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Question Number 9172 by nazar last updated on 22/Nov/16

The present age of a father and that of his son are in the ratio 7 : 1. After 4 years, the ratio will be 4 : 1. What is the son′s present age (in years) ?

$$\mathrm{The}\:\mathrm{present}\:\mathrm{age}\:\mathrm{of}\:\mathrm{a}\:\mathrm{father}\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{of}\:\mathrm{his}\:\mathrm{son}\:\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{7}\::\:\mathrm{1}.\:\mathrm{After} \\ $$$$\mathrm{4}\:\mathrm{years},\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{will}\:\mathrm{be}\:\mathrm{4}\::\:\mathrm{1}.\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{son}'\mathrm{s}\:\mathrm{present}\:\mathrm{age}\:\left(\mathrm{in}\:\mathrm{years}\right)\:? \\ $$

Commented by tawakalitu last updated on 22/Nov/16

Let the age of the father be = x Let the age of the son be = y From the first statement: x : y = 7 : 1 ∴ x/y = 7/1 ∴ x = 7y From the second statement: (x + 4) : (y + 4) = 4 : 1 Therefore, ((x + 4)/(y + 4)) = (4/1) Cross multiply x + 4 = 4(y + 4) 7y + 4 = 4y + 16 [Note: x = 7y] Collect the like terms 7y − 4y = 16 − 4 ∴ 3y = 12 Divide both side by 3 ∴ y = 12/3 ∴ y = 4 Since, x = 7y and y = 4 ∴ x = 7(4) ∴ x = 28 Hence, the son′s present age is 4 years old. DONE !

$$\mathrm{Let}\:\mathrm{the}\:\mathrm{age}\:\mathrm{of}\:\mathrm{the}\:\mathrm{father}\:\mathrm{be}\:=\:\mathrm{x} \\ $$$$\mathrm{Let}\:\mathrm{the}\:\mathrm{age}\:\mathrm{of}\:\mathrm{the}\:\mathrm{son}\:\mathrm{be}\:=\:\mathrm{y} \\ $$$$\mathrm{From}\:\mathrm{the}\:\mathrm{first}\:\mathrm{statement}:\:\mathrm{x}\::\:\mathrm{y}\:=\:\mathrm{7}\::\:\mathrm{1} \\ $$$$\therefore\:\mathrm{x}/\mathrm{y}\:=\:\mathrm{7}/\mathrm{1} \\ $$$$\therefore\:\mathrm{x}\:=\:\mathrm{7y} \\ $$$$\mathrm{From}\:\mathrm{the}\:\mathrm{second}\:\mathrm{statement}: \\ $$$$\left(\mathrm{x}\:+\:\mathrm{4}\right)\::\:\left(\mathrm{y}\:+\:\mathrm{4}\right)\:=\:\mathrm{4}\::\:\mathrm{1} \\ $$$$\mathrm{Therefore}, \\ $$$$\frac{\mathrm{x}\:+\:\mathrm{4}}{\mathrm{y}\:+\:\mathrm{4}}\:=\:\frac{\mathrm{4}}{\mathrm{1}} \\ $$$$\mathrm{Cross}\:\mathrm{multiply} \\ $$$$\mathrm{x}\:+\:\mathrm{4}\:=\:\mathrm{4}\left(\mathrm{y}\:+\:\mathrm{4}\right) \\ $$$$\mathrm{7y}\:+\:\mathrm{4}\:=\:\mathrm{4y}\:+\:\mathrm{16}\:\:\:\:\:\:\:\:\:\:\:\left[\mathrm{Note}:\:\mathrm{x}\:=\:\mathrm{7y}\right] \\ $$$$\mathrm{Collect}\:\mathrm{the}\:\mathrm{like}\:\mathrm{terms} \\ $$$$\mathrm{7y}\:−\:\mathrm{4y}\:=\:\mathrm{16}\:−\:\mathrm{4} \\ $$$$\therefore\:\mathrm{3y}\:=\:\mathrm{12} \\ $$$$\mathrm{Divide}\:\mathrm{both}\:\mathrm{side}\:\mathrm{by}\:\mathrm{3} \\ $$$$\therefore\:\:\mathrm{y}\:=\:\mathrm{12}/\mathrm{3} \\ $$$$\therefore\:\:\mathrm{y}\:=\:\mathrm{4} \\ $$$$\mathrm{Since},\:\mathrm{x}\:=\:\mathrm{7y}\:\:\mathrm{and}\:\:\mathrm{y}\:=\:\mathrm{4} \\ $$$$\therefore\:\:\:\mathrm{x}\:=\:\mathrm{7}\left(\mathrm{4}\right) \\ $$$$\therefore\:\:\:\mathrm{x}\:=\:\mathrm{28} \\ $$$$\mathrm{Hence},\:\mathrm{the}\:\mathrm{son}'\mathrm{s}\:\mathrm{present}\:\mathrm{age}\:\mathrm{is}\:\mathrm{4}\:\mathrm{years}\:\mathrm{old}. \\ $$$$\mathrm{DONE}\:! \\ $$

Answered by RasheedSoomro last updated on 23/Nov/16

This problem also can be solved using only one variable. At present Let the son′s age is x years The father′s age will be 7x years. After four years The father′s and son′s ages will be 7x+4 and x+4 respectively. Their ratio will be 4 : 1 7x+4 : x+4 = 4 : 1 7x+4=4x+16 3x=12 x=4 (The son′s present age) 7x=28 (The father′s present age

$$\mathrm{This}\:\mathrm{problem}\:\mathrm{also}\:\mathrm{can}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{using} \\ $$$$\mathrm{only}\:\mathrm{one}\:\mathrm{variable}. \\ $$$$\mathrm{At}\:\mathrm{present} \\ $$$$\mathrm{Let}\:\mathrm{the}\:\mathrm{son}'\mathrm{s}\:\:\mathrm{age}\:\mathrm{is}\:\:\:\mathrm{x}\:\:\mathrm{years} \\ $$$$\mathrm{The}\:\mathrm{father}'\mathrm{s}\:\:\mathrm{age}\:\mathrm{will}\:\mathrm{be}\:\:\mathrm{7x}\:\mathrm{years}. \\ $$$$ \\ $$$$\mathrm{After}\:\mathrm{four}\:\mathrm{years} \\ $$$$\mathrm{The}\:\mathrm{father}'\mathrm{s}\:\mathrm{and}\:\mathrm{son}'\mathrm{s}\:\mathrm{ages}\:\mathrm{will}\:\mathrm{be}\:\mathrm{7x}+\mathrm{4}\:\mathrm{and}\:\mathrm{x}+\mathrm{4} \\ $$$$\mathrm{respectively}. \\ $$$$\mathrm{Their}\:\mathrm{ratio}\:\mathrm{will}\:\mathrm{be}\:\:\:\mathrm{4}\::\:\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{7x}+\mathrm{4}\::\:\mathrm{x}+\mathrm{4}\:=\:\mathrm{4}\::\:\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{7x}+\mathrm{4}=\mathrm{4x}+\mathrm{16} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3x}=\mathrm{12} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}=\mathrm{4}\:\:\:\left(\mathrm{The}\:\mathrm{son}'\mathrm{s}\:\mathrm{present}\:\mathrm{age}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{7x}=\mathrm{28}\:\left(\mathrm{The}\:\mathrm{father}'\mathrm{s}\:\mathrm{present}\:\mathrm{age}\right. \\ $$

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