Multiple choice

The ratio of age between $A$ and $B$ is $6 : 5$ and the age of each $C$ and $D$ is $\dfrac {9}{10}$ times that of $B$. Age of $F$ is less than $A$ but greater than $B$. The ratio of ages between $B$ and $E$ is $2 : 3$. Also age of $A$ is $3$ years less than $E$. What is the ratio of ages of $A$ and $F$ if all the ages are in integers?

  1. $12 : 11$
  2. $9 : 7$
  3. $24 : 19$
  4. $12 : 13$
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A Correct answer
Explanation

A/B = 6/5, C=D=0.9B. B/E = 2/3. A = E - 3. Substituting E = 1.5B into A = E - 3 gives A = 1.5B - 3. Since A/B = 6/5, A = 1.2B. Thus 1.2B = 1.5B - 3, so 0.3B = 3, B = 10. Then A = 12, E = 15. F is between A(12) and B(10), so F = 11. Ratio A:F = 12:11.

AI explanation

Using the ratio of ages between B and E of 2 to 3, let their ages be 2x and 3x. Since A is 3 years younger than E, the age of A is 3x minus 3. Using the ratio of A to B of 6 to 5, we get the equation 5 times (3x minus 3) equals 12x, which solves to x equals 5. Therefore, the ages of A and B are 12 and 10 years respectively, making the ages of C and D both 9 years. Since F is less than 12 but greater than 10 and must be an integer, F is 11 years old. The ratio of the ages of A to F is therefore 12 to 11.

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