Maharashtra Board Class 6 Maths part 2 Chapter 11 Ratio Proportion PDF Download

Ratio - Proportion

In the previous classes, we have learnt to compare two numbers. We shall now learn another way to do the same.

Let's say Nilima is 12 years old and Ramesh is 6. How to compare their ages?

Ramesh did so by finding out the difference.

Nilima did it by saying how many times she is as old as Ramesh.

Nilima's age is twice as much as Ramesh's. We can give the same information by saying that Nilima's and Ramesh's ages are in the proportion 2:1. It is read as 'Two is to one'.

In mathematics, the proportion of two numbers can also be expressed as their ratio. The proportion 2:1 is written as \(\frac{2}{1}\) in the form of a ratio.

Examples of Proportion in Daily Life

Example: Jankiamma's idlis and dosas are delicious. For idlis, she uses udad dal and rice in the proportion 1 cup dal to 2 cups of rice. But for dosas, the proportion is 1 cup dal to 3 cups of rice. That is, for idlis the proportion of dal and rice is 1:2 or the ratio is \(\frac{1}{2}\) whereas for dosas, the proportion is 1:3 or the ratio is \(\frac{1}{3}\).

Example: Margaret makes great biscuits. She uses 3 cups of wheat flour with 2 cups of sugar. It means that the proportion of sugar and flour in the biscuits is 2:3 or that the ratio is \(\frac{2}{3}\).

Teacher's Note

Ratio is used in many cooking recipes in Indian kitchens. For example, when making dal rice, we use a ratio of 1 cup dal to 3 cups of rice.

Exam Trick

Remember: Ratio means comparing two things. Always write the first thing in the numerator and the second thing in the denominator. For example, ratio of 2 to 3 is written as 2:3 or \(\frac{2}{3}\).

Points to Remember

A proportion shows how two numbers compare to each other.
A ratio can be written as 2:1 or as a fraction \(\frac{2}{1}\).
When we say the proportion is 2:1, it means the first thing is twice the second thing.
Ratios are used in cooking, mixing colors, and many other daily activities.

Example: Flowers were distributed among the girls in equal proportions.

Fill in the empty boxes:

\(\frac{3}{12} = \frac{1}{4}\)

It means that every girl got 4 flowers.

The proportion of girls and flowers is 'One is to every four'. It is written as 1:4 or their ratio is written as \(\frac{1}{4}\).

Example: Every student finds the ratio of his or her own age to that of his/her grandmother's.

John's age is 10 years and his grandmother's 65. John said the ratio was \(\frac{10}{65}\) for him.

\(\frac{10}{65} = \frac{10 \div 5}{65 \div 5} = \frac{2}{13}\)

We can make use of equivalent fractions to write the ratio in the simplest form.

Example: Nikhil brought 12 guavas and 16 chikoos.

(1) Find the ratio of guavas to chikoos (2) Find the ratio of chikoos to guavas

Ratio of guavas to chikoos Ratio of chikoos to guavas

\(= \frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}\) \(= \frac{16}{12} = \frac{16 \div 4}{12 \div 4} = \frac{4}{3}\)

∴ Ratio of guavas to chikoos is \(\frac{3}{4}\) ∴ Ratio of chikoos to guavas is \(\frac{4}{3}\)

In the figure, colour some boxes with any colour you like and leave some blank.

(1) Count all the boxes and write the number

(2) Count the coloured ones and write the number

(3) Count the blank ones and write the number

(4) Find the ratio of the coloured boxes to the blank ones

(5) Find the ratio of the coloured boxes to the total boxes

(6) Find the ratio of the blank boxes to the total boxes

Teacher's Note

You can teach ratio using real objects in the classroom. For example, show the ratio of boys to girls in your class as 2:3.

Exam Trick

To simplify a ratio, divide both numbers by their greatest common factor (GCF). For example, 12:16 becomes 3:4 when you divide both by 4.

Points to Remember

A ratio shows the relationship between two numbers.
Ratios are written as a:b or as a fraction \(\frac{a}{b}\).
We can simplify ratios by dividing both numbers by the same number.
The order of numbers in a ratio is very important.

Some Important Points about Ratio

Example: The weight of the large block of jaggery is 1 kg and a smaller lump weighs 200 g. Find the ratio of the weight of the lump of jaggery to that of the block.

\(\frac{\text{Weight of the lump}}{\text{Weight of the block}} = \frac{200}{1}\)

Is this right?

Is the weight of the lump 200 times that of the block?

What mistake have we made?

First we must measure both quantities in the same units.

It would be convenient to use grams here.

1kg = 1000 grams

∴ The block weighs 1000 g and the lump, 200 g.

\(\frac{\text{Weight of the lump}}{\text{Weight of the block}} = \frac{200}{1000} = \frac{2 \times 100}{10 \times 100} = \frac{2}{10} = \frac{1 \times 2}{5 \times 2} = \frac{1}{5}\)

Thus, the ratio of the weight of the lump of jaggery to that of the block is \(\frac{1}{5}\).

When finding the ratio of two quantities of the same kind, their measures must be in the same units.

A ratio can be used to write an equation. Then it is easier to solve the problem.

Example: A hostel is to be built for schoolgoing girls. Two toilets are to be built for every 15 girls. If 75 girls will be living in the hostel, how many toilets will be required in this proportion?

Let us consider the proportion or ratio of toilets and girls. Let us suppose x toilets will be needed for 75 girls. The ratio of the number of toilets to the number of girls is \(\frac{2}{15}\). Let us write this in two ways and form an equation.

∴ \(\frac{x}{75} = \frac{2}{15}\)

∴ \(\frac{x}{75} \times 75 = \frac{2}{15} \times 75\) (Multiplying both sides by 75)

∴ x = 2 × 5

= 10

∴ 10 toilets will be required for 75 girls

Teacher's Note

In India, schools must follow rules about how many toilets are needed for a certain number of students. This is an example of using ratios in real life.

Exam Trick

Always convert both quantities to the same units before finding the ratio. For example, convert 1 kg to 1000 grams if you are comparing with grams.

Points to Remember

Both quantities must be in the same units before you find a ratio.
You can write ratios as equations to solve problems.
Cross multiply when solving ratio equations.
Always simplify the ratio to its lowest form.
Check your answer by seeing if it makes sense.

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