Ratio & Proportion

The formula for ratio is expressed as a : b ⇒ a/b, where,

• a = the first term or antecedent.
• b = the second term or consequent.

For example, ratio 2 : 7 is also represented as 2/7, where 2 is the antecedent and 7 is the consequent.

Now, in order to express a proportion for the two ratios, a : b and c : d, we write it as $a:b::c:d⟶\frac{a}{b}=\frac{c}{d}$

• The two terms b and c are called mean terms.
• The two terms a and d are known as extreme terms.
• In a: b = c : d, the quantities a and b should be of the same kind with the same units, whereas, c and d may be separately of the same kind and of the same units. For example, 5 kg : 15 kg = Rs. 75 : Rs. 225
• The proportion formula can be expressed as, a/b = c/d or a : b : : c : d.
• In a proportion, the product of the means = the product of the extremes. Therefore, in the proportion formula a: b : : c : d, we get b × c = a × d. For example, in 5 : 15 :: 75 : 225, we will get 15 × 75 = 5 × 225

The difference between ratio and proportion can be seen in the following table.

Important Notes on Ratio and Proportion

• Any two quantities with the same units can be compared.
• Two ratios are said to be in proportion only if they are equal.
• To check whether two ratios are equal and are in proportion, we can also use the cross-product method.
• If we multiply and divide each term of a ratio by the same number, the ratio remains the same.
• For any three quantities, if the ratio between the first and the second is equal to the ratio between the second and the third, then these are said to be in a continued proportion.
• Similarly, in the case of any four quantities in a continued proportion, the ratio between the first and the second is equal to the ratio between the third and the fourth.

Important Question

Ratio And Proportion Examples

Example 1: 

Are the ratios 4:5 and 8:10 said to be in Proportion?

Solution:

4:5= 4/5 = 0.8 and 8: 10= 8/10= 0.8

Since both the ratios are equal, they are said to be in proportion.

Example 2:

Are the two ratios 8:10 and 7:10 in proportion?

Solution:

8:10= 8/10= 0.8 and 7:10= 7/10= 0.7

Since both the ratios are not equal, they are not in proportion.

Example 3: 

Given ratio are-

a:b = 2:3

b:c = 5:2

c:d = 1:4

Find a: b: c.

Solution:

Multiplying the first ratio by 5, second by 3 and third by 6, we have

a:b = 10: 15

b:c = 15 : 6

c:d = 6 : 24

In the ratio’s above, all the mean terms are equal, thus

a:b:c:d = 10:15:6:24

Example 4:

Check whether the following statements are true or false.

a] 12 : 18 = 28 : 56

b] 25 people : 130 people = 15kg : 78kg

Solution: 

a] 12 : 18 = 28 : 56

The given statement is false.

12 : 18 = 12 / 18 = 2 / 3 = 2 : 3

28 : 56 = 28 / 56 = 1 / 2 = 1 : 2

They are unequal.

b] 25 persons : 130 persons = 15kg : 78kg

The given statement is true.

25 people : 130 people = 5: 26

15kg : 78kg = 5: 26

They are equal.

Example 5:

The earnings of Rohan is 12000 rupees every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 rupees. Find the ratio of the savings.

Solution: 

Savings of Rohan per month = Rs (12000-9960) = Rs. 2040

Yearly income of Anish = Rs. 191520

Hence, the monthly income of Anish = Rs. 191520/12 = Rs. 15960.

So, the savings of Anish per month = Rs (15960 – 9960) = Rs. 6000

Thus, the ratio of savings of Rohan and Anish is Rs. 2040: Rs.6000 = 17: 50.

Example 6:

Twenty tons of iron is Rs. 600000 (six lakhs). What is the cost of 560 kilograms of iron?

Solution: 

1 ton = 1000 kg
20 tons = 20000 kg
The cost of 20000 kg iron = Rs. 600000
The cost of 1 kg iron = Rs{600000}/ {20000}
= Rs. 30
The cost of 560 kg iron = Rs 30 × 560 = Rs 16800

Example 7:

The dimensions of the rectangular field are given. The length and breadth of the rectangular field are 50 meters and 15 meters. What is the ratio of the length and breadth of the field?

Solution: 

Length of the rectangular field = 50 m

Breadth of the rectangular field = 15 m

Hence, the ratio of length to breadth = 50: 15

⇒ 50: 15 = 10: 3.

Thus, the ratio of length and breadth of the rectangular field is 10:3.

Example 8:

Obtain a ratio of 90 centimetres to 1.5 meters.

Solution: 

The given two quantities are not in the same units.

Convert them into the same units.

1.5 m = 1.5 × 100 = 150 cm

Hence, the required ratio is 90 cm: 150 cm

⇒ 90: 150 = 3: 5

Therefore, the ratio of 90 centimetres to 1.5 meters is 3: 5.

Example 9:

There exists 45 people in an office. Out of which female employees are 25 and the remaining are male employees. Find the ratio of

a] The count of females to males.

b] The count of males to females.

Solution: 

Count of females = 25

Total count of employees = 45

Count of males = 45 – 25 = 20

The ratio of the count of females to the count of males

= 25 : 20

= 5 : 4

The count of males to the count of females

= 20 : 25

= 4 : 5

Example 10:

Write two equivalent ratios of 6: 4.

Answer: 

Given Ratio : 6: 4, which is equal to 6/4.

Multiplying or dividing the same numbers on both numerator and denominator, we will get the equivalent ratio.

⇒(6×2)/(4×2) = 12/8 = 12: 8

⇒(6÷2)/(4÷2) = 3/2 = 3: 2

Therefore, the two equivalent ratios of 6: 4 are 3: 2 and 12: 8

Example 11:

Out of the total students in a class, if the number of boys is 5 and the number of girls is 3, then find the ratio between girls and boys.

Solution: 

The ratio between girls and boys can be written as 3:5 (Girls: Boys). The ratio can also be written in the form of factor like 3/5.

Example 12: 

Two numbers are in the ratio 2 : 3. If the sum of numbers is 60, find the numbers.

Solution:

Given, 2/3 is the ratio of any two numbers.

Let the two numbers be 2x and 3x.

As per the given question, the sum of these two numbers = 60

So, 2x + 3x = 60

5x = 60

x = 12

Hence, the two numbers are;

2x = 2 x 12 = 24

and

3x = 3 x 12 = 36

24 and 36 are the required numbers.

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