# Proportion – Definition – Properties – Examples

## Proportion

**When two ratios or (fractions) are equal, then they are in Proportion.**

A proportion is an equality of ratios.

**The symbol “::” or “=” to equal represent the two ratios.**

It can be written in two ways

1. Colon form

2. Fractional form

The terms a, b, c, and d are known as the terms of the proportion.

**Both can be read as “a is to b and c is to d”.**

a :b : : c : d, a, b, c and are the first, second, third and fourth terms of proportion respectively.

The first and fourth terms a and d are called extremes or extreme terms and the second and third terms b and c are called means or mean terms (middle terms).

Mathematically, if four quantities a, b, c, and d are in proportion, then the ratio a : b must be equal to c : d.

i.e. a : b = c : d

**The product of extremes = The product of the means**

i.e. **ad = bc (The cross product rule)**

Proportion can include numbers or variables or both.

**How to solve Proportions?**

If given ratios are proportional, it is easy to calculate.

**Step 1:** Multiply first term with the fourth term: a x d

**Step 2:** Multiply second term with the third term: b x c

**Step 3: **If the product of extreme terms is equal to product of mean terms, then the ratios are proportional

i.e. a x d = b x c

1. Using a colon a : b = c : d

2. Two equal fractions a/b = c/d

When two ratios are equal then there cross product, of the ratios are equal.

Therefore, a : b = c : d, a x d = b x c

The ratios 15/25 and 3/5 are proportional and are write as

The ratio **15/25 = 3/5 **

is read as fifteen is to twenty five as three is to five.

**Example:** **when we say that 4, 5, 40 and 50 are in proportion which is written as **

4 : 5 :: 40 : 50

and read as 4 is to 5 as 40 is to 50

or written as 4 : 5 = 40 : 50

4/5 = 40/50

cross product of ratios is

4 x 50 = 5 x 40

200 = 200

both are equal.

**Example: we can say 2, 3, 10, and 15 are in proportion which is written as**

2 : 3 :: 10 : 15

and read as 2 is to 3 as 10 is to 15.

or written as 2 : 3 = 10 : 15

2/3 = 10/15

cross product of ratios is

2 x 15 = 3 x 10

30 = 30, both are equal.

Here, the cross products are not equal, so this shows that the ratios are not in proportion.

**When ratio is 3 : 5 and 4 : 7**

Check the cross product of the given ratios.

3 x 7 = 4 x 5

The products are 21 and 20.

Here, the cross products are not same, so this ratios are not in proportion.

** 21 ≠ 20,** This shows that ratios are not in proportion.

**When ratio is 5 : 4 and 8 : 7**

Cross product 5 x 7 = 35, and 8 x 4 = 32.

Here, the cross products are not same, so this shows that the ratios are not in proportion.

**Find the third proportion to 6 and 12.**

Let the third proportional be c

Then, **b² = ac**

**(12)² = 6 x c**

**144 = 6 x c**

**c = 144/6**

**c = 24**

**Find the mean proportion between 5 and 45**.

Let the mean proportion between 5 and 45 is p.

apply formula b² = ac

Therefore, **p x p = 5 x 45 = 225**

**(p²) = 225**

**p = 15**

Hence, the mean proportion between 5 and 45 is 15

**How to tell two ratios are in form a proportion**

**First method: Same scale used on top and bottom**

Here we see that a scale 5 used on top and bottom both, because multiply both 6 and 7 by same number this is a proportion.

**Second method :** **Simplify ratios**

In this example, we divide the ratio 30 : 35, by 5 and we see that the ratios are equivalent. Therefore, they are in the proportional form.

**Third method :** **Cross product**

In cross product method we multiply the numbers that are diagonal to each other, if the products are equal, the two ratios form a proportions.

**3 x 12 = 36 and 4 x 9 = 36**

**Example: Solve k/20 = 5/4**

Apply cross multiplication we get

**k x 4 = 5 x 20 **

**4k = 100**

**k = 100/4**

**k = 25**

**Example: Solve 3/12 = 5/k**

Apply cross multiplication we get

**3 x k = 12 x 5 **

**3k = 60**

**k = 60/3**

**k = 2**0

**Continued proportions**

Three quantities a, b, and c are said to be in continued proportion if a: b = b : c.

The term c is called the third proportion of a and b, and b is called the mean proportion of terms a and c.

In fractional form ab = bc,

Here, we see that b² = ac.

**Verify the ratio 3 : 6 :: 6 : 12 is proportion.**

This is continued proportion, therefore the formula

**a x c = b x b**

Here, **a : b : c = 3 : 6 : 12**

Multiply first and third terms

**3 x 12 = 36**

square of middle term = **(6²) = 36**

Therefore, the ratio 3 : 6 : 12 is in proportion

If two ratios are equal, then their reciprocals must also equal as long as they exist.

Reciprocal of a/b = c/d ⇨ b/a = d/c

**a/b = c/d ⇨ a x d (Product of extremes) = b x c (Product of means)**