(i) 36:90 =   36    =    36 ​÷ 18    =     2          (As the H.C.F. of 36 and 90 is 18.)
                   90          90 ​÷ 18            5
Since the H.C.F. of 2 and 5 is 1, the simplest form of 36:90 is 2:5.

(ii) 324:144 =   324   =    324 ​÷ 36    =    9       (As the H.C.F. of  324 and 144 is 36.)
                          144          144 â€‹÷ 36           4
 Since the H.C.F. of 9 and 4 is 1, the simplest form of 324:144 is 9:4.

(iii) 85:561 =  85   =   85 ​÷ 17   =      5         (As the H.C.F. of 85 and 561 is 17.)
                        561      561 â€‹â€‹÷ 17           33
Since the H.C.F. of 5 and 33 is 1, the simplest form of 85:561 is 5:33.

(iv) 480:384 =    480    =    480 ​÷ 96    =   5       (As the H.C.F. of 480 and 384 is 96.)
                            384          384 â€‹â€‹÷ 96          4
Since the H.C.F. of 5 and 4 is 1, the simplest form of 480:384 is 5:4.

(v) 186:403 =     186   =   186 ÷ 31    =   6        (As the H.C.F. of 186 and 403 is 31.)
                             403        403 ÷ 31        13
Since the H.C.F. of 6 and 13 is 1, the simplest form of 186:403 is 6:13.

(vi) 777:1147 =  777  â€‹÷ 37      =   21           (As the H.C.F. of 777 and 1147 is 37.)
                            1147  â€‹÷  37         31
Since the H.C.F. of 21 and 31 is 1, the simplest form of 777:1147 is 21:31.

Question 4:

(i) Rs 6.30:Rs 16.80
             6.30     =    63    =   63 â€‹÷ 21     =   3        (H.C.F. of 63 and 168 is 21.)
            16.80         168        168  â€‹÷ 21        8
    Ratio = 3 : 8
(ii)3 weeks:30 days = 21days:30 days          (1 week = 7 days)
          21     =   21 â€‹÷ 3   =   7          (H.C.F. of 21 and 30 is 3.)
          30          30 â€‹ â€‹÷ 3       10
     Ratio = 7 : 10
 (iii) 3 m 5 cm:35 cm = 305 cm:35 cm      (1 m = 100 cm)
            305  =   305  â€‹÷ 5   =   61        (H.C.F. of 305 and 35 is 5.)
             35         35  â€‹÷ 5          7
      Ratio = 61:7
(iv) 48 min:2 hours 40 min = 48 min:160 min        (1 hour = 60 mins)
             48   =    48  â€‹÷ 16    =    3         (H.C.F. of 48 and 160 is 16.)
            160        160  â€‹÷ 16       10
      Ratio = 3:10
(v) 1 L 35 mL:270 mL = 1035 mL:270 mL         (1 L = 1000 mL)
             1035   =   1035  â€‹÷ 45    =   23        (H.C.F. of 1035 and 270 is 45.)
              270           270  â€‹÷ 45          6
       Ratio = 23:6
(vi) 4 kg:2 kg 500 g = 4000 g:2500 g        (1 kg= 1000 g)
            4000   =   40  =  40  â€‹÷ 5   =   8     (H.C.F. of 40 and 25 is 5.)
            2500         25      25  â€‹÷ 5        5
      Ratio = 8:5

Question 5:

Mr Sahai's earning = Rs 16800
Mrs Sahai's earning = Rs 10500
(i) Ratio = 16800:10500 = 168:105 =  168  â€‹÷ 21  =   8          (H.C.F. of 168 and 105 is 21.)
                                                           105 â€‹ â€‹÷ 21           5
Mr Sahai's income:Mrs Sahai's income = 8:5
(ii)Ratio = 10500:16800 = 105:168 =  105  ​÷ 21   =    5        (H.C.F. of 168 and 105 is 21.)
                                                         168 ​ â€‹÷ 21          8
Mrs Sahai's income:Mr Sahai's income = 5:8

(iii) Total income = 16800 + 10500 = Rs  27300
   Ratio = 16800:27300 = 168:273 =  168   =  168  â€‹÷ 21   =  8  (H.C.F. of 168 and 273 is 21.)
                                                        273        273  â€‹÷ 21     13
Mrs Sahai's income:Total income = 8:13

Question 6:

Rohit's income = Rs 15300
Rohit's savings = Rs 1224
(i) Income:Savings = 15300:1224 = 15300 ​÷ 612   =  25        (H.C.F. of 15300 and 1224 is 612.)
                                                                  1224  â€‹÷ 612         2 
    Income:Savings = 25:2
(ii) Monthly expenditure = Rs (15300 $-$ 1224) = Rs 14076
    Income:Expenditure = 15300:14076 =  15300  ÷ 612  =   25       (H.C.F. of 15300 and 14076 is 612.)
                                                                       14076  â€‹÷ 612        23
     Income:Expenditure = 25:23
(iii) Expenditure : Savings = 14076:1224 =  14076  ÷ 612  =  23       (H.C.F. of 14076 and 1224 is 612.)
                                                                           1224  â€‹÷ 612         2
      Expenditure:Savings = 23:2
                    

Question 7:

Number of male:Number of female = 5:3
Let the number be x.
Number of male = 5x
​Number of female = 3x
Number of male workers = 115
   Now, 5x = 115
         ⇒  x =   115   = 23
                       5
Number of female workers in the mill = 3x = 3 × 23 = 69

Question 8:

Boys:Girls = 9:5
Let the number of boys = 9x 
Let the number of girls = 5x
Total strength of the school = 448
According to given condition, we have:
                                                      9x + 5x = 448
                                              ⇒         14x = 448
                                               ⇒            x =   448   = 32
                                                                      14
Number of boys = 9x = 9 × 32 = 288
Number of girls = 5x = 5 â€‹× 32 = 160

Question 9:

Kamal:Madhu = 7:2
Sum of the ratio terms = 7 + 2 = 9
Kamal's share =   7   × 1575 =   11025   = Rs 1225
                            9                         9
Madhu's share =   2   × 1575 =   3150   = Rs 350
                            9                       9

Question 10:

A:B:C = 3:5:7
Sum of the ratio terms = 3 + 5 +7 = 15
A's share =   3    × 3450 =   10350   = Rs 690
                   15                       15

B's share =   5   × 3450 =   17250   = Rs 1150
                   15                        15

 C's share =     7   × 3450 =   24150  = Rs 1610      
                      15                       15

Question 11:

Two number are in the ratio 11:12.
Let the numbers be 11x and 12x.
Given:     11x + 12x = 460
         ⇒          23x = 460
        ⇒               x =   460   =  20
                                    23
First number = 11x = 11 × 20 = 220
Second number = 12x = 12 × 20 = 240
Hence, the numbers are 220 and 240.

Question 12:

Ratio of the two parts of line segment = 4:3
Sum of the ratio terms = 4 + 3 = 7
First part =   4   × 35 cm = 4 × 5 cm = 20 cm
                    7
Second part =    3   × 35 cm = 3 × 5 cm = 15 cm
                         7

Question 13:

Number of bulbs produced each day = 630
Out of 10 bulbs, 1 is defective.
Number of defective bulbs =  630  = 63
                                             10

$\therefore$ Number of defective bulbs produced each day = 63

Question 14:

Price of pencil = Rs 96 per score 
Price of ball pen = Rs 50.40 per dozen
Price per unit of pencil =  96  = 4.8
                                       20
Price per unit of ball pen =  50.40  =  4.2
                                            12
   Ratio =    4.8   =   48   =   48  â€‹÷ 6    =    8  
                   4.2        42        42  â€‹÷  6         7
Price of a pencil:Price of a ball pen = 8:7

Question 15:

Length:Width = 5:3
Let the length and the width of the field be 5x m and 3x m, respectively.
Width = 42 m
3x = 42
 x =   42   = 14
          3
$\therefore$ Length = 5x = 5 × 14 = 70 metres

Question 16:

Income:Savings = 11:2
Let the income and the saving be Rs 11x and Rs 2x, respectively.
Saving = Rs 1520
  2x = 1520
   x =   1520   = 760
              2
$\therefore$ Income = Rs 11x =Rs (11 × 760) = Rs 8360
   Expenditure = Income $-$ Saving
                      = Rs (8360 $-$ 1520 )
                      = Rs 6840

Question 17:

Income:Expenditure = 7:6
Let the income and the expenditure be Rs 7x and Rs 6x, respectively.
Income = Rs 14000
7x = 14000
x =    14000  =  2000
             7
Expenditure = Rs 6x = Rs 6 × 2000 = Rs 12000
$\therefore$ Saving = Income $-$ Expenditure
          = Rs (14000 $-$ 12000)
         = Rs 2000

Question 18:

Let the weight of zinc be x kg.
Ratio of zinc and copper = 7:9
Weight of copper in the alloy = 11.7 kg
     7   =    x     
     9       11.7
⇒  x =  11.7 × 7   =  81.9   = 9.1
                9                9
Weight of zinc = 9.1 kg

Question 19:

A bus covers 128 km in 2 hours.
Speed of the bus =   Distance  =  128 km   = 64 km/ hr
                               Time               2 hr

A train covers 240 km in 3 hours.
Speed of the train =  Distance   =  240   = 80 km /hr
                                 Time             3

Ratio of their speeds = 64:80 =  64  =   64 ÷ 16   =   4  
                                                 80       80 ÷ 16        5
$\therefore$ Ratio of the speeds of the bus and the train = 4:5

Question 20:

(i) (3:4) or (9:16) 

Making the denominator equal:

        3 × 4   =  12  and 12   >   9 
        4 × 4       16         16       16

 $\therefore$ (3:4) > (9:16)

(ii) (5:12) or (17:30) 

Making the denominator equal:

        5 × 5   =    25   and   17 × 2    =    34     
      12 × 5         60            30 × 2          60
⇒    25   <    34         
      60         60
 $\therefore$ (5:12) < (17:30)

(iii) (3:7) or (4:9) 

Making the denominator equal:

          3 × 9  =   27   and   4 × 7   =  28  
          7 × 9       63            9 â€‹× 7       63
⇒       27  <   28      
         63       63

(3:7) < (4:9)

(iv) (1:2) or (13:27)

Making the denominator equal:

        1× 27   =   27   and   13 × 2   =   26  
        2 × 27       54            27 â€‹× 2        54

⇒   27  >  26          
     54       54

(1:2) > (13:27)

Question 1:

(i)   24   =   24 ​÷ 8  =   3   =    3 × 4  =  12      
      40         40 â€‹÷ 8      5          5  × 4      20

(ii)    36  =   36  â€‹÷ 9  =  4   =   4 × 3  =   12       
        63        63  â€‹÷ 9      7        7 × 3        21

(iii)   5   =   5 × 4   =  20  =   5 × 7   =   35  
        7        7 × 4       28        7 × 7         49

Question 2:

(i) 4, 6, 8, 12
    4  =    4 â€‹÷ 2   =   2 ;     8  =    8  â€‹÷ 4   =   2  
    6        6  â€‹÷ 2       3       12       12  â€‹÷ 4       3
Hence, 4:9::8:12 are in proportion.

(ii) 7, 42, 13, 78
      7    =  7  â€‹÷ 7   =   1 ;     13   =  13  â€‹÷ 13     =    1  
      42      42  â€‹÷ 7      6        78        78  â€‹÷ 13           6
Hence, 7:42::13:78 are in proportion.

(iii) 33, 121, 9, 96
      33   =   33  â€‹÷ 11   =   3   ;     9   =   9  â€‹÷ 3   =   3  
     121      121  â€‹÷ 11      11         96       96  â€‹÷ 3      32
 Hence, 33:121::9:96 are not in proportion.

(iv) 22, 33, 42, 63
     
$\frac{22}{33}=\frac{22÷11}{33÷11}=\frac{2}{3} \mathrm{and} \frac{42}{63}=\frac{42÷21}{63÷21}=\frac{2}{3}$

Hence, 22:33 :: 42 : 63 are not in proportion.
 
(v) 32, 48, 70, 210
       32   =   32  â€‹÷ 6  =   7 ;    70   =   70  â€‹÷ 70   =  1 
       48        48  â€‹÷ 6       8     210      210  â€‹÷ 70      3
    Hence, 32:48::70:210 are not in proportion.


 (vi) 150, 200, 250, 300
       150   =   150  â€‹÷ 50   =  3;   250  =  250  â€‹÷ 50  =   5  
      200         200  â€‹÷ 50        4   300      300  â€‹÷ 50       6
    Hence, 150:200::250:300 are not in proportion.

Question 3:

(i) 60:105::84:147
      60   =   60  â€‹÷ 15   =   4         (H.C.F. of 60 and 105 is 15.)
     105       105  â€‹÷ 15       7
       84   =   84  â€‹÷ 21  =   4         (H.C.F. of 84 and 147 is 21.)
      147     147  â€‹÷ 21        7
    Hence, 60:105::84:147 are in proportion.
(ii) 91:104::119:136
      91   =  91  â€‹÷ 13   =   7         (H.C.F. of 91 and 104 is 13.)
     104       104  â€‹÷ 13      8  
      119  =   119  â€‹÷ 17  =   7      (H.C.F. of 11 and 136 is 17.)
     136        136  â€‹÷ 17       8
  Hence, 91:104::119:136 are in proportion.
(iii) 108:72::129:86
      108  =   108  â€‹÷ 36   =   3        (H.C.F. of 108 and 72 is 36.)
      72           72 â€‹ â€‹÷ 36        2
       129   =   â€‹129  â€‹÷  43  =  3       (H.C.F. of 129 and 86 is 43.)
        86          86   â€‹÷ 43        2
   Hence, 108:72::129:86 are in proportion.
(iv) 39:65::141:235
       39   =   39  â€‹÷ 13  =   3        (H.C.F. of 39 and 65 is 13.)
       65         65  â€‹÷ 13      5
      141   =   141  â€‹÷  47   =   3     (H.C.F. of 141 and 235 is 47.)
      235         235  â€‹÷ 47        5
   Hence, 39:65::141:235 are in proportion.

Question 4:

(i) 55:11::x:6
      Product of extremes = Product of means
                          55 × 6 = 11 × x
⇒                            11x = 330
⇒                               x =  330   = 30
                                          11
(ii) 27:x::63:84
      Product of extremes = Product of means
                      27 â€‹× 84 = x â€‹× 63
 ⇒                         63x = 2268
 ⇒                             x =  2268  = 36
                                           63
(iii) 51:85::57:x 
      Product of extremes = Product of means
                           51 × x = 85 × 57
 ⇒                           51x = 4845
 ⇒                               x =   4845   = 95
                                             51
(iv) x:92::87:116
     Product of extremes = Product of means
                       x ×  116 = 92 â€‹× 87
 ⇒                      116x = 8004
 ⇒                           x  =   8004  = 69
                                       116

Question 5:

(i) 51:68::85:102
   Product of means = 68 × 85 = 5780
   Product of extremes = 51 × 102 = 5202
   Product of means ≠ Product of extremes
    Hence, (F).
(ii) 36:45::80:100
  Product of means = 45 â€‹× 80 = 3600
  Product of extremes = 36 × 100 = 3600
  Product of means = Product of extremes 
   Hence, (T).
(iii) 30 bags:18 bags::Rs 450:Rs 270
       or 30:18::450:270
     Product of means = 18 × 450 = 8100
     Product of extremes = 30 â€‹× 270 = 8100
     Product of means = Product of extremes 
     Hence, (T).
(iv) 81 kg:45 kg::18 men:10 men
      or 81:45::18:10
     Product of means = 45 × 18 = 810
     Product of extremes = 81 × 10 = 810
     Product of means = Product of extremes
      Hence, (T).
(v) 45 km:60 km::12 h:15 h
     or,45:60::12:15
     Product of means = 60 × 12 = 720
     Product of extremes = 45 × 15 = 675
     Product of means ≠ Product of extremes 
      Hence, (F).
(vi) 32 kg:Rs 36::8 kg:Rs 9
     Product of means = 36 × 8 = 288
     Product of extremes = 32 × 9 = 288
     Product of means = Product of extremes
     Hence, (T).
    

Question 6:

(i) 25 cm:1 m and Rs 40:Rs 160 (or) 25 cm:100 cm and Rs 40:Rs 160
      25  =  25 ​÷ 25  =  1  and  40  =  40 ÷ 40  =  1
    100      100 â€‹â€‹÷ 25     4         160    160 ​÷ 40      4 
       Hence, they are in proportion.

(ii) 39 litres:65 litres and 6 bottles:10 bottles
       39   =  39 ​÷ 13   =   3    and   6   =   6 ​÷ 2   =   3 
       65        65 â€‹â€‹÷ 13       5            10      10 â€‹÷ 2        5
      Hence they are  in proportion.

(iii) 200 mL:2.5 L and Rs 4:Rs 50 (or) 200 mL:2500 mL and Rs 4:Rs 50
        200   =   2   and   4   =   4 ​÷ 2   =    2  
       2500      25           50       50 ÷ 2       25
     Hence, they are in proportion.

(iv) 2 kg:80 kg and 25 g:625 kg  (or)  2 kg:80 kg and 25 g:625000 g
        2   =  2 ​÷ 2    =   1    and   25      =   25 ​÷ 25     =     1   
       80      80 â€‹÷ 2       40         625000    625000 â€‹â€‹÷ 25     25000
        Hence, they are not in proportion.

Question 7:

Let the 3rd term be x.
Thus, 51:68::x:108
 We know:
                      Product of extremes = Product of means
                               51 × 108 = 68 × x
             ⇒                     5508 = 68x
             ⇒                          x =  5508  = 81
                                                   68
Hence, the third term is 81.

Question 8:

Let the second term be x.
Then. 12:x::8:14
We know:
  Product of extremes = Product of means
                                 12 × 14 = 8x
            ⇒                       168 = 8x
​            ⇒                           x =   168  = 21
                                                    8
 Hence, the second term is 21.

Question 9:

(i) 48:60, 60:75
      Product of means = 60 × 60 = 3600
      Product of extremes = 48 × 75 = 3600
Product of means = Product of extremes
       Hence, 48:60::60:75 are in continued proportion.

(ii) 36:90, 90:225
     Product of means = 90 × 90 = 8100
     Product of extremes = 36 × 225 = 8100
Product of means = Product of extremes
      Hence, 36:90::90:225 are in continued proportion.

(iii) 16:84, 84:441
    Product of means = 84 × 84 = 7056
    Product of extremes = 16 × 441 = 7056
Product of means = Product of extremes
    Hence, 16:84::84:441 are in continued proportion.
                             

Question 10:

Given: 9:x::x:49
We know:
  Product of means = Product of extremes
                                     x × x = 9 × 49
                       ⇒               x² = 441
                       ⇒               x² = (21)²
                       ⇒                x = 21

Question 11:

Let the height of the pole = x m
Then, we have:
      x:20::6:8
Now, we know:
        Product of extremes = Product of means
                                   8x = 20​ × 6
                                     x =  120  = 15
                                              8
​Hence, the height of the pole is 15 m.

Question 1:

5:3::x:6
We know:
   Product of means = Product of extremes
                               3x = 5 â€‹× 6
                            ⇒ x =  30  = 10
                                        3
$\therefore$x = 10

Question 2:

Cost of 14 m of cloth = Rs 1890
Cost of 1 m of cloth =  1890  = Rs 135
                                     14
Cost of 6 m of cloth = 6​ × 135 = Rs 810

Question 3:

Cost of dozen soaps = Rs 285.60
Cost of 1 soap =  285.60 
                            12
Cost of 15 soaps = 15​ ×  285.60  =  4284  = Rs 357
                                         12            12

Question 4:

Cost of 9 kg of rice = Rs 327.60
Cost of 1 kg of rice =  327.60  
                                     9 
Cost of 50 kg of rice = 50​ ×  327.60  =  16380  = Rs 1820
                                               9              9
Hence, the cost of 50 kg of rice is Rs 1820.

Question 5:

Weight of 22.5 m of uniform iron rod = 85.5 kg
Weight of 1 m of uniform iron rod =  85.5  kg
                                                        22.5
Weight of 5 m of uniform iron rod = 5​ ×  85.5  =  427.5  = 19 kg
                                                              22.5       22.5
Thus, the weight of 5 m of iron rod is 19 kg.

Question 6:

Oil contained by 15 tins = 234 kg
Oil contained by 1 tin =  234  kg
                                       15
Oil contained by 10 tins = 10 ×  234  =  2340  = 156 kg
                                                  15         15

Question 7:

Distance covered by a car in 12 L diesel = 222 km
Distance covered by it in 1 L diesel =  222  km
                                                          12
Distance covered by it in 22 L diesel = 22 ×  222  =  4884  = 407 km
                                                                   12          12

Question 8:

Cost of transporting 25 tonnes of weight = Rs 540
Cost of transporting 1 tone of weight =  540 
                                                             25
Cost of transporting 35 tonnes of weight = 35​ ×  540  =  18900  = Rs 756
                                                                          25          25

Question 9:

Let the weight of copper be x g.
​Then, 4.5:3.5::18.9:x
        Product of extremes = Product of means
              4.5 × x  = 3.5 × 18.9
          ⇒ x =  66.15  = 14.7
                       4.5
So, the weight of copper is 14.7 g.

Question 10:

Number of inland letters whose total cost is Rs 87.50 = 35
Number of inland letters of whose cost is Re 1 =   35    
                                                                         87.50
Number of inland letters whose cost is Rs 315 = 315​ ×   35    =  11025  = 126
                                                                                  87.50      87.50
Hence, we can buy 126 inland letters for Rs 315.

Question 11:

Number of bananas that can be purchased for Rs 104 = 48 (4 dozen)
Number of bananas that can be purchased for Re 1 =  48 
                                                                               104
Number of bananas that can be purchased for Rs 6.50 = 6.50 ×  48   =  312  = 3
                                                                                               104       104
Hence, 3 bananas can be purchased for Rs 6.50.

Question 12:

Number of chairs that can be bought for Rs 22770 = 18
Number of chairs that can be bought for Re 1 =    18   
                                                                        22770
Number of chairs that can be bought for Rs 10120 = 10120 ×    18     =  182160  = 8
                                                                                             22770       22770

Question 13:

(i) Time taken by the car to travel 195 km = 3 hours
   Time taken by it to travel 1 km =   3   hours
                                                      195
  Time taken by it to travel 520 km = 520 ×   3   =  1560  = 8 hours
                                                                  195      195

(ii) Distance covered by the car in 3 hours = 195 km
    Distance covered by it in 1 hour =  195  = 65 km
                                                          3
   Distance covered by it in 7 hours = 7 × 65 = 455 km

Question 14:

(i) Earning of a labourer in 12 days = Rs 1980
    Earning of the labourer in 1 day =  1980  = Rs 165
                                                        12
    Earning of the labourer in 7 days = 7​ × 165 = Rs 1155
(ii) Number of days taken by the labourer to earn Rs 1980 = 12 days
     Number of days taken by him to earn Re 1 =  12  days
                                                                       1980
     Number of days taken by him to earn Rs 2640 = 2640 ×  12   =  31680  = 16 days
                                                                                         1980      1980

Question 15:

Weight of 65 books = 13 kg
(i) Weight of 1 book =  13  kg
                                   65
    Weight of 80 books = 80 ×  13   =  1040   = 16 kg
                                              65         65

(ii) Number of books weighing 13 kg = 65
    Number of books weighing 1 kg =  65  = 5 
                                                        13
   Number of books weighing 6.4 kg = 6.4 × 5 = 32

Question 16:

Number of boxes containing 6000 pens = 48
Number of boxes containing 1 pen =   48  
                                                       6000
Number of boxes containing 1875 pens = 1875 ×   48    =   90000  = 15
                                                                          6000        6000
15 boxes are needed for 1875 pens.

Question 17:

Number of days taken by 24 workers to build a wall = 15 days
Number of days taken by 1 worker to build the wall = 15 × 24 = 360 days         (less worker means more days)
Number of days taken by 9 workers to build the wall =  360  = 40 days
                                                                                     9

Question 18:

Number of men required to complete the work in 26 days = 40
Number of men required to complete the work in 1 day = 40 × 26 = 1040 men  (less men more days)
Number of men required to complete the work in 16 days =  1040  = 65
                                                                                                   16

Question 19:

Number of days the provisions will last for 550 men = 28 days
Number of days the provisions will last for 1 man = 28 × 550 = 15400 days  (less men means more days)
Number of days the provisions will last for 700 men =  15400  = 22 days
                                                                                    700
The provision will last for 22 days. 

Question 1:

Number of days for which the given quantity of rice is sufficient for 60 persons = 3 days
Number of days for which it is sufficient for 1 person = 3 × 60 = 180 days      (less men means more days )
Number of days for which it is sufficient for 18 persons =  180  = 10 days
                                                                                       18

Question 2:

(d) 4 : 5
92:115 =   92 ​÷ 23   =  4  (As H.C.F. of 92 and 115 is 23.)
                115 ​÷ 23       5

Question 3:

(a) 95
57:x::51:85
    57  =   51 
     x        85
⇒ x =  57 × 85  
               51
⇒ x =  4845  = 95
            51

Question 4:

(a) 63
25:35::45:x
          25  =  45 
          35       x
⇒ x =  35 × 45  =  1575  = 63
              25             25

Question 5:

(c) 28
4:5::x:35
⇒  4  =  x  
     5      35
⇒ x =  4 × 35  = 4 × 7 = 28
              5

Question 6:

(b) ad = bc
Given:
a, b, c, d are in proportion.
a:b::c:d
     a  =  c  
     b      d
⇒ ad = bc

Question 7:

(b) b² = ac
Given:
a, b, c are in proportion.
a:b::b:c
    Product of means = Product of extremes
⇒​ b^​2 = ac

Question 8:

(b) (5 : 8) < (3 : 4)

We can write
$(5:8) = \frac{5}{8} \mathrm{and} (3:4) = \frac{3}{4}$
Making the denominator equal:
 5   and   3 × 2  =   6       
 8           4 × 2        8
As 6 > 5,   5   <   3    
                 8        4

Question 9:

(a) Rs 440
A:B = 8:11
Sum of ratio terms = 8 + 11 = 19
B's share =  11  × 760 =  8360  = Rs 440
                   19                  19

Question 10:

(d) 147
Ratio = 5:7
Let x be any number such that we have:
   5x + 7x = 252
⇒ 12x = 252
⇒ x =  252  = 21
           12
Now, 5x = 5 × 21= 105
7x = 7 × 21 = 147

The largest number is 147.

Question 11:

(b) 50 cm
The sides of the triangle are in the ratio 1:3:5.
Let x be any number such that the sides are 1x cm, 3x cm and 5x cm.
          1x + 3x + 5x = 90
       ⇒ 9x = 90
       â€‹⇒ x =  90  = 10
                   9
First side = 1x = 1 â€‹× 10 = 10 cm
Second side = 3x = 3 â€‹× 10 = 30 cm
Third side = 5x = 5 × 10 = 50 cm
The length of the largest side is 50 cm.

Question 12:

(c) 2856
Ratio of boys and girls = 12:5
Let x be any number such that the number of boys and girls are 12x and 5x, respectively.
Number of girls = 840
      5x = 840
 ⇒ x =  840  = 168
             5
Number of boys = 12x = 12 × 168 = 2016
Number of girls = 840
Total strength of the school = 2016 + 840 = 2856

Question 13:

(b) Rs 161
Cost of 12 pens = Rs 138
Cost of 1 pen = Rs  138  
                              12
Cost of 14 pens = Rs  138  × 14 = Rs 1932  = Rs 161
                                     12                     12

Question 14:

(b) 45 days
Time taken by 24 workers to build a wall = 15 days
Time taken by 1 worker to build a wall = 24 × 15 = 360 days        (clearly less workers will take more time to build a wall)
Time taken by 8 workers to build a wall =  360  = 45 days
                                                                       8

Question 15:

(a) 52
Number of men required to finish the work in 26 days = 40
Number of men required to finish it in 1 day = 40 × 26 = 1040 men          (More men means less days)
Number of men required to finish it in 20 days =  1040  = 52
                                                                                  20

Question 16:

(b) 185 km
Distance covered in 6 L of petrol = 111 km
Distance covered in 1 L of  petrol =  111 km
                                                       6
Distance covered in 10 L of petrol =  111  × 10 =  1110  = 185 km
                                                              6                    6

Question 17:

(a) 22 days
Number of days for which 550 men had provisions = 28 days
Number of days for which 1 man had provisions = 28 × 550 = 15400 days (more men means less days)
Number of days for which 700 men had provisions =  15400  = 22 days
                                                                                        700

Question 18:

(c) 90°
Ratio of the angles of a triangle is 3:1: 2
Let x be any number such that the three angles are (3x)$°$, (1x)$°$ and (2x)$°$.
We know, the sum of the angles of a triangle is 180$°$.
        3x + 1x + 2x = 180
      ⇒ 6x = 180
​      ⇒ x =  180  = 30
                  6
$\therefore$  (3x )$°$ = (3 â€‹× 30)$°$ = 90^o
​     (1x)$°$ = (1​ × 30)$°$ = 30^o
​     (2x)$°$ = (2 × 30)$°$ = 60^o
The measure of the largest angle is 90^o​.

Question 19:

(b) 45 m
Length:Breadth = 5:4
Let x be any number such that the length and the breadth are 5x and 4x, respectively.
Now , 4x = 36
            x =  36  = 9
                    4
Length = 5x = 5 × 9 = 45 m

Question 20:

(a) 13 : 15

Speed =  Distance  
                Time 
Speed of the bus =  195 km = 65 km/hr
                                  3 hr
Speed of the train =  300 km  = 75 km/hr
                                    4 hr
Ratio =  65  =  65 ÷ 5  =  13  = 13:15
             75       75 ÷ 5      15

Question 21:

(c) Rs 198
Cost of 5 bars of soap = Rs 82.50
Cost of 1 bar of soap =  82.50  = Rs 16.5
                                         5
Cost of 12 (1 dozen) bars of soap = 16.5 × 12 = Rs 198

Question 22:

(b) Rs 750
Cost of 30 packets of 8 pencils each = Rs 600
Cost of 1 packet of 8 pencils =  600   = Rs 20
                                                 30
Cost of  1 pencil = Rs  20   
                                      8
Cost of 1 packet of 12 pencils = 12​ ×  20  =  240  = Rs 30
                                                              8         8
Cost of 25 packets of 12 pencils each = 25 × 30 = Rs 750

Question 23:

(a) Rs 344
Cost of rail journey of 75 km = Rs 215
Cost of rail journey of 1 km = Rs  215 
                                                   75
Cost of rail journey of 120 km = 120​ ×  215   = 25800 = Rs 344
                                                              75          75

Question 24:

(d) 8
Let the third term be x.
Then, we have:
12:21::x:14
We know:
    Product of means = Product of extremes
      21x = 12 × 14
  ⇒ 21x = 168
  ⇒ x =  168  = 8
              21
The third term is 8

Question 1:

(b) 15 h
Time taken by 10 boys to dig a pitch = 12 hours
Time taken by 1 boy to dig a pitch = 12 × 10 = 120 hours        (less boys means more time)
Time taken by 8 boys to dig a pitch =  120 = 15 hours
                                                                8

Question 2:

(a) 90 cm:1.05 m (or) 90 cm:105 cm             (1 m = 100 cm)
        90   =  90 ​÷ 15  =   6         (H.C.F. of 90 and 105 is 15.) 
       105      105 â€‹÷ 15      7
     $\therefore$ 6:7

(b) 35 minutes to an hour (or) 35 minutes:60 minutes      (1 hour = 60 minutes)
        35  =  35 ​÷ 5  =  7       (H.C.F. of 35 and 60 is 5.)
        60       60 â€‹÷ 5     12  
   $\therefore$  7:12

(c) 150 mL to 2 L (or) 150 L:2000 L                (1 L= 1000 mL)
         150  =   150 ​÷ 50  =   3    (HCF of 150 and 2000 is 50)
        2000      2000​ â€‹÷50       40
    $\therefore$ 3:40

(d) 2 dozens to a score (or) 24:20       (1 dozen = 12 and 1 score = 20)
        24  =   24 ​÷ 4  =  6     (H.C.F. of 24 and 20 is 4)
        20        20​ â€‹÷ 4      5
   $\therefore$ 6:5

Question 3:

Ratio of zinc and copper in an alloy is = 7:9
Let the weight of zinc and copper in it be (7x) and (9x), respectively.
Now, the weight of a copper = 12.6 kg   (given)
∴ 9x = 12.6
⇒  x = 12.6  = 1.4
             9
∴ Weight of zinc = 7x = 7​ × 1.4 = 9.8 kg

Question 4:

Given:
  A:B:C = 2:3:5
Sum of ratio = 2 + 3 + 5 = 10
Total money = Rs 1400
Then, share of A =   2   × Rs 1400 = Rs  2800  = Rs 280
                             10                             10
Share of B =  3  â€‹× Rs 1400 = Rs  4200  = Rs 420
                    10                             10
Share of C =  5  â€‹× Rs 1400 = Rs  7000  = Rs 700
                    10                            10

Question 5:

We can write:
$(5:6) = \frac{5}{6} \mathrm{and} (3:4) = \frac{3}{4}$
By making their denominators same: (Taking the L.C.M. of 6 and 4, which is 24.)
Consider, 5:6
         5 â€‹× 4  =  20 
         6 â€‹× 4      24
              
And,  3 â€‹× 6  =  18 
         4 â€‹× 6       24
As 20 > 18
Clearly, (5:6) > (3:4)

Question 6:

Number of men needed to finish a piece of work in 26 days = 40
Number of men needed to finish it in 1 day = 26 × 40 = 1040    (less days means more men)
Number of men needed to finish it in 16 days =  1040  = 65
                                                                          16 

Question 7:

Number of days for which provisions last for 425 men = 30 days
Number of days for which provisions last for 1 men = 30 × 425 = 12750 days. (less men means more days)
Number of days for which provisions last for 375 men = 12750  = 34 days
                                                                                   375
Hence, provisions will last for 34 days for 375 men.

Question 8:

Given:
36:x::x:16
We know:
Product of means = Product of extremes 
  x × x = 36 × 16
⇒ x² = 576
⇒ x² = 24²
⇒ x = 24

Question 9:

Consider 48:60::60:75

Product of means = 60 × 60 = 3600
Product of extremes = 48 × 75 = 3600
So product of means = Product of extremes
Hence, 48, 60, 75 are in continued proportion.

Question 10:

(c) 60
Ratio = 3:5
Let x be any number such that we have:
    3x + 5x = 96
 ⇒ 8x = 96
 ⇒ x =  96  = 12
             8
The numbers are:
     3x = 3 â€‹× 12 = 36
     5x = 5 â€‹× 12 = 60
The largest number = 60

Question 11:

(b) 4 : 5
Speed of the car =  Distance  =  288 km  = 72 km/hr
                                 Time            4 hr
 
Speed of the train =  Distance  =  540 km  = 90 km/hr
                                   Time             6 hr
 
Ratio of their speeds = 72:90
where, 72  = 72 ​÷ 18  =  4      (H.C.F. of 72 and 90 is 18.)
           90      90 ​÷ 18      5

Question 12:

(c) 14
Let the 4th term be x, such that we have:
     12:21::8:x
 Now, we know:
     Product of extremes = Product of means
                                    12x = 21 × 8 
                                       x =  168  = 14
                                               12

Question 13:

(d) 4 : 5
92:115
 92  =  92 ​÷ 23   =  4         (H.C.F. of 92 and 115 is 23)
115     115 â€‹÷ 23      5

Question 14:

(a) 95
Given :  
57:x::51:85
We know:
Product of means = Product of extremes
                                      51x = 57 × 85
                                          x =   4845  = 95
                                                     51

Question 15:

(c) 36
Given:
4:5::x:45
We know:
Product of mean = Product of extremes
                                      5x = 4 â€‹× 45
                                       x =    180   = 36
                                                  5

Question 16:

(b) b² = ac
Given:
a, b, c are in proportion, such that we have:
  a:b::b:c
Now, we know:
  Product of means = Product of extremes
                         b â€‹× b = a â€‹× c
                              b² = ac

Question 17:

(b) 15 hrs
Time taken by 10 boys to dig a pitch = 12 hours
Time taken by 1 boy to dig a pitch = 12 × 10 = 120 hours   (Less boys would take more hours.)
Time taken by 8 boys to dig a pitch =  120  = 15 hours
                                                               8

Question 18:

 (b) 185 km
Distance covered by a car in 8 litres of petrol = 148 km
Distance covered by it in 1 litre of petrol =  148  km
                                                                    8 
Distance covered by it in 10 litres of petrol = 10 × 148 = 1480 = 185 km
                                                                              8         8

Question 19:

(i)
           $$\begin{gathered} \mathrm{Let} \frac{14}{21} = \frac{x}{3} \\ \mathrm{Thus}, \mathrm{we} \mathrm{have}: \\ 21x = 14 \times 3 \Rightarrow x = \frac{14 \times 3}{21} = 2 \\ \therefore \frac{14}{21} = \frac{2}{3} \\ \mathrm{Again}, \mathrm{let} \frac{2}{3}=\frac{6}{y} \\ \mathrm{Thus}, \mathrm{we} \mathrm{have}: \\ 2y = 6 \times 3 \Rightarrow y = \frac{6 \times 3}{2} = 9 \\ \therefore \frac{2}{3} = \frac{6}{9} \\ \therefore \frac{14}{21} = \frac{2}{3} = \frac{6}{9} \end{gathered}$$

(ii) 90 cm:1.5 m (or) 90 cm:150 cm          (1 m = 100 cm)
      90  =  9  =  9 ÷ 3  =  3              (H.C.F. of 9 and 15 is 3.)
     150     15    15 â€‹â€‹÷ 3      5
  
(iii) If 36:81::x:63
      Product of means = Product of extremes
                          81x = 36 × 63
                             x =  2268 
                                      81
                             x = 28

(iv) Given:
      25, 35, x are in proportion.
       25:35::35:x
     Now, we know:
     Product of extremes = Product of means
                          25 × x = 35 â€‹× 35
                            25x = 1225
                              x =  1225  = 49
                                       25

(v) Given:
     9, x, x, 49 are in proportion.
         9:x::x:49
     Now, we know:
     Product of means = Product of extremes
                             x â€‹× x = 9 â€‹× 49
                                   x² = 441
                                   x² = 21²
                                    x = 21