Question

Shalin's mother is thrice the age of Shalin. After 5 years, the ratio of their ages will be 5:2. Find their present ages.

Answer 1

Hint:
1) Let's assume that their present ages are x years and y years.
2) After 5 years, their ages will then become \[\left( {x{\text{ }} + {\text{ }}5} \right)\] years and \[\left( {y{\text{ }} + {\text{ }}5} \right)\] years respectively.
3) We can form equations using the given information and solve them to get the values of x and y.

Complete step by step solution:
Let's say that Shalin's current age is x years and her mother's current age is y years.
Therefore, in 5 years, their ages will become \[\left( {x{\text{ }} + {\text{ }}5} \right)\] years and \[\left( {y{\text{ }} + {\text{ }}5} \right)\] years respectively.
The following table lists both their ages for easier comparison:

	Shalin	Shalin's Mother	Relation
Current age:	\[x\]	$y$	$3x = y$
After 5 years:	\[\left( {x{\text{ }} + {\text{ }}5} \right)\]	\[\left( {y{\text{ }} + {\text{ }}5} \right)\]	\[\left( {x{\text{ }} + {\text{ }}5} \right):\left( {y{\text{ }} + {\text{ }}5} \right){\text{ }} = {\text{ }}2:5\]

Using the given relations:
Shalin's mother is currently 3 times as Shalin:
⇒\[y{\text{ }} = {\text{ }}3x\] ... (1)
After 5 years, the ratio of her mother's age to Shalin's age becomes 5:2.
⇒ \[\left( {x{\text{ }} + {\text{ }}5} \right):\left( {y{\text{ }} + {\text{ }}5} \right){\text{ }} = {\text{ }}2:5\]
⇒ $\frac{{x + 5}}{{y + 5}} = \frac{2}{5}$
On multiplying both sides by \[5\left( {y{\text{ }} + {\text{ }}5} \right),\] we will get:
⇒ \[5\left( {x{\text{ }} + {\text{ }}5} \right){\text{ }} = {\text{ }}2\left( {y{\text{ }} + {\text{ }}5} \right)\]
⇒ \[5x{\text{ }} + {\text{ }}25{\text{ }} = {\text{ }}2y{\text{ }} + {\text{ }}10\]
⇒ \[5x{\text{ }} - {\text{ }}2y{\text{ }} = {\text{ }} - 15\] ... (2)
Substituting the value of y from equation (1) in equation (2), we get:
⇒ \[5x{\text{ }} - {\text{ }}2\left( {3x} \right){\text{ }} = {\text{ }} - 15\]
⇒ \[5x{\text{ }} - {\text{ }}6x{\text{ }} = - 15\]
⇒ \[ - x{\text{ }} = {\text{ }} - 15\]
⇒ \[x{\text{ }} = {\text{ }}15\]

∴ Shalin's current age is x = 15 years and Shalin's mother's current age is \[y{\text{ }} = {\text{ }}3x{\text{ }} = {\text{ }}3\left( {15} \right){\text{ }} = {\text{ }}45\] years.

Note:
1) The difference between people's age always remains a constant; currently, in the past or in the future.
2) With increasing age (in the future), the ratio between the ages gets smaller.
3) The question can also be solved by using only a single variable.