Question

The sum of age of four brothers A, B, C and D is 4.5 times the age of C. The sum of age of B. C and D is 2 times the age of A. The ratio of age of A to C is​

Answer 1

★ Concept

Here, the above question (general age problem) is based on the concept of Ratio and Proportion. It's given that the sum of ages of four brothers is 4.5 times the age of 'C'. So, let the ages of these four brothers A, B, C and D be 'a', 'b', 'c', & 'd' respectively. Back to the question, From the very first condition, we have eqⁿ a + b + c + d = 4.5c. Moreover, it's futher stated that the sum of age of B, C and D is 2 times the age of 'A', this means another equation (b + c + d = 2a). Now, we get two equations, by substituting the values from eqⁿ (2) in eqⁿ (1) we get the ratio of ages of A to C.

Let's proceed with Calculation !!

$\rule{190pt}{1pt}$

Assume that the ages of these Four Brothers namely A, B, C & D is 'a', 'b', 'c', & 'd' respectively.

$\sf{ Condition : \orange{1}}$

The sum of ages of four brothers is 4.5 times the age of 'C'. Mathematically,

$\sf a + b + c + d = 4.5c \rightarrow( \red1)$

$\sf{ Condition : \orange{2}}$

The sum of age of B, C and D is 2 times the age of 'A'. Mathematically,

$\sf \: b + c + d = 2a \: \rightarrow( \red2)$

Substitute the values of eqⁿ (2) in eqⁿ (1), we have

$\sf a + (\purple{b + c + d}) = 4.5c$

$\sf a + 2a= 4.5c$

$\sf \: 3a \: = 4.5c$

Therefore, the ratio of ages of A to C is thus,

$\sf \dfrac{a}{c} = \dfrac{4.5}{3}$

$\sf \dfrac{a}{c} = \dfrac{ \cancel{45}}{ \cancel{30}} \: = \underline{\boxed{ \bf\red{\dfrac{3}{2} }}}$

$\sf \: a : c \: = 3 : 2$

The Ratio of age of A to C is 3 : 2.

$\underline{ \rule{190pt}{2pt}}$

Additional Information

Ratio is a comparison of two quantities of similar kind having same units. The general structure for ratio is a : b (read as a 'is to' b) - 'a' and 'b' are the terms of the ratio, 'antecedent' and 'consequent'. Below are provided some important points :-

1) Ratio has no units.

2) If the terms of ratio are multiplied or divided by the same non zero number, the ratio remain unchanged or unaltered.

3) In the above question, the assumption of variables totally depends on us (You can take another variables for the representation of their respective ages).