# Continued Proportion

Quantities are said to be in continued proportion if the first is related to the second, the second is related to the third and the third is related to the forth, and so on.

If $a,b,c,d,e$ are in continued proportion, then

$\frac{{\text{a}}}{{\text{b}}} = \frac{{\text{b}}}{{\text{c}}} = \frac{{\text{c}}}{{\text{d}}} = \frac{{\text{d}}}{{\text{e}}}$

Example:
Three people A, B and C earned a profit of 70000 Rs in a business. Their share of the profit is as follows

A : B = 4 : 2
B : C = 10 : 5
Find the share of each person.

Solution:
Total amount of profit = 70000

Ratio of A, B and C is
$\underline {\begin{array}{*{20}{c}} {\text{A}}&{\text{:}}&{\text{B}}&{\text{:}}&{\text{C}} \\ {\text{4}}&{\text{:}}&{\text{2}}&{}&{} \\ {}&{}&{{\text{10}}}&{\text{:}}&{\text{5}} \end{array}}$
$\underline {\begin{array}{*{20}{c}} {40}&:&{20}&:&{10} \end{array}}$

Sum of ratio = 40 + 20 + 10 = 70

Share of A = $\frac{{40}}{{70}}$x 70000 = 40000

Share of B = $\frac{{20}}{{70}}$x 70000 = 20000

Share of C = $\frac{{10}}{{70}}$x 70000 = 10000

Example:
Divide 2562 among three friends X, Y and Z, such that the ratio between their share is
X : Y = 4 : 5
Y : Z = 8 : 10

Solution:
Let the amount to be divided = 2562

Ratio of their share
$\underline {\begin{array}{*{20}{c}} {\text{X}}&{\text{:}}&{\text{Y}}&{\text{:}}&{\text{Z}} \\ {\text{4}}&{\text{:}}&{\text{5}}&{}&{} \\ {}&{}&{\text{8}}&{\text{:}}&{{\text{10}}} \end{array}}$
$\underline {\begin{array}{*{20}{c}} {32}&{}&:&{}&{40}&{}&:&{}&{50} \end{array}}$

Sum of ratios = 32 + 40 + 50 + 122

Share of X = $\frac{{32}}{{122}}$x 2562 = 672

Share of Y = $\frac{{40}}{{122}}$x 2562 = 840

Share of Z = $\frac{{50}}{{122}}$x 2562 = 1052