Learn Ratios, Proportions And Unitary Method In Maths

Ratio in arithmetic

Ratio is a term used to compare two quantities. A Ratio of two numbers tells how many times one quantity is to another quantity.

Ratio and its example

When two quantities, which are of the same kind and have the same units of measurement, are compared by dividing one quantity to another quantity, it is called ratio.
∴ ratio of two quantities a and b can be written as $\frac{a}{b}$ , where b ≠ 0
It is denoted by : symbol
Ratio of and b is written as a : b and read as a ratio b
a and b are called terms of ratio.

In ratio a : b, the first term a is called antecedent.
The later term b is called consequent.

Example of ratio

What is the ratio of pears in two bags with weights 15 kg and 20 kg?
The ratio of their weight = $\frac{15}{20}$
which is further = $\frac{3}{4}$ (∵ the ratio must be expressed in its lowest terms)
So, the ratio is 3:4.

Proportion and its example

When two ratios are equal, that implies the two ratios are in proportion also.
It means the equality of two ratios is called proportion.
The symbol of proportion is ::

To understand it, consider four quantities a, b, c and d.
If the ratio of first two quantities a and b is equal to ratio of last two quantities c and d , then four quantities a, b, c and d are said to be in proportion.
It is written as $\frac{a}{b} = \frac{c}{d}$
∴ the proportion of above ratios can be written as a : b :: c : d

In a : b :: c : d, the first term a and fourth term d are called extremes.
The second term b and third term c are called means.

Note

If four terms are in proportion then:
Product of extremes = Product of means

Example of proportion

Is this a correct 15 : 45 :: 40 : 120 proportion?
Here, 15 and 120 are called extremes and 45 and 40 are called means.
The product of extremes = 15 × 120 = 1800
The product of means = 45 × 40 = 1800
i.e. The product of extremes and means both are equal to 1800
the numbers 15, 45, 40 and 120 in proportion.

What is Continued Proportion?

Let there are three quantities a, b and c. If the ratio between first and second quantity is equal to ratio between second and third quantity. It implies that the three quantities are in continued proportion.

It is written as following:
a : b = b : c
Second quantity, here b, is called mean proportional between first quantity a and third quantity c.
Third quantity, here c, is called third proportional to first quantity a and second quantity b.

Example of continued proportion

Are the numbers 4,6 and 9 in continued proportion?
Reduce $\frac{4}{6}$ , which can be written as $\frac{2}{3}$
Reduce $\frac{6}{9}$ , which can be written as $\frac{2}{3}$
∴ both ratios $\frac{4}{6}$ and $\frac{6}{9}$ are equal to $\frac{2}{3}$
∴ $\frac{4}{6} = \frac{6}{9}$
or 4 : 6 = 6 : 9
Hence, 4, 6 and 9 are in continued proportion
Also, 6 is mean proportional between 4 and 9.
9 is third proportional to 4 and 6.

What is the unitary method?

This method is used to find the value of any number of units of a quantity when the value of more than one unit of the same quantity is known. With this method first the value of a single unit is calculated and then that value is multiplied by the number of units for which the value is to be calculated. This method can solve ratios and proportions problems such as find the price of 8 apples when the price of 6 apples is known.
This method is called as Unitary method because it involves an intermediate step to finding the value of a single unit.

Steps of unitary method

Step 1: Find the value of a single unit by dividing the given total value by the total number of units.
Step 2: Multiply the number obtained in step 1 with the total number of units to know their value.

Examples of unitary method problems

Example 1: If the cost of 20 toy cars is $220. Find the cost of 5 such toy cars.
Step 1: Calculate the cost of single unit i.e. one toy car by dividing the cost of 20 toy cars i.e. $220 by the total number of toy cars i.e. 20
Cost of 20 toy cars = $220
Cost of 1 toy car = $\frac{220}{20} = 11$
Step 2: Multiply 11 obtained in step 1 with the 5 toy cars.
∴ Cost of 5 toy cars = 11 × 5 = $55

Frequently Asked Questions

1) What is ratio?

When two quantities of the same kind and same unit of measurement are divided then the value we get is called ratio. Ratio is represented by the symbol :. Ratio of two quantities a and b is written as a:b and read as a ratio b, where a and b are called as terms.

2) What is antecedent in ratio?

Antecedent is the first term of a ratio. For example a is the antecedent in ratio a:b.

3) What is consequent in ratio?

Consequent is the second term of a ratio. For example b is the consequent in ratio a:b.

4) What is the proportion?

Proportion means that the ratios are equal. Proportion is represented by the symbol ::. If two ratios a:b and c:d are equal, i.e. a ÷ b = c ÷ d then we can say a,b,c and d are in proportion.

Solved Examples

1) Find the ratio of 35 minutes to 3 hours.

1 hour = 60 min
3 hrs = 3 × 60
= 180 mins
Ratio of 35 minutes to 180 mins = $\frac{35}{180}$
= $\frac{7}{36}$

2) The number of girls and boys in a school are 300 and 450 respectively. Express the ratio of boys to girls.

Number of boys = 450
Number of girls = 300
Ratio of boys to girls = $\frac{450}{300}$
= $\frac{3}{2}$

3) There are 100 workers in a factory. Out of 100 workers, 60 are male workers. Find the ratio of male workers to female workers.

Total number of workers = 100
Number of male workers = 60
Number of female workers = 100 - 60 = 40
Ratio of male workers to female workers = $\frac{60}{40}$
= $\frac{6}{4}$
= $\frac{3}{2}$

4) Does ratio 3:6 and 5:10 form a proportion?

$3 : 6 = \frac{3}{6} = \frac{1}{2}$
$5 : 10 = \frac{5}{10} = \frac{1}{2}$
∴ $\frac{3}{6} = \frac{5}{10}$
Hence, 3:6 and 5:10 are in proportion.

5) The ratio of length and width of a sheet is 3:4. Find the width of the sheet if its length is 15 cm.

Let width of sheet = x cm
∴ Ratio of length to width = 15:x
Also, ratio of length to width = 3:4 (given)
3:4 = 15:x
$\frac{3}{4} = \frac{15}{x}$
3 × x = 4 × 15
3x = 60
$\frac{60}{3}$
x = 20 cm

6) Express

4 \frac{1}{2} : 3 \frac{1}{3}

as the simplest ratio.

$\frac{9}{2} : \frac{10}{3}$
Ist method:
Cross multiply 9 by 3 and 2 by 10
= $\frac{9 \times 3}{2 \times 10}$
= $\frac{27}{20}$
⇒ 27:20
IInd method:
Take the LCM of denominators.
i.e. 2 and 3
∴ LCM = 6
$\frac{9}{2} \times 6 = 27$
$\frac{10}{3} \times 6 = 20$
⇒ 27:20

7) Express

\frac{2}{3} : \frac{3}{2} : \frac{1}{5}

as the simplest ratio.

Take the LCM of 3, 2 and 5.
LCM = 30
$\frac{2}{3} \times 30 = 10$
$\frac{3}{2} \times 30 = 45$
$\frac{1}{5} \times 30 = 6$
⇒ 10 : 45 : 6

8) Divide 100 into two parts in the ratio 2:3.

Since, 2 + 3 = 5
Ist part = $\frac{2}{5} \times 100 = 40$
IInd part = $\frac{3}{5} \times 100 = 60$

9) Divide $1200 among three people in the ratio of 2:3:5. Find the share of each person.

Since, 2 + 3 + 5 = 10
Ist person's share = $\frac{2}{10} \times 1200 = 240$
IInd person's share = $\frac{3}{10} \times 1200 = 360$
IIIrd person's share = $\frac{5}{10} \times 1200 = 600$

10) Divide $2900 among two persons in the ratio of

2 \frac{1}{2} : 2 \frac{1}{3}

$2 \frac{1}{2} : 2 \frac{1}{3} = \frac{5}{2} : \frac{7}{3}$
Take the LCM of 2 and 3
LCM = 6
$\frac{5}{2} \times 6 = 15$
$\frac{7}{3} \times 6 = 14$
⇒ 15:14
Since, 15 + 14 = 29
Ist person's share = $\frac{15}{29} \times 2900 = $ 1500$
IInd person's share = $\frac{14}{29} \times 2900 = $ 1400$

11) If the sides of a triangle are in the ratio 1:2:3 and its perimeter is 120cm. Find the length of each side.

Let x, 2x and 3x be sides of the triangle.
∴ perimeter = x + 2x + 3x = 120
6x = 120
$x = \frac{120}{6}$
x = 20
First side = x = 20cm
Second side = 2x = 2 × 20 = 40cm
Third side = 3x = 3 × 20 = 60cm

12) A man earns $2400 in 15 days. Calculate his earnings for 1 week.

The man's earning in 15 days = $2400
The man's earning in 1 day = $\frac{2400}{15} = $ 160$
∴ The man's earning in 1 week or 7 days = 160 × 7 = $1120

13) The cost of 14 juice cans is $42. What will be the cost of 30 such juice cans?

Cost of 14 juice cans = $42
Cost of 1 juice can = $\frac{42}{14} = $ 3$
∴ Cost of 30 juice cans = 3 × 30 = $90

Fill in Blanks Worksheet

Blanks - 1

A ratio is a comparison of one quantity to another quantity of the same kind and same ___.
The first term of the ratio is called ___.
If A's age is 15 years and the ratio of ages of A:B = 5:4. Then the age of B is ___ years.
In a leap year, there are 140 days of holidays. Then, the ratio of days of holidays and total number of days in the year is 70 : ___.
The third proportional to 5 and 10 is ___.
The ratio of boys to girls in a class is 2:3. If the total number of girls are 300, then the number of boys in the class are ___.
The angles of a triangle are in the ratio of 1:2:3. The measure of the smallest angle is ___.
4kg : 500g = 8 : x, then the value of x is ___.
4 : 5 :: x : 20, then the value of x is ___.
The weight of 5 boxes is 150kg. Then the weight of 2 boxes will be ___kg.

Help box

30°

unit

200

183

antecedent

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Word Problems Worksheet

Word Problems - 1

Solve the following questions.

The weight of 1 box of apples is 5kg. Find the weight of 29 such boxes.
A bus consumes 10 litres of petrol to cover a distance of 120km. How much distance will the bus cover with 15 litres of petrol?
The weight of 1 box of apples is 5kg. Find the weight of 29 such boxes.
There are 5 shelves in a book rack of a library and 100 books can be placed on a shelf. How many books does this library have, if there are a total of 10 such racks in the library.
The cost of 1 pack of pencils is $24. One pack contains a total of 12 pencils. What is the cost of 20 pencils?
A boat ride costs $200 for 2 persons on a boat. How much money will be spent if 6 people decide to take the boat ride?

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Multiple Choice Questions Worksheet

MCQ - 1

1) A ratio equivalent to 3:4 is