Question

Six years ago, the ratio of the ages of Ram and Shyam was 6:5. Four years in future from today the ratio of their ages will be 11:10, what is Shyam's present age?​

Answer 1

Answer:-

$\red{\bigstar}$ Shyam's present age $\large\leadsto\boxed{\rm\green{16 \: years}}$

• Given:-

• Six years ago, the ratio of the ages of Ram and Shyam was 6:5.
• Four years in future from today the ratio of their ages will be 11:10.

• To Find:-

• The present age of Shyam.

• Solution:-

Let the present ages of Ram be 'x' and Shyam be 'y'.

• According to the question:-

✴ Six years ago ratio of Ram and Shyam's age was 6:5.

→ $\\ \sf \dfrac{x-6}{y-6} = \dfrac{6}{5}$

→ $\sf 5(x-6) = 6(y-6)$

→ $\sf 5x - 30 = 6y - 36$

→ $\sf 5x - 6y - 30 + 36 = 0$

→ $\sf 5x - 6y + 6 = 0$

→ $\bf 5x - 6y = -6 \dashrightarrow\bf\red{[eqn.i]}$

Now,

✴ Four years in future the ratio will be 11:10.

→ $\\ \sf \dfrac{x+4}{y+4} = \dfrac{11}{10}$

→ $\sf 10(x+4) = 11(y+4)$

→ $\sf 10x + 40 = 11y + 44$

→ $\sf 10x - 11y + 40 - 44 = 0$

→ $\sf 10x - 11y - 4 = 0$

→ $\sf 10x - 11y = 4 \dashrightarrow\bf\red{[eqn.ii]}$

• Multiplying eqn[i] by 2:-

→ $\sf 5x - 6y = -6$

→ $\sf (5x - 6y = -6) \times 2$

→ $\bf 10x - 12y = -12 \dashrightarrow\bf\red{[eqn.iii]}$

• Subtracting eqn[iii] from [ii]:-

→ $\sf (10x - 11y) - (10x - 12y) = 4 - (-12)$

→ $\sf 10x - 11y - 10x + 12y = 4 + 12$

★ $\large{\bf\pink{y = 16}}$

• Substituting the value of y in eqn[i]:-

→ $\sf 5x - 6y = -6$

→ $\sf 5x - 6 \times 16 = -6$

→ $\sf 5x - 96 + 6 = 0$

→ $\sf 5x - 90 = 0$

→ $\sf 5x = 90$

→ $\sf x = \dfrac{90}{5}$

★ $\large{\bf\pink{x = 18}}$

Therefore, the present ages of

• Ram = 18 years

• Shyam = 16 years