Basic Concept of Ratios

We will discuss here about the basic concept of ratios.

In our day-to-day life, we compare one quantity with another quantity of the same kind by using the method of subtraction and method of division. Let us take a simple example.

The height of Jessica is 1 m 96 cm and that of Maria is 1 m 92 cm. The difference in their height is:

196 cm -192 cm = 4 cm

Thus, we say Jessica is 4 cm taller than Maria.

Similarly, suppose the weight of Jessica is 50 kg and the weight of Maria is 40 kg. We can compare their weight by division i.e.,

\(\frac{\textrm{Weight of Jessica}}{\textrm{Weight of Maria}}\) = \(\frac{\textrm{50 kg}}{\textrm{40 kg}}\) = \(\frac{5}{4}\)

So, the weight of Jessica is \(\frac{5}{4}\) times the weight of Maria.

When we compare two quantities of the same kind (with respect to magnitude) by division, we can say that we have formed a ratio, we denote ratio using the symbol (:)

Definition: The ratio of two like quantities a and b is the fraction \(\frac{a}{b}\), which indicates how many times b is the quantity a. In other words, their ratio indicates their relative sizes.

If x and y are two quantities of the same kind and with the same units such that y ≠ 0; then the quotient \(\frac{x}{y}\) is called the ratio between x and y.

Let the weights of two persons be 40 kg and 80 kg. Clearly, the weight of the second person is double the weight of the first person because 80 kg = 2 × 40 kg.

Therefore, \(\frac{\textrm{Weight of the first person}}{\textrm{Weight of the second person}}\) = \(\frac{\textrm{40 kg}}{\textrm{80 kg}}\) = \(\frac{1}{2}\)

We say, the ratio of the weight of the first person to the weight of the second person is \(\frac{1}{2}\) or 1 : 2.

The ratio of two like quantities a and b is the quotient a ÷ b, and it is written as a : b (read a is to b).

In the ratio a : b, a and b are called terms of the ratio, a is called the antecedent or first term, and b is called the consequent or second term. Then, ratio of two quantities = antecedent : consequent.

Example: The ratio of heights of two persons A and B whose heights are 6 ft and 5 ft is \(\frac{6 ft}{5 ft}\), i.e., \(\frac{6}{5}\) or 6 : 5. Here, 6 is the antecedent and 5 is the consequent.

Solved Examples on Word Problems on Basic Concept of Ratios:

1. There are 30 oranges and 18 apples in a fruit basket.

(i) What is the ratio of the number of oranges to the number of apples?

(ii) What is the ratio of the number of apples to the number of fruits in the basket?

Solution:

(i) Number of oranges= 30

Number of apples = 18

The ratio of the number of oranges to the number of apples

                    = 30 : 18 

                    = \(\frac{30}{18}\)

                    = \(\frac{30 ÷ 6}{18 ÷ 6}\); [Since HCF of 30 and 18 is 6]

                    = \(\frac{5}{3}\)

                    = 5 : 3

(ii) Number of apples = 18

Total number of fruits = 30 + 18 = 48

The ratio of the number of apples to the total number of fruits

                    = 18 : 48

                    = \(\frac{18}{48}\)

                    = \(\frac{18 ÷ 6}{48 ÷ 6}\); [Since HCF of 18 and 48 is 6]

                    = \(\frac{3}{8}\)

                    = 3 : 8

2. In a class, there are boys and 40 girls. Find the ratio of the number of boys to the number of girls.

Solution:

Ratio of the number of boys to the number of girls

                    = 20 : 40

                    = \(\frac{20}{40}\)

                    = \(\frac{20 ÷ 20}{40 ÷ 20}\); [Since HCF of 20 and 40 is 20]

                    = \(\frac{1}{2}\)

                    = 1 : 2

Hence, the ratio of the number of boys to the number of girls is 1 : 2.

3. Write the ratio of the following:

(i) 1 m to 50 cm

(ii) 500 g to 2 kg

Solution:

(i) Here, two terms are 1 m and 50 cm respectively so, we have to express both the terms in the same units of measurement.

Now, we convert metre into centimetre.

So, m = 100 cm

Therefore, the ratio of 1 m to 50 cm = \(\frac{\textrm{100 cm}}{\textrm{50 cm}}\) 

                                                    = \(\frac{2}{1}\)

                                                    = 2 : 1

(ii) Here, we convert 2 kilograms into grams.

Now, we convert kilograms into grams.

We know that 1 kg = 1000 g

So, 2 kg = 2 × 1000 g

             = 2000 g

Therefore the ratio of 500 g to 2 kg = \(\frac{\textrm{500 g}}{\textrm{2000 g}}\) 

                                                   = \(\frac{1}{4}\)

                                                   = 1 : 4

Worksheet on Basic Concept of Ratios:

1. Write the ratio for each of the following:

(i) 60 paise to 3

(ii) 2 m to 1 cm

(iii) 1 hour to 40 minutes

(iv) 4 to 250 paise

(v) 1.5 km to 500 m

(vi) 1 year to 4 months

Answer:

1. (i) 1 : 5

(ii) 200 : 1

(iii) 3 : 2

(iv) 8 : 5

(v) 3 : 1

(vi) 3 : 1

● Ratio and proportion

• Basic Concept of Ratios
• Important Properties of Ratios
• Ratio in Lowest Term
  
• Types of Ratios
• Comparing Ratios
• Arranging Ratios
  
• Dividing into a Given Ratio
  
• Divide a Number into Three Parts in a Given Ratio
• Dividing a Quantity into Three Parts in a Given Ratio
  
• Problems on Ratio
  
• Worksheet on Ratio in Lowest Term
  
• Worksheet on Types of Ratios
  
• Worksheet on Comparison on Ratios
• Worksheet on Ratio of Two or More Quantities
  
• Worksheet on Dividing a Quantity in a Given Ratio
• Word Problems on Ratio
  
• Proportion
  
• Definition of Continued Proportion
  
• Mean and Third Proportional
  
• Word Problems on Proportion
  
• Worksheet on Proportion and Continued Proportion
  
• Worksheet on Mean Proportional
  
• Properties of Ratio and Proportion

10th Grade Math

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