Question

The ratio of ages (in years) of 14 and BIO years ago, was 23:12. The ratio of their ages after 15 ve
ye
be 35:24. What is the ratio of present ages of A and B ?​

Answer 1

Answer:-

Given:

Ratio of ages of A & B before 9 years = 23 : 12

Ratio of ages of A & B after 15 years = 35 : 24

Let the age of A be A years and that of B be B years.

• Age of A before 9 years = (A - 9) years

• Age of B before 9 years = (B - 9) years.

So,

$\implies \sf \: \dfrac{A - 9}{B - 9} = \dfrac{23}{12} \\ \\ \implies \sf \: 12(A - 9) = 23(B - 9) \\ \\ \implies \sf \:12A - 108 = 23B - 207 \\ \\ \implies \sf \:12A = 23B - 207 + 108 \\ \\ \implies \sf \:12A= 23B - 99 \\ \\ \implies \boxed{ \sf \:A = \dfrac{23B - 99}{12} - - \: equation(1) }$

Also,

• A's age after 15 years = (A + 15) years.

• B's age after 15 years = (B + 15) years.

$\ \: \implies \sf \: \frac{A + 15}{B + 15} = \frac{35}{24} \\ \\ \implies \sf \:24(A + 15) = 35(B + 15) \\ \\ \implies \sf \:24A + 360 = 35B + 525$

Substitute the value of A from equation (1).

$\implies \sf \:24 \bigg( \dfrac{23B - 99}{12} \bigg) + 360 = 35B + 525 \\ \\ \implies \sf \:46B- 198 + 360 = 35B + 525 \\ \\ \implies \sf \:46B - 35B = 525 + 198 - 360 \\ \\ \implies \sf \:11B = 363 \\ \\ \implies \sf \:B = \frac{363}{11} \\ \\ \implies \boxed{ \sf \:B = 33}$

Substitute the value of B in equation (1).

$\ \: \implies \sf \: A = \frac{23(33) - 99}{12} \\ \\ \implies \sf \: A = \frac{759 - 99}{12} \\ \\ \implies \sf \: A = \frac{660}{12} \\ \\ \implies \boxed{ \sf \: A = 55}$

Hence,

• Present age of A = 55 years

• Present age of B = 33 years

Now,

Ratio of A's age to B's age = 55/33 = 5 : 3

∴ The ratio of ages of A & B is 5 : 3.

Answer 2

$\sf Question :$

The ratio of ages (in years) of A and B 9 years ago, was 23:12. The ratio of their ages after 15 years will be 35:24. What is the ratio of present ages of A and B?

$\sf Answer :$

Let the Age of A be 23x and of B be 12x 9 years ago from Present Age.

$\underline{\bigstar\:\textsf{According to the given Question :}}$

$:\implies\sf \dfrac{A+9+15}{B+9+15}=\dfrac{35}{24}\\\\\\:\implies\sf \dfrac{23x+9+15}{12x+9+15}=\dfrac{35}{24}\\\\\\:\implies\sf \dfrac{23x+24}{12x+24}=\dfrac{35}{24}\\\\\\:\implies\sf \dfrac{23x+24}{12(x+2)}=\dfrac{35}{24}\\\\\\:\implies\sf \dfrac{23x+24}{x+2}=\dfrac{35}{2}\\\\\\:\implies\sf (23x + 24) \times 2 = 35 \times (x + 2)\\\\\\:\implies\sf 46x + 48 = 35x + 70\\\\\\:\implies\sf 46x - 35x = 70 - 48\\\\\\:\implies\sf 11x = 22\\\\\\:\implies\sf x = \dfrac{22}{11}\\\\\\:\implies\sf x = 2$

$\rule{180}{1.5}$

$\underline{\bigstar\:\textsf{Required ratio of Present Age of A \& B:}}$

$\dashrightarrow\sf \:\:\dfrac{A}{B}=\dfrac{23x+9}{12x+9}\\\\\\\dashrightarrow\sf \:\: \dfrac{A}{B}=\dfrac{23(2)+9}{12(2)+9}\\\\\\\dashrightarrow\sf \:\: \dfrac{A}{B}=\dfrac{46+9}{24+9}\\\\\\\dashrightarrow\sf \:\: \dfrac{A}{B}=\dfrac{55}{33}\\\\\\\dashrightarrow\sf \:\: \dfrac{A}{B}=\dfrac{5}{3}\\\\\\\dashrightarrow \: \:\underline{\boxed{\sf A:B = 5:3}}$