Ratio and proportion is the heart of arithmetic. If you understand this chapter properly you can solve virtually any problem in arithmetic.

If two numbers are in the ratio 2:3 means for every two units of the first number, second has 3 units. This is a mere comparison between numbers, and actual numbers may be way bigger than these numbers. If you multiply or divide a ratio the comparison does not change. i.e., 2:3 is same as 4:6.

If two numbers are in the ratio a:b then this ration has to be multiplied with a number K, to get actual numbers. This K is called multiplication factor (MF)

If two ratios are equal then we say they are in proportion. then $a:b::c:d \Rightarrow a \times d = b \times c$

or $\displaystyle\frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c$.

Chain rule is comes in handy when there are many variables need to compare with the given variable. We can understand this rule by observing a practice problem.

If 12 carpenters working 6 hours a day can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8 hours a day?

Let us prepare small table to understand the problem.

Now with repect to the Chairs we need to understand how each variable is related.

If the number of men got increased (i.e., 12 to 18), do they manufacture more chairs or less chairs is the question we have to ask ourselves. If the answer is "more" then the higher number between 12, 18 will go to the numerator and other will go to denominator and vice versa. Here answer is "more"

So $460 \times \displaystyle\frac{{18}}{{12}}$

Next we go to Hours. If the number of hours they work each day got increases then do they manufactures more chairs or less chairs? Answer is more

So $460 \times \displaystyle\frac{{18}}{{12}} \times \frac{8}{6}$

Last, If the number of day they work increases then .... answer is more.

So $460 \times \displaystyle\frac{{18}}{{12}} \times \frac{8}{6} \times \frac{{36}}{{24}} = 1380$

a. 10

b. 15

c. 20

d. 25

a. 1 : 2

b. 2 : 3

c. 3 : 4

d. 3 : 8

a. 5 : 6

b. 7 : 8

c. 8 : 7

d. 14:13

a. 42 litres

b. 56 litres

c. 60 litres

d. 77 litres

a. Rs.1000

b. Rs.1200

c. Rs.1500

d. Rs.2000

a. 17

b. $16\displaystyle\frac{2}{3}$

c. $18\displaystyle\frac{1}{2}$

d. 15

a. 120

b. 90

c. 40

d. 80

a. $\displaystyle\frac{4}{{12}}$

b. $\displaystyle\frac{5}{8}$

c. $\displaystyle\frac{8}{{12}}$

d. Data inadequate

If two numbers are in the ratio 2:3 means for every two units of the first number, second has 3 units. This is a mere comparison between numbers, and actual numbers may be way bigger than these numbers. If you multiply or divide a ratio the comparison does not change. i.e., 2:3 is same as 4:6.

If two numbers are in the ratio a:b then this ration has to be multiplied with a number K, to get actual numbers. This K is called multiplication factor (MF)

If two ratios are equal then we say they are in proportion. then $a:b::c:d \Rightarrow a \times d = b \times c$

or $\displaystyle\frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c$.

**Chain Rule:**

Chain rule is comes in handy when there are many variables need to compare with the given variable. We can understand this rule by observing a practice problem.**Solved Example:**If 12 carpenters working 6 hours a day can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8 hours a day?

Let us prepare small table to understand the problem.

Now with repect to the Chairs we need to understand how each variable is related.

If the number of men got increased (i.e., 12 to 18), do they manufacture more chairs or less chairs is the question we have to ask ourselves. If the answer is "more" then the higher number between 12, 18 will go to the numerator and other will go to denominator and vice versa. Here answer is "more"

So $460 \times \displaystyle\frac{{18}}{{12}}$

Next we go to Hours. If the number of hours they work each day got increases then do they manufactures more chairs or less chairs? Answer is more

So $460 \times \displaystyle\frac{{18}}{{12}} \times \frac{8}{6}$

Last, If the number of day they work increases then .... answer is more.

So $460 \times \displaystyle\frac{{18}}{{12}} \times \frac{8}{6} \times \frac{{36}}{{24}} = 1380$

**Exercise**

**1.**The ratio between two numbers is 5 : 8. If 8 is subtracted from both the numbers, the ratio becomes 1 : 2. The smaller number is:a. 10

b. 15

c. 20

d. 25

**2.**The ratio between Sumit's and Prakash's age at present is 2:3. Sumit is 6 years younger than Prakash. The ratio of Sumit's age to Prakash's age after 6 years will be :a. 1 : 2

b. 2 : 3

c. 3 : 4

d. 3 : 8

**3.**The ratio between the ages of Kamala and Savitri is 6:5 and the sum of their ages is 44 years. The ratio of their ages after 8 years will be :a. 5 : 6

b. 7 : 8

c. 8 : 7

d. 14:13

**4.**In a mixture of 60 litres, the ratio of milk and water is 2:1. What amount of water must be added to make the ratio 1:2?a. 42 litres

b. 56 litres

c. 60 litres

d. 77 litres

**5.**A's money is to B's money as 4:5 and B's money is to C's money as 2:3. If A has Rs.800, C hasa. Rs.1000

b. Rs.1200

c. Rs.1500

d. Rs.2000

**6.**15 litres of a mixture contains 20% alcohol and the rest water. If 3 litres of water is mixed in it, the percentage of alcohol in the new mixture will be :a. 17

b. $16\displaystyle\frac{2}{3}$

c. $18\displaystyle\frac{1}{2}$

d. 15

**7.**Vinay got thrice as many marks in Maths as in English. The proportion of his marks in Maths and History is 4:3. If his total marks in Maths, English and History are 250, what are his marks in English ?a. 120

b. 90

c. 40

d. 80

**8.**One-fourth of the boys and three-eighth of the girls in a school participated in the sports. What fractional part of the total student population of the school participated in the annual sports ?a. $\displaystyle\frac{4}{{12}}$

b. $\displaystyle\frac{5}{8}$

c. $\displaystyle\frac{8}{{12}}$

d. Data inadequate