At present, the respective ratio between the ages of A and B is 2:9 and that between A and C is 1:5. 4 years ago, the sum of the ages of A, B, and C was 72 years. What is the prese... At present, the respective ratio between the ages of A and B is 2:9 and that between A and C is 1:5. 4 years ago, the sum of the ages of A, B, and C was 72 years. What is the present age of C? Read more

Steps to Solve

1. Define Variables for Ages Let the present ages of A, B, and C be represented as:

• $A = 2x$ (based on the ratio of A to B)
• $B = 9x$
• $C = 5y$ (based on the ratio of A to C)

2. Establish Relationships Between Variables From the ratio $A:C = 1:5$, we can express $y$ in terms of $x$ by using the relationship from the first equation: $$ \frac{A}{C} = \frac{2x}{5y} = \frac{1}{5} \implies y = \frac{2x}{5}. $$

3. Calculate Present Ages of A, B, and C Substituting $y = \frac{2x}{5}$ into the expression for $C$: $$ C = 5y = 5\left(\frac{2x}{5}\right) = 2x. $$ Now we have:

• $A = 2x$
• $B = 9x$
• $C = 2x$

4. Use Information About Ages 4 Years Ago Four years ago, the ages will be:

• $A - 4 = 2x - 4$
• $B - 4 = 9x - 4$
• $C - 4 = 2x - 4$

According to the problem, the sum of their ages was 72 years: $$ (A - 4) + (B - 4) + (C - 4) = 72. $$ This simplifies to: $$ (2x - 4) + (9x - 4) + (2x - 4) = 72. $$

5. Solve for x Combining the equation: $$ 2x + 9x + 2x - 12 = 72. $$ Simplified gives: $$ 13x - 12 = 72 \implies 13x = 84 \implies x = \frac{84}{13}. $$

6. Calculate Present Age of C Substituting $x$ back into the expression for $C$: $$ C = 2x = 2 \times \frac{84}{13} = \frac{168}{13} \approx 12.92. $$ Therefore: $$ C \approx 12.92. $$