#### description

- Proportion
- Continued or mean proportion

#### definition

**Proportion:** If two ratios are equal, then they are said to be in proportion. We use the symbol ‘::’ or ‘=’ to equate the two ratios.

**Continued proportion: **Four quantities are said to be in continued proportion if the ratio of the first term and second term be equal to the ratio of the second term and the third term be equal to the ratio of the third term and fourth term. If w, x, y, and z are four quantities such that w: x = x: y = y: z i.e., `w/x = x/y = y/z`, they are said to be continued proportion.

#### notes

**Proportion:**

- Bhavika has 28 marbles and Vini has 180 flowers. They want to share these among themselves. Bhavika gave 14 marbles to Vini and Vini gave 90 flowers to Bhavika. But Vini was not satisfied. She felt that she had given more flowers to Bhavika than the marbles given by Bhavika to her. What do you think? Is Vini correct?
- To solve this problem both went to Vini’s mother Pooja. Pooja explained that out of 28 marbles, Bhavika gave 14 marbles to Vini. Therefore, the ratio is 14: 28 = 1: 2. And out of 180 flowers, Vini had given 90 flowers to Bhavika. Therefore, the ratio is 90: 180 = 1: 2. Since both the ratios are the same, so the distribution is fair.
- If two ratios are equal, then they are said to be in proportion.
- If two ratios are equal, we say that they are in proportion and use the symbol ‘::’ or ‘=’ to equate the two ratios.

**Continued or Mean Proportion:**

Three quantities are said to be in continued proportion if the ratio of the first term and second term be equal to the ratio of the second term and third term.

**Continued proportion:** Three numbers ‘a’, ‘b’, and ‘c’ are said to be continued proportion if a, b and c are in proportion.

Thus, if a, b and c are in continued-proportion, then a, b, b, c are in proportion, that means; a: b: : b: c

⇒ Product of extremes = Product of means

⇒ a x c = b x b

⇒ a x c = b²

Continued proportion is also known as mean proportional.

If ‘b’ is a mean proportional between a and c then b^{2} = ac.

Suppose, the three quantities x, y and z are said to be in continued proportion if x: y = y: z, i.e., `x/y = y/z`

Similarly, four quantities are said to be in continued proportion if the ratio of the first term and second term be equal to the ratio of the second term and the third term be equal to the ratio of the third term and fourth term.

If w, x, y, and z are four quantities such that w: x = x: y = y: z i.e., `w/x = x/y = y/z`, they are said to be continued proportion.

- For example, we can say 2, 4, 180, and 360 are in the proportion which is written as 2: 4:: 180: 360 and is read as 2 is to 4 as 180 is to 360.
- If two ratios are not equal, then we say that they are not in proportion.

For example, the two ratios 2: 5 and 180: 45 are not equal, i.e. 2: 5 ≠ 180: 45. Therefore, the four quantities 2, 5, 180 and 45 are not in proportion - In a statement of proportion, the four quantities involved when taken in order are known as respective terms. The first and fourth terms are known as extreme terms. Second and third terms are known as middle terms.

#### Example

Are the ratios 25g: 30g and 40 kg: 48 kg in proportion?

25 g: 30 g = `25/30 `= 5: 6

40 kg: 48 kg = `40/48` = 5: 6

So, 25: 30 = 40: 48.

Therefore, the ratios 25 g: 30 g and 40 kg: 48 kg are in proportion, i.e. 25: 30:: 40: 48

The middle terms in this are 30, 40 and the extreme terms are 25, 48.

#### Example

Are 30, 40, 45, and 60 in proportion?

Ratio of 30 to 40 = `30/40` = 3: 4.

Ratio of 45 to 60 = `45/60` = 3: 4.

Since, 30: 40 = 45: 60

Therefore, 30, 40, 45, and 60 are in proportion.

#### Example

Do the ratios 15 cm to 2 m and 10 sec to 3 minutes form a proportion?

15 cm : 2 m :: 10 sec : 3 min

15 cm : 2 × 100 cm :: 10 sec : 30 × 60 sec

15 : 200 :: 10 : 1800

3 : 40 :: 1 : 180

No, they donot form a proportion

Ratio of 15 cm to 2 m = 15: 2 × 100 (1 m = 100 cm)

= 3: 40

Ratio of 10 sec to 3 min = 10: 3 × 60 (1 min = 60 sec)

=1: 18

Since, 3: 40 ≠ 1: 18, therefore, the given ratios do not form a proportion.

#### 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 suppose x toilets will be needed for 75 girls.

The ratio of the number of toilets to the number of girls is `2/15`.

∴ `x/75 = 2/15`

∴ `x/75 xx 75 = 2/15 xx 75`................(Multiplying both sides by 75)

∴ x = 2 × 5

∴ x = 10

∴ 10 toilets will be required for 75 girls.