Ratio

What is Ratio?

Ratio - It is a way to compare one quantity to another quantity of the same kind .

It is denoted by : sign, e.g. a:b

Ratio can also be written in fractional form, e.g. a:b = a/b

For example:
The ratio of Rs. 15 and Rs. 25 is 15/25 = 3/5, i.e. 3 : 5
The ratio of 10 minutes and 1 hour = 10/60 = 1/6, i.e. 1 : 6

Basic Concepts of Ratio

Concept 1: Relative values

Ratios just give an idea of the relative values and not the actual magnitude.

For example:
If the ratio of boys and girls in a class is 3 : 5, then it just means that:
Number of boys / Number of girls = 3/5

It does not mean that there are 3 boys and 5 girls in the class. We cannot tell for sure the number of boys and girls in the class.

Have a look at all the following cases, which have the ratio of boys to girls as 3 : 5.

Concept 2: Remove common factors

Ratio of any number of quantities is expressed after removing any common factors that all the terms of the ratio have.

For example, if there are two quantities 6 and 9:
6 : 9 = 2 : 3
(after taking out the common factor 3 between them)

If there are three quantities 6, 14 and 18:
6 : 14 : 18 = 3 : 7 : 9
(after taking out the common factor 2 between them)

Concept 3

If A : B : C = 1/3 : 1/5 : 1/7
Then, A : B : C = 35 : 21 : 15
(we just multiplied by the LCM 105)

But if A/3 = B/5 = C/7
Then, A : B : C = 3 : 5 : 7

Concept 4: Finding original quantities from a ratio

Let there be two quantities (whose absolute values are A and B respectively) which are in the ratio a : b.

We know that some common factor k(>0) would have been removed from A and B to get the ratio a : b. So, we can write the original values of the two quantities as:
A = ak and
B = bk

For example, if the marks of two persons are in the ratio 5 : 3, we can write their individual marks as 5k and 3k respectively.

Let’s consider another example:
If it is given that the ratio of boys and girls in a class is 3 : 5
The only information that we can gather is that if there are 3k boys, then there will be 5k girls.
If any further data is given such that we get any equation involving k and hence can find the value of k, then we can ascertain the exact number of boys and girls in the class.
E.g. If it is given that the twice the number of boys minus the number of girls is 10. we can form the equation 6k - 5k = 10, which gives k = 10. So, there are 30 boys and 50 girls in the class.

Q. In a parking lot, the cars and bikes are in the ratio 7:11. There are 56 more bikes parked than the number of cars. What is the number of cars parked in the parking lot?

(a) 91   (b) 98    (c) 147   (d) 154

Explanations :
1 2

Q. In an animal farm, the ratio of cows to pigs was 1:3. After 50 more cows were brought to the farm, the ratio of cows to pigs became 1:1. What is the total number of cows in the farm now?

(a) 75   (b) 100    (c) 41250   (d) Cannot be determined

Explanations :
1 2

Concept 5

If two quantities are in the ratio a : b, and the sum of those two quantities = S, then

First quantity = $\frac{a}{(a + b)}$ × S

Second quantity = $\frac{b}{(a + b)}$ × S

Q. Sum of two numbers is 42. If the two numbers are in the ratio 4:3, then find the smaller number.

Explanations :
1 2

Q. A class has boys and girls in the ratio 7:3. Two girls were absent on one particular day and the total strength of the class on that day was 198. What is the percentage of girls in the class?

(a) 70%   (b) 28%    (c) 30%   (d) 32%

Explanation:

Types of Ratios

Ratio of greater inequality

The ratio a : b where a > b (example 5 : 4)

Ratio of lesser inequality

The ratio a : b where a < b (example 4 : 5)

Ratio of equality

The ratio a : b where a = b (example 1 : 1)

Some other kinds of ratios

• Duplicate ratio of a and b = $a^2$ : $b^2$

• Triplicate ratio of a and b = $a^3$ : $b^3$

• Sub duplicate ratio of a and b = √a : √b

• Sub triplicate ratio of a and b = $(𝑎)^{1/3}$ : $(b)^{1/3}$

• Compound ratio of a : b, c : d, e : f = $\frac{ace}{bdf}$

Properties of Ratios

Property 1: Multiplication/Division

A ratio remains the same if both antecedent and consequent are multiplied or divided by the same non-zero number.

$\frac{a}{b}$ = $\frac{pa}{pb}$ = $\frac{qa}{qb}$ (where p, q ≠ 0)

$\frac{a}{b}$ = $\frac{\frac{a}{p}}{\frac{b}{p}}$ = $\frac{\frac{a}{q}}{\frac{b}{q}}$ (where p, q ≠ 0)

For example: 5/1 = 10/2 = 15/3 = 50/10 etc.

Property 2: Addition

When we add the same quantity x (positive) to both the terms of the ratio, we have the following results:

If a > b, then (a + x) : (b + x) < a : b
E.g. if a/b = 11/10 = 1.1. Then on adding 90 to both, we get 101/100 = 1.01
(the value keeps getting closer to 1, as we keep on add positive values to both)

If a < b, then (a + x) : (b + x) > a : b
E.g. if a/b = 9/10 = 0.90. Then on adding 90 to both, we get 99/100 = 0.99
(the value keeps getting closer to 1, as we keep on add positive values to both)

If a = b, then (a + x) : (b + x) = a : b = 1 : 1

Q. Two numbers are in the ratio 4:5. If 7 is added to each, the ratio of the two numbers becomes 5:6. Find the larger number.

Explanations :
1 2

Q. Ratio of the ages of grandfather, father and the son in a family is 6:3:1. Which of the following can be the ratio among their ages 5 years from now?

(a) 13: 7: 3
(b) 15: 5: 2
(c) 9: 4: 2
(d) 12: 7: 2

Explanation: