Question

Divide 156 in 4 parts such that they are in continued
proportion and the sum of the first and third parts is in
a ratio of 1:5 to the sum of the second and the fourth
part.​

Answer 1

Given:  Number 156.

To find: 4 numbers such that they are in continued

proportion and the sum of the first and third parts is in

a ratio of 1:5 to the sum of the second and the fourth

part.​

Solution:

Continued proportion means the numbers will be in a Geometric Progression.

i.e. The ratio between 1st and 2nd number = Ration between 2nd and 3rd number = Ratio between 3rd and 4th number.

This ratio is called as Common Ratio.

Let this ratio be $r$.

Let the first term be $a$.

2nd term = $ar$

3rd term = $ar^{2}$

4th term = $ar^{3}$

As per question:

$a+ar+ar^{2} +ar^{3} = 156 ....... (1)$

and

the sum of the first and third parts is in  a ratio of 1:5 to the sum of the second and the fourth  part.​

$\Rightarrow \dfrac{(a+ar^2)}{(ar+ar^3)} = \dfrac{1}{5}\\\Rightarrow \dfrac{a(1+r^2)}{ar(1+r^2)} = \dfrac{1}{5}\\\Rightarrow \dfrac{1}{r} = \dfrac{1}{5}\\\Rightarrow r = 5$

Putting value of $r$ in equation (1):

$a+a \times 5+a \times 5^{2} +a \times 5^{3} = 156\\\Rightarrow a(1+5+25+125) = 156\\\Rightarrow 156a = 156\\\Rightarrow a = 1$

So, the numbers are 1, 5, 25 and 125.

Answer 2

Answer:

Meaning of Continued proportion is that numbers will be in Geometric Progression(GP).