Question

find the ratio of the ages of a and b is 9:4 after 7 years this ratio becomes 5:3 find the present age​

Answer 1

$\bold{\underline{\underline{Answer:}}}$

Present age of A = 18 years

Present age of B = 8 years.

$\bold{\underline{\underline{Step\:by\:step\:explanation:}}}$

Given :

• The ratio of the ages of A and B is 9:4
• After 7 years this ratio becomes 5:3

To find :

• Present age of A
• Present age of B

Solution :

Let x be the common multiple of the ratio 9:4

•°• Present age of A = 9x years

Present age of B = 4x years

$\bold{\underline{\underline{As\:per\:the\:question:}}}$

• After 7 years this ratio becomes 5:3

Ages after 7 years :

Age of A = 9x + 7 years

Age of B = 4x + 7 years

Ratio = 5:3

Constituting it mathematically,

$\rightarrow$ $\bold{\dfrac{9x+7}{4x+7}}$ = $\bold{\dfrac{5}{3}}$

Cross multiplying,

$\rightarrow$ $\bold{3(9x+7)=5(4x+7)}$

$\rightarrow$ $\bold{27x+21=20x+35}$

$\rightarrow$ $\bold{27x-20x=35-21}$

$\rightarrow$ $\bold{7x=14}$

$\rightarrow$ $\bold{x={\dfrac{14}{7}}}$

$\rightarrow$ $\bold{x=2}$

Substitute x = 2 in value of the ratios,

Present age of A,

$\rightarrow$9x

Substitute value of x,

$\rightarrow$ $\bold{9\times\:2}$

$\rightarrow$ $\bold{18}$

•°• Present age of A = 18 years

Present age of B,

$\rightarrow$ 4x

$\rightarrow$ $\bold{4\times\:2}$

$\rightarrow$ $\bold{8}$

•°• Present age of B = 8 years.

Answer 2

$\bold{\underline {\underline {\red{Method \:1}}}}$

Let present age of A be x years and present age of B be y years.

Ratio of ages of A and B is 9:4.

=> $\sf{\dfrac{x}{y}\:=\:\dfrac{9}{4}}$

=> $\sf{x\:=\:\dfrac{9y}{4}}$ ___ (eq 1)

After 7 years -

• Age of A = (x + 7) years
• Age of B = (y + 7) years

According to question,

=> $\sf{\dfrac{x\:+\:7}{y\:+\:7}\:=\:\dfrac{5}{3}}$

=> $\sf{3x\:+\:21\:=\:5y\:+\:35}$

=> $\sf{3(\frac{9y}{4})\:+\:21\:=\:5y\:+\:35}$

=> $\sf{\frac{27y}{4}\:+\:21\:=\:5y\:+\:35}$

=> $\sf{\frac{27y}{4}\:-\:21\:=\:35\:-\:21}$

=> $\sf{\frac{27y\:-\:20y}{4}\:=\:14}$

=> $\sf{7y\:=\:56}$

=> $\sf{y\:=\:8}$

Substitute value of y in (eq 1)

=> $\sf{x\:=\:\dfrac{9(8)}{4}}$

=> $\sf{x\:=\:18}$

•°• Present age of A is 18 years.

Present age of B is 8 years.

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$\bold{\underline {\underline {\red{Method \:2}}}}$

Let present age of A be 9M years and present age of B be 4M years.

After 7 years, the ratio of their ages becomes 5:3.

After 7 years -

• Age of A = (9M + 7) years
• Age of B = (4M + 7) years

According to question,

=> $\sf{\dfrac{9M\:+\:7}{4M\:+\:7}\:=\:\dfrac{5}{3}}$

=> $\sf{3(9M\:+\:7)\:=\:5(4M\:+\:7)}$

=> $\sf{27M\:+\:21\:=\:20M\:+\:35}$

=> $\sf{27M\:-\:20M\:=\:35\:-\:21}$

=> $\sf{7M\:=\:14}$

=> $\sf{M\:=\:2}$

Present age of A = 9M

=> 9(2)

=> 18

•°• Present age of A is 18 years.

Present age of B = 4M

=> 4(2)

=> 8

•°• Present age of B is 8 years.