Definition of Continued Proportion:

Three quantities are said to be in continued proportion; if the ratio between the first and the second is equal to the ratio between the second and the third.

Suppose, if we have three qualities such that the ratio of first to second is equal to the ratio of second to third, we say that the three qualities are in continued proportion. The middle term is called the mean proportional between the first the third terms.

i.e. a, b and c are in continued proportion, if a : b = b : c

The second quantity is called the **mean proportional** between the first and the third

i.e. in a : b = b : c; b is the mean proportional between a and c.

The third quantity is called the **third proportional** to the first and the second

i.e. in a : b = b : c; c is the third proportional to a and b.

For example, let us consider the numbers 6, 12, 24.

Here the ratio of first quantity to the second = 6 : 12 = 1 : 2

And ratio of second quantity to the third = 12 : 24 = 1 : 2

We see that 6 : 12 = 12 : 24

Thus, 6, 12, 24 are in continued proportion.

The second quantity 12 is the mean proportional and third quantity 24 is the third proportional.

Solved example on continued proportion:

**1.**
Find the mean proportion between 4 and 9.

**Solution:**

Let the mean proportion be x

Therefore, 4 : x = x : 9

⇒ x × x = 4 × 9

⇒ x^{2} = 36

⇒ x^{2} = 6^{2}

⇒ x = 6

**2.**
Find, m, if 7, 14, m are in continued proportion.

**Solution:
**

x, y and z are in continued proportion xz = y^{2}

Let 7, 14, and m be x, y and z respectively.

Therefore, 7m = 14^{2}

or, 7m = 196

or, m = 196/7

Therefore, m = 28.

Hence, m = 28.

**3.**
Find the third proportional to 12 and 30.

**Solution:**

Let x be the third proportional

Therefore, 12 : 30 = x : 30

⇒ 12 × x = 30 × 30

⇒ 12x = 900

⇒ x = 900/12

⇒ x = 75

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