**Definition of Continued 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.

Suppose, the three quantities x, y and z are said to be in continued proportion if x : y = y : z, i.e., \(\frac{x}{y}\) = \(\frac{y}{z}\).

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 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., \(\frac{w}{x}\) = \(\frac{x}{y}\) = \(\frac{y}{z}\), they are said to be in continued proportion.

**For example,**

(i) The numbers 4, 6 and 9 are in continued proportion because

\(\frac{4}{6}\) = \(\frac{6}{9}\)

or, 6\(^{2}\) = 4 × 9.

(ii) The numbers 2, 4 and 6 are not in continued proportion because

\(\frac{2}{4}\) ≠ \(\frac{4}{6}\) .

(iii) The numbers 2, 4, 8 and 16 are in continued proportion because

\(\frac{2}{4}\) = \(\frac{4}{8}\) = \(\frac{8}{16}\).

Solved examples on continued proportion of three or four quantities:

1. If k, 8, 16 are in continued proportion then find k.

Solution:

k, 8 and 16 are in continued proportion.

⟹ k : 8 = 8 : 16

⟹ \(\frac{k}{8}\) = \(\frac{8}{16}\)

⟹ k × 16 = 8\(^{2}\)

⟹ 16k = 64

⟹ k = \(\frac{64}{16}\)

⟹ k = 4

Therefore, the value of k = 4.

2. Quantities m, 2, 10 and n are in continued proportion then find the values of m and n.

Solution:

m, 2, 10 and n are in continued proportion.

⟹ m : 2 = 2 : 10 = 10 : n

⟹ \(\frac{m}{2}\) = \(\frac{2}{10}\) = \(\frac{10}{n}\)

⟹ \(\frac{m}{2}\) = \(\frac{2}{10}\) and \(\frac{2}{10}\) = \(\frac{10}{n}\)

⟹ m × 10 = 2\(^{2}\) and 2 × n = 10\(^{2}\)

⟹ 10m = 4 and 2n = 100

⟹ m = \(\frac{4}{10}\) and n = \(\frac{100}{2}\)

⟹ m = 0.4 and n = 50

Therefore, the value of m = 0.4 and n = 50

● **Ratio and proportion**

**Basic Concept of Ratios****Important Properties of Ratios****Ratio in Lowest Term****Types of Ratios****Comparing Ratios****Arranging Ratios****Dividing into a Given Ratio****Divide a Number into Three Parts in a Given Ratio****Dividing a Quantity into Three Parts in a Given Ratio****Problems on Ratio****Worksheet on Ratio in Lowest Term****Worksheet on Types of Ratios****Worksheet on Comparison on Ratios****Worksheet on Ratio of Two or More Quantities****Worksheet on Dividing a Quantity in a Given Ratio****Word Problems on Ratio****Proportion****Definition of Continued Proportion****Mean and Third Proportional****Word Problems on Proportion****Worksheet on Proportion and Continued Proportion****Worksheet on Mean Proportional****Properties of Ratio and Proportion**

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