Question 1: Express the following ratios in simplest form:

i) 18:30      ii) 7.5 \colon 9      iii) 6 \frac{2}{3} \colon 7 \frac{1}{2}      iv) \frac{1}{6}  \colon \frac{ 1}{9} \colon \frac{ 1}{12}      v) 7 \colon 5 \colon \frac{ 9}{2}       vi) 3 \frac{1}{5}   \colon 5 \frac{1}{3} \colon 6 \frac{2}{3}  

Answer:

i) 18:30 = \frac{18}{30} = \frac{3}{5}

ii) 7.5 \colon 9 = \frac{7.5}{ 9} = \frac{75}{ 90} = \frac{5}{6}

iii) 6 \frac{2}{3} \colon 7 \frac{1}{2} = \frac{20}{3} \times \frac{2}{15} = \frac{8}{9}

iv) \frac{1}{6}  \colon \frac{ 1}{9} \colon \frac{ 1}{12} = \frac{1}{6} \times 36 \colon \frac{ 1}{9} \times 36 \colon \frac{ 1}{12} \times 36=6 \colon 4 \colon 3

v) 7 \colon 5 \colon \frac{ 9}{2} = 7 \times 2 \colon 5 \times 2 \colon \frac{ 9}{2} \times 2=14 \colon 10 \colon 9

vi) 3 \frac{1}{5}   \colon 5 \frac{1}{3} \colon 6 \frac{2}{3} = \frac{16}{5}   \colon \frac{16}{3}   \colon \frac{20}{3} = \frac{16}{5} \times 15 \colon \frac{16}{3} \times 15 \colon \frac{20}{3} \times 15 =48 \colon 80 \colon 100=12 \colon 20 \colon 25

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Question 2: Express the following ratios in simplest form:

i) \ 75 \ paisa : 4 \ Rupees       ii) 1 \ m \ 8 \ cm : 72 \ cm   

iii) 1 \ hour 15 \ minutes : 45 \ minutes      iv) 2 \ kg \ 750 \ g : 3 \ kg   

v) 1 \ year \ 9 \ months : 2 \ years \ 4 \ months 

Answer:

i) \ 75 \ paisa : 4 \ Rupees = \frac{75}{400}   = \frac{3}{16}  

ii) 1 \ m \ 8 \ cm : 72 \ cm = \frac{108}{72}   = \frac{3}{2}  

iii) 1 \ hour 15 \ minutes : 45 \ minutes = \frac{75}{45}   = \frac{5}{3}  

iv) 2 \ kg \ 750 \ g : 3 \ kg = \frac{2750}{3000}   = \frac{11}{12}  

v) 1 \ year \ 9 \ months : 2 \ years \ 4 \ months = \frac{21}{28}   = \frac{3}{4}  

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Question 3: Which ratio is greater:

i) (4 : 9) \ or \ (3:7)      ii) \Big( 2 \frac{1}{3} \colon 3 \frac{1}{3} \Big) \ or \ (3.6\colon 4.8)

iii) \Big( \frac{1}{2} \colon \frac{1}{3} \Big) \ \ or \  \ \Big( \frac{1}{6} \colon \frac{1}{4} \Big)      iv) \Big( 3 \frac{1}{3} \colon 4 \frac{1}{6} \Big) \ \ or \ \ (0.9 \colon 1)

Answer:

i) (4 : 9) \ or \ (3:7) = \frac{4}{9} \ \ or \ \ \frac{3}{7} =   \frac{28}{63} \ \ or \  \   \frac{27}{63} . Therefore (4:9) is greater than (3:7)

ii) \Big( 2 \frac{1}{3} \colon 3 \frac{1}{3} \Big) \ or \ (3.6\colon 4.8) = \Big( \frac{7}{3} \times \frac{3}{10} \Big) \ \ or \ \frac{3.6}{4.8} = \frac{7}{10} \ \ or \  \ \frac{3}{4} = \frac{14}{20} \ \ or \  \ \frac{15}{20}

Therefore (3.6\colon 4.8) is greater than \Big( 2 \frac{1}{3} \colon 3 \frac{1}{3} \Big)

iii) \Big( \frac{1}{2} \colon \frac{1}{3} \Big) \ \ or \  \ \Big( \frac{1}{6} \colon \frac{1}{4} \Big) = \frac{3}{2} \ \ or \  \ \frac{2}{3}

Therefore \Big( \frac{1}{2} \colon \frac{1}{3} \Big) is greater than \Big( \frac{1}{6} \colon \frac{1}{4} \Big)

iv) \Big( 3 \frac{1}{3} \colon 4 \frac{1}{6} \Big) \ \ or \ \ (0.9 \colon 1) = \frac{4}{5} \ \ or \ \ \frac{9}{10} = \frac{8}{10} \ \ or \  \ \frac{9}{10}

Therefore (0.9 \colon 1) is greater than \Big( 3 \frac{1}{3} \colon 4 \frac{1}{6} \Big)

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Question 4: Arrange the following ratios in ascending order of magnitude:

i) (2:3), (5:9), (11:15)     ii) (5:7), (9:14), (20:21) and (3:8)

Answer:

i) (2:3), (5:9), (11:15)

\frac{2}{3} \colon \frac{5}{9} \colon \frac{11}{15} = \frac{30}{45} \colon \frac{25}{45} \colon \frac{33}{45}    Therefore \frac{5}{9} < \frac{2}{3} < \frac{11}{15}

ii) (5:7), (9:14), (20:21) and (3:8)

\frac{5}{7} \colon \frac{9}{14} \colon \frac{20}{21} \colon \frac{3}{8} = \frac{120}{168} \colon \frac{108}{168} \colon \frac{160}{168} \colon \frac{63}{168}    Therefore \frac{3}{8} < \frac{9}{14} < \frac{5}{7} < \frac{20}{21}

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Question 5: Divide Rs. 142.20 between A and B in the ratio \frac{1}{4} \colon   \frac{1}{5}

Answer:

A : B = 5:4

A’s Share = 142.20 \times   \frac{5}{9} = 79 Rs

B’s share = 142.20 \times   \frac{4}{9} = 63.2 Rs

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Question 6: Divide Rs. 3726 among A, B, C in the ratio of \frac{1}{3} \colon \frac{1}{4} \colon \frac{1}{6}

Answer:

A : B: C = \frac{1}{3} \colon \frac{1}{4} \colon \frac{1}{6} = 4: 3: 2

A’s share = \frac{4}{9} \times 3726 = 1656 Rs

B’s share = \frac{3}{9} \times 3726 = 1242 Rs

C’s share = \frac{2}{9} \times 3726 = 828 Rs

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Question 7: Divide Rs. 810 among A, B, C in the ration of  \frac{1}{4} : \frac{2}{5} : 1   \frac{3}{8}  

Answer:

A : B: C = \frac{1}{4} \colon \frac{2}{5} \colon \frac{11}{8} = 10: 16: 55

A’s share = \frac{10}{81} \times 810 = 100 Rs

B’s share = \frac{16}{81} \times 810 = 160 Rs

C’s share = \frac{55}{81} \times 810 = 550 Rs

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Question 8: Divide Rs. 1050 between A and B in the ratio 2   \frac{2}{3} : 6   \frac{2}{3}

Answer:

A : B =   \frac{8}{3} : \frac{20}{3} \ or \ 8:20

A’s share =   \frac{8}{28} \times 1050 = 300 Rs

B’s share =   \frac{20}{28} \times 1050 = 750 Rs

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Question 9: Divide Rs. 747 among A, B, C such that 4A = 5B = 7C

Answer:

A : B : C =   \frac{a}{4} :   \frac{a}{5} :   \frac{a}{7} = 35 : 28 : 20

A’s share =   \frac{35}{83} \times 747 = 315 Rs

B’s share =   \frac{28}{83} \times 747 = 252 Rs

C’s share =   \frac{20}{83} \times 747 = 180 Rs

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Question 10: A big contains one rupee, 50  p and 25  p coins in the ration of 5 : 6 : 8  amounting to Rs 210 . Find the number of coins in each type.

Answer:

Coins: x  One Rupee : 50  paisa : 25  paisa = 5x : 6x : 8x

Amount: One Rupee : 50  paisa : 25  paisa = 500x : 300x : 200x

Therefore 500x+300x+200x=21000 \ or \ x=21

Therefore the coins are in ratio of 5x:6x:8x  or 105:126:168

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Question 11: Find A : B : C when:

i)    A : B = 2 : 5 and B : C = 7 : 9

ii)   A : B = 3 : 4 and B : C = 6 : 11

iii)  A : B =    \frac{1}{2} :   \frac{1}{3} \ and \ B : C =    \frac{1}{4} :   \frac{1}{5}

Answer:

i)    A : B = 2 : 5 and B : C = 7 : 9

A : B = 2 : 5 and B : C = 7 :9

LCM of 5 and 7 = 35 . Therefore we can say

A: B = 14 : 35

B: C = 35 : 45

Therefore A : B : C = 14 : 35 : 45

ii)   A : B = 3 : 4 and B : C = 6 : 11

A : B = 3 : 4 and B : C = 6 : 11

LCM of 4 and 6 = 12 . Therefore we can say

A : B = 9 : 12

B : C = 12 : 22

Therefore A : B : C = 9 : 12 : 22

iii)  A : B =    \frac{1}{2} :   \frac{1}{3} \ and \ B : C =    \frac{1}{4} :   \frac{1}{5}

A : B = 3 : 2 and B : C = 5 : 4

LCM of 2 and 5 = 10 . Therefore we can say

A : B = 15 : 10

B : C = 10 : 08

Therefore A : B : C = 15 : 10 : 08

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Question 12: If A : B = 4 : 9 and A : C = 2 : 3 , find B : C and A : B : C .

Answer:

\frac{A}{B} = \frac{4}{9}            \frac{A}{C} = \frac{2}{3}

Therefore, \frac{B}{C} = \frac{A}{C} \times \frac{B}{A} = \frac{2}{3} \times \frac{9}{4} = \frac{3}{2}   \ or \ B : C = 3 : 2

Now,   A : B = 4 : 9 and B : C = 3 : 2

LCM of 9 and 3 = 9 ,       Therefore we can say

A : B = 4 : 9           B : C = 9 : 6

Therefore A : B : C = 4 : 9 : 6

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Question 13: If A : C = 5 : 8 and B : C = 5 : 6 , find A : B and A : B : C

Answer:

\frac{A}{C} = \frac{5}{8}            \frac{B}{C} = \frac{5}{6}

Therefore, \frac{A}{B} = \frac{A}{C} \times \frac{C}{B} = \frac{5}{8} \times \frac{6}{5} = \frac{3}{4} \ or \ A:B = 3: 4

Now,   A : C = 5 : 8 and A : B = 3 : 4

LCM of 5 and 3 = 15 ,   Therefore we can say

A : C = 15 : 24

A : B = 15 : 20

Therefore A : B : C = 15 : 20 : 24

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Question 14: Two numbers are in the ration of 6 : 11 . On adding 2 to the first and 7 to the second, their rations become 8 : 15 . Find the numbers.

Answer:

Let the two numbers be a and b . Therefore

\frac{a}{b} = \frac{6}{11} or b= \frac{11}{6} a

\Rightarrow \frac{a+2}{b+7} = \frac{8}{15}

\Rightarrow a+2= \frac{8}{15} (b+7)

\Rightarrow a+2= \frac{8}{15} ( \frac{11}{6} a + 7)

\Rightarrow a+2= \frac{8}{15} \frac{11}{6} a + \frac{8}{15} \times 7

\Rightarrow a- \frac{88}{90} a= \frac{56}{15} -2

\Rightarrow a= \frac{90}{2} \times \frac{26}{15} = 78

\Rightarrow b= \frac{11}{6} \times 78 = 143

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Question 15: A sum of money is divided between A and B in the ratio 5 : 7 . If B’s share is Rs. 665 , find the total amount.

Answer:

Let the total money to be divided be x

\frac{7}{12} \times x = 665

x= \frac{12}{7} \times 665=1140 Rs.

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Question 16: Two numbers are in the ratio of 7:4 . If their difference is 72 , find the numbers.

Answer:

Let the two numbers be a and b

Given \frac{a}{b} = \frac{7}{4} or b= \frac{4}{7} a and (a - b) = 72

Substituting (a- \frac{4}{7} a)=72

a= \frac{7}{3} \times 72=168

b= \frac{4}{7} \times 168=96

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Question 17: A certain sum of money is divided between A, B and C in the ration of 5 : 6 : 7 . If A’s share is Rs. 175 , find the total amount and also the shares of each one of B and C .

Answer:

Let the total money to be divided be x

\frac{5}{18} \times x=175

x= \frac{18}{5} \times 175 = 630 Rs.

B’s share = \frac{6}{18} \times 630 = 210 Rs.

C’ s share = \frac{7}{18} \times 630 = 245 Rs.

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Question 18: The ratio of the number of boys and number of girls in a school of 1440 students is 7:5 . If 40 new boys are admitted, find how many new girls can be admitted to make the ratio 4:3 .

Answer:

Let the number of boys and girls be a and b respectively.

a+b=1440

\frac{a}{b} = \frac{7}{5} or b= \frac{5}{7} a

a+ \frac{5}{7} a=1440 hence a (boys)=840

b (girls)=1440-840=600

Let the number of girls added = x

\frac{880}{600+x} = \frac{4}{3}

Solve for x = 60 . Therefore we need to add 60 girls to make the new ratio to 4 : 3

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Question 19: The sum of three numbers is 212 . If the ratio of the first to the second is 13 : 16 and that of the second to the third is 2 : 3 , then find the numbers.

Answer:

Let the three numbers be A, B and C .

A : B = 13 : 16

B : C = 2 : 3

LCM of 16 and 2 is 16 .

Therefore B : C = 16 : 24

Therefore A : B : C = 13 : 16 : 24

Hence:

A = \frac{13}{53} \times 212=52           B = \frac{16}{53} \times 212=64            C = \frac{24}{53} \times 212=96

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Question 20: Find the number which when added to each term of the ratio 27 : 35 , changes the ratio to 4 : 5

Answer:

Let x be added to both the terms

\frac{27+x}{35+x} = \frac{4}{5}

x=5

Question 21: The present ages of Mr. X and his Son are in the ratio of 17 : 19 . If the ratio of their ages 9 years ago was 7 : 3 , then find their present ages

Answer:

Let the present age of Mr. X be a and his son be b . Therefore

\frac{a}{b} = \frac{17}{9} or b= \frac{9}{17}

\frac{(a-9)}{(b-9)} = \frac{7}{3}

3a-27=7b-63

3a-27=7 \times \frac{9}{17} a-63

Solving for a=51 and b=27

Question 22: Salaries of A, B and C are in the ratio of 2 : 3: 5 . If the increment of 15\%, 10\% and 20\% are allowed respectively in their salaries, then what will be the new ratio of their salaries.

Answer:

If A’s salary was 200 , B’s salary was 300 and C’s salary was 500 , then the ratios of their salaries would be 2 : 3 : 5 .

With increments, A’s salary becomes 230 , B’s salary becomes 330 and C’s salary becomes 600 .

Therefore the new ratio would be A : B : C = 23 : 33 : 60

Question 23: A got 58 marks out of 75 in Physics and 97 marks out of 120 in Math. In which of the two subjects did he perform better?

Answer:

\frac{Physics}{Total} = \frac{58}{75}

\frac{Math}{Total} = \frac{97}{120}

LCM of 75 and 120 = 600 Therefore

\frac{Physics}{Total} = \frac{58}{75} \times \frac{8}{8} = \frac{464}{600}

\frac{Math}{Total} = \frac{97}{120} \times \frac{5}{5} = \frac{485}{600}

Therefore A performed better in Math.

Question 24: A mixture consists of only two components A and B . In 60 liters of this mixture, the components A and B are in the ratio of 2 :1 . What quantity of component B $ has to be added in this mixture so that the new ratio is 1 : 2 ?

Answer:

Quantity of A in the mixture = 60 \times \frac{2}{3} =40 liters

Quantity of B in the mixture = 60 \times \frac{1}{3} =20 liters

Let the quantity of B added to the mixture is x

\frac{40}{(20+x)} = \frac{1}{2} or x=60 liters