Question

Six years hence a man's age will be three times the age of his son
and three years ago he was nine times as old as his son . the
present age of the man is
Ya) 28 (b) 32
(c) 30
(d) 34​

Answer 1

Basic Concept Used :-

Writing Systems of Linear Equations from Word Problem

1. Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find. ...

2. Translate the problem to an equation.

• Assign a variable (or variables) to represent the unknown.

• Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Let's solve the problem now!!!

$\begin{gathered}\begin{gathered}\bf\: Let-\begin{cases} &\sf{present \: age \: of \: man \: = \: x \: years} \\ &\sf{present \: age \: of \: son = \: y \: years} \end{cases}\end{gathered}\end{gathered}$

After 6 years,

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{present \: age \: of \: man \: = \: x + 6 \: years} \\ &\sf{present \: age \: of \: son = \: y + 6\: years} \end{cases}\end{gathered}\end{gathered}$

According to statement,

Father age will be three times of son age

$\rm :\longmapsto\:x + 6 = 3(y + 6)$

$\rm :\longmapsto\:x + 6 = 3y + 18$

$\bf\implies \:x = 3y + 12 - - - (1)$

3 years ago

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{present \: age \: of \: man \: = \: x - 3 \: years} \\ &\sf{present \: age \: of \: son = \: y - 3\: years} \end{cases}\end{gathered}\end{gathered}$

According to statement

Father age will be 9 times of son age

$\rm :\longmapsto\:x - 3 = 9(y - 3)$

$\rm :\longmapsto\:x - 3 = 9y -27$

$\rm :\longmapsto\:x = 9y -24$

$\rm :\longmapsto\:3y + 12 = 9y -24 \: \: \: \: \: \: \: \: \: \{ \: using \: (1) \}$

$\rm :\longmapsto\:9y - 3y = 24 + 12$

$\rm :\longmapsto\:6y= 36$

$\bf\implies \:y = 6$

On substituting y = 6 in equation (1), we get

$\rm :\longmapsto\:x = 3 \times 6 + 12$

$\rm :\longmapsto\:x = 18 + 12$

$\bf\implies \:x = 30$

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{present \: age \: of \: man \: = \: 30\: years} \\ &\sf{present \: age \: of \: son = \: 6 \: years} \end{cases}\end{gathered}\end{gathered}$

Hence, option (c) is correct.