When two ratios are equal, they are said to be in proportion.

To verify whether two ratios are in proportion, we simplify the two ratios first and then we determine whether they are equal or not. If both the simplified ratios are equal, they are said to be in proportion. If the simplified ratios are not equal, then the ratios are not in proportion. We use the symbols " :: " or " = " to denote a proportion.

Consider two ratios in proportion, $\frac{a}{b}$ = $\frac{c}{d}$ , ( a:b :: c:d). Here, we have a x d = c x d.

In a statement of proportion, the first and fourth terms are known as extreme terms and the second and third terms are known as middle terms. Thus, if two ratios are in proportion, the product of the extreme terms = product of the middle terms.

## Proportion Definition

Two ratios are said to be in proportion if they are equal. By definition, a, b, c and d is called a proportion, if a:b = c:d.

Consider the number of boys and girls in a class. Let there be 30 boys and 15 girls.

The ratio of boys to girls = 30:15 = 2:1

The above two ratios are the same. We say that these ratios are in proportion. That is, 60:30 is in proportion to 30:15.

This is written as 60 : 30 :: 30 : 15

## Mean Proportion

If x is the mean proportion between a and b, then the mean proportion is x = $\sqrt{ab}$

### Examples on Mean Proportion

Given below are some examples on mean proportion.

Example 1:

Find the mean proportion of 2.5 and 0.9

Solution:

If b is the mean proportion between a and c, then the mean proportional is b = $\sqrt{ac}$

## Direct Proportion

If two quantities are so related to each other that an increase (or decrease) in the first causes an increase (or decrease) in the second, then the two quantities are said to vary directly.

### Examples of Direct Proportion:

• The cost of an article varies directly with the number of articles. [More cost, More articles]
• The work done directly varies with the number of men at work.
• The distance covered by a car directly varies with its speed.
• More the money deposited in the bank, the more the interest earned.

### Direct Proportion Rule:

If two quantities ‘x' and ‘y' vary directly with each other then always

$\frac{x}{y}$ = k (Constant)

### Examples on Direct Proportion

Given below are some examples on Direct Proportion.

Example 1:

The cost of 5 meters of a particular quality of cloth is $210. Tabulate the cost of 2, 4, 10 and 13 meters of cloth of the same type. Solution: Suppose the length of a cloth is x meters and its cost, in is$y.

 x 2 4 5 10 13 y y2 y3 210 y4 y5

As the length of the cloth increases, the cost of the cloth also increases in the same ratio. It is a case of direct proportion.

We make use of the relation of type $\frac{x1}{y1}$ = $\frac{x2}{y2}$

(i) Here x1 = 5 y1 = 210 and x2 = 2

Therefore, $\frac{x1}{y1}$ = $\frac{x2}{y2}$

$\frac{5}{210}$ = $\frac{2}{y2}$

5y2 = 2 x 210

y2 = $\frac{420}{5}$

y2 = 64

(ii) x3 = 4, then $\frac{5}{210}$ = $\frac{4}{y3}$

5 y3 = 4 x 210

y3 = $\frac{840}{5}$ = 168

(iii) x4 = 10, then $\frac{5}{210}$ = $\frac{10}{y4}$

5 y4 = 10 x 210

y4 = $\frac{2100}{5}$ = 420

(iv) x5 = 13, then $\frac{5}{210}$ = $\frac{13}{y5}$

5 y5 = 12 x 210

y5 = $\frac{(13\times210)}{5}$ = 546

Example 2:

An electric pole, 14 meters high, casts a shadow of 10 meters. Find the height of a tree that casts a shadow of 15 meters under similar conditions.

Solution:

Let the height of the tree be x meters. We form a table as shown below:

 Height of the Object (in metres) 14 x Length of the Shadow (in metres) 10 15

Note: The more the height of the object, the more the length of the shadow.

Hence, this is the case of direct proportion. That is,

$\frac{x1}{y1}$ = $\frac{x2}{y2}$

We have

$\frac{14}{10}$ = $\frac{x}{15}$

$\frac{(14\times15)}{10}$ = x

21 = x

Thus, height of the tree is 21 m

Example 3:

If 32 horses consume 112 kg of grass in a certain period, how much grass will be consumed by 11 horses during the same period?

Solution:

Fewer Horses, lesser the consumption of grass [Direct Variation]

32 horses consume = 112 kg

1 horse will consume = $\frac{112}{32}$ kg

11 horses will consume = $\frac{(112\times11)}{32}$

= 38. 5 kg.

Example 4:

If the weight of 65 coffee packets of the same size is 26 kg. What is the weight of 25 such packets?

Solution:

The fewer the packets, the lesser the weight [Direct Variation]

Weight of 65 packets = 26 kg

Weight of 1 packet = $\frac{26}{65}$ kg

Therefore, the weight of 25 packets = $\frac{(26\times25)}{65}$ kg

= 10 kg

## Indirect Proportion

If two quantities are so related to each other that an increase (or decrease) in the first causes a decrease (or increase) in the other, then the two quantities are said to vary indirectly.

### Examples of Indirect Proportion:

• Time taken by a car to cover a certain distance varies indirectly with the speed of the car. [More the speed, the less is the time taken]
• Time taken to finish a work is inversely proportional to the number of men at work. [The more the men, the less is the time taken to finish the work]

### Indirect Proportion Rule:

If two quantities ‘x' and ‘y' inversely vary with each other then xy = k (constant)

### Examples on Indirect Proportion:

Given below are some examples on indirect proportion.

Example 1:6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used?

Solution:

Let the desired time to fill the tank be x minutes. Thus, we have the following table.

 Number of Pipes 60 5 Time in Minutes 80 x

Lesser the number of pipes the more will be the time required to fill the tank. So, this is a case of inverse proportion.
Hence, 80 × 6 = x × 5
Therefore, x = 96
Thus, time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.

Example 2:

There are 100 students in a hostel. The Food provision available for them is for 20 days. How long will these provisions last, if 25 more students join the group?

Solution:

Suppose the provisions last for y days when the number of students is 125.

We have the following table.

 Number of Students 100 125 Time in Days 20 y

Note the more the number of students, the sooner the provisions would exhaust. Therefore, this is a case of inverse proportion.

So, 100 × 20 = 125 × y

Therefore, y = 16

Thus, the provisions will last for 16 days, if 25 more students join the hostel.

Example 3:

If 15 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?

Solution:

Let the number of workers employed to build the wall in 30 hours be y. We have the following table.

 Number of Hours 48 30 Number of workers 15 y

Obviously the more the number of workers, the faster they will build the wall. So, the number of hours and number of workers vary in inverse proportion.

So, 48 × 15 = 30 × y

Therefore, y = 24

Therefore, to finish the work in 30 hours, 24 workers are required.

## Properties of Proportion

If a, b, c and d is called a proportion, then

• $\frac{a}{b}$ = $\frac{c}{d}$

• $\frac{a}{c}$ = $\frac{b}{d}$

• $\frac{(a^2)}{(b^2)}$ = $\frac{(c^2)}{(d^2)}$

If 1 is added to both sides of the equation $\frac{a}{b}$ = $\frac{c}{d}$,

We get, $\frac{a}{b}$ + 1 = $\frac{c}{d}$ + 1

$\frac{(a+b)}{b}$ = $\frac{(c+d)}{d}$

If 1 is subtracted from both sides of the equation $\frac{a}{b}$ = $\frac{c}{d}$,

We get, $\frac{a}{b}$ - 1 = $\frac{c}{d}$ - 1

$\frac{(a-b)}{b}$ = $\frac{(c-d)}{d}$

Now, since $\frac{(a+b)}{b}$ = $\frac{(c+d)}{d}$ and $\frac{(a-b)}{b}$ = $\frac{(c-d)}{d}$of these equations, division of the left sides is equal to the division of the right side.

$\frac{(a+b)/b}{(a-b)/b}$ = $\frac{(c+d)/d}{(c-d)/d}$

$\frac{(a+b)}{b}$ x $\frac{b}{(a-b)}$ = $\frac{(c+d)}{d}$ x $\frac{d}{(c-d)}$

$\frac{(a+b)}{(a-b)}$ = $\frac{(c+d)}{(c-d)}$

### Examples on Properties of Proportions:

Given below are some examples that help us understand the properties of proportions.

Example 1:

If x:y = 5:3 , then what is (x+y):(x-y) ?

Solution:

From the proportion, $\frac{(x+y)}{(x-y)}$ = $\frac{(a+b)}{(a-b)}$

Then, (x+y):(x-y) = (5 + 3):( 5-3)

= 8:2

= 4:1

Example 2:If (x+y):(x-y) = 3:2 , then what is x:y ?

Solution:

If x:y = a:b ,

Then,

(x+y):(x-y) = (a+b):(a-b)

$\frac{(x+y)+(x-y)}{(x+y)-(x-y)}$ = $\frac{(3+2)}{(3-2)}$

$\frac{(x+x)}{(y+y)}$ = $\frac{5}{1}$

$\frac{2x}{2y}$ = $\frac{5}{1}$

So,

$\frac{x}{y}$ = $\frac{5}{1}$
(or)

x:y = 5:1

### Ratios of Three Numbers:

If x:y:z = a:b:c, then $\frac{x}{a}$ = $\frac{y}{b}$ = $\frac{z}{x}$
In other words, the numbers x, y, z and the numbers a, b, c are proportional if and only if $\frac{x}{a}$ = $\frac{y}{b}$ = $\frac{z}{c}$

### Examples on Ratios of Three Numbers:

The length, breadth and height of a rectangular box are in the ratio 2:3:5 and its volume 3750 cc. Find its dimensions.

Solution:

Lets assume the length of the rectangular box is 2k cm, breadth is 3k cm and height is 5k cm.

Then, volume = 2k x 3k x 5k = 30$k^{3}$

So,

30$k^{3}$ = 3750

$k^{3}$ = $\frac{3750}{30}$

= 125

k = 5

Thus, Length = 2 x 5 = 10cm

Breadth = 3 x 5 = 15cm

Height = 5 x 5 = 25 cm

## How to Solve Proportions

A statement of equality of two ratios is called a proportion. 3, 10, 15 and 50 are in proportion which is written as 3:10:: 15:50 and is read as 3 is to 10 as 15 is to 50 or it is written as 3:10 = 15:50.

### Facts about Proportions:

• The first and fourth terms are called extremes
• The second and the third terms are known as the means.
• Product of Extremes = Product of Means
• If a:b :: c:d, then d is called as the fourth proportional to a, b, c.
• If a:b :: b:c, then
1. a, b, c are in continued proportion.
2. c is called the third proportional to a and b.
3. b is the mean proportional between a and c. And, the formula to find the mean proportional is $b = \sqrt{ac}$

### Proportion Examples

Given below are some examples on proportions.

Example 1:

Do the ratio of 18 seconds to 3 minutes and 20 cm to 2 meters form a proportion?

Solution:

The ratio of 18seconds to 3 minutes = 18 sec: 3 min

= 8 : (3*60), (Writing both quantities in the same units.)

= $\frac{18}{3\times60}$

= $\frac{18}{180}$

= $\frac{1}{10}$

= 1:10

The ratio of 20 cm to 2 meters = 20:200, (Writing both quantities in same units.)

= $\frac{20}{200}$

= $\frac{1}{10}$

= 1:10

The given ratios are in proportion.

Alternate Method:

We have the ratios of quantities taken in same unit as 18 seconds to 180 seconds and 20 cm to 200 meters.

The product of the extreme numbers = 18 X 200 = 3600

The product of the middle numbers = 180 X 20 = 3600

So, the product of extreme numbers = the product of the middle numbers. Hence, the given ratios are in proportion.

Example 2:

Find the value of the variable x such that 7:84 :: 2:x

Since two ratios are in proportion, we have the product of extreme numbers = the product of the middle numbers.

Therefore, we have 7 x = 84 X 2

= 168

Thus, the value of x = $\frac{168}{7}$

= 24

## Proportion Word Problems

Given below are some word problems that explain how to convert statements to proportions and then solve them.

Example 1:

John picked 18 oranges and 24 apples and Sam picked 15 oranges and some apples. The ratio of oranges to apples picked by both John and Sam were the same. Find the number of apples picked by Sam.

Solution:

Let x be the number of apples picked by Sam.

John picked 18 oranges and 24 apples.

The ratio of number of oranges to that of apples = 18:24

= $\frac{18}{24}$

= $\frac{3}{4}$ , by writing in the simplest form.

Given that the ratio of number of oranges to that of apples picked by John and Sam are same.

The ratio of number of oranges to that of apples picked by Sam = $\frac{15}{x}$

Therefore, we have $\frac{15}{x}$ = $\frac{3}{4}$

Cross multiplying, we get, 60 = 3x

So, the value of x = 20

Hence, Sam picked 20 apples

Example 2:

Experiments show that the ratio of the weights of objects on the moon and on the earth are in proportion. If the weight of a dog on the moon is 240lb and the weight of the object in earth is 40lb, find the weight of a man on the moon when he weighs 60lb on earth.

Solution:

The ratio of the weight of the dog on moon and that on earth = 240:40

= 6:1, if we write the ratio in reduced form

Let x lb be the weight of the man on the moon.

Therefore, x : 60 = 6:1

$\frac{x}{60}$ = $\frac{6}{1}$

Cross multiplying, we get, x = 360

Hence, the weight of the man on moon = 360 lb