Learn Ratios & Properties Of Proportions With Examples

Ratios

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

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 fraction $\frac{a}{b}$ , 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.

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.

Antecedent

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

Consequent

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

Proportions

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

Extremes

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

Means

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

Cross product rule

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

Let's say a, b, c and d are in proportion, i.e.
$\frac{a}{b} = \frac{c}{d}$ , where a, d are extremes and b, c are means
then, according to cross product rule a × d = b × c

Example of cross product rule in 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
∴ as per cross product rule, if product of means is equal to product of extremes then numbers 15, 45, 40 and 120 in proportion also.

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.

So, the continued proportion of a, b and c is written as a : b = b : c

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

Mean proportional

The second quantity b in the continued proportion a : b = b : c is called mean proportional between first quantity a and third quantity c.

The value of mean proportional can be calculated as following:

a : b = b : c can be written as $\frac{a}{b} = \frac{b}{c}$
or $b^{2} = a \times c$
$b = \sqrt{ac}$

Third proportional

The third quantity c in the continued proportion a : b = b : c is called third proportional to first quantity a and second quantity b. The value of third proportional can be calculated as following:
$c = \frac{b^{2}}{a}$

Example of mean and third proportional

Write mean and third proportional of the continued proportion 7 : 21 = 21 : 63
Here, 4, 6 and 9 are in continued proportion
Also, 21 is mean proportional between 7 and 63.
63 is third proportional to 7 and 21.

Properties of proportion

There are main four properties of proportion, which are applied to two ratios when they are equal.
Let's say the two ratios $\frac{a}{b}$ and $\frac{c}{d}$ are equal.
i.e. $\frac{a}{b} = \frac{c}{d}$

Then these two ratios will satisfy the following four properties of proportions:

Alternendo

$\frac{a}{c}$ and $\frac{b}{d}$ will be in proportion.
i.e. $\frac{a}{c} = \frac{b}{d}$

Invertendo

$\frac{b}{a}$ and $\frac{d}{c}$ will be in proportion.
i.e. $\frac{b}{a} = \frac{d}{c}$

Componendo

$\frac{a + b}{b}$ and $\frac{c + d}{d}$ will be in proportion.
i.e. $\frac{a + b}{b} = \frac{c + d}{d}$

Dividendo

$\frac{a - b}{b}$ and $\frac{c - d}{d}$ will be in proportion.
i.e. $\frac{a - b}{b} = \frac{c - d}{d}$

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. It is called as Unitary method because it involves an intermediate step to finding the value of a single unit.

Steps to solve word problems

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.

Example of unitary method problems

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

What is direct proportion?

If a value on the left side of the proportion increases or decreases with the change in the value on the right side of the proportion at the same rate, then the ratios are in the direct proportion.

What is inverse proportion?

If a value on the left side of the proportion increases while decreasing the value on the right side or decreases while increasing the value on the right side of the proportion, then the ratios are in the indirect proportion.

Solved Examples

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 = $\frac{35}{180}$
= $\frac{7}{36}$

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 = $\frac{450}{300}$
= $\frac{3}{2}$

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 = $\frac{60}{40}$
= $\frac{6}{4}$
= $\frac{3}{2}$

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.

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

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

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

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

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

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 = $\frac{5}{10} \times 1200 = 600$

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$

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

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

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