Five years ago, A age was 1/3 of B's age at that time. The ratio of B's age six years hence to A's age twelve years hence will be 7:4 .What will be the ratio between A's age three years ago and B's age three years after ?

Options

A. `17:31`
B. `17:58`
C. `17:53`
D. `3:4`

Explanation

To solve the problem step by step, we will define the ages of A and B and use the given conditions to form equations.

### Step 1: Define Variables
Let:
- A's age 5 years ago = \( x \)
- B's age 5 years ago = \( 3x \) (since A's age was \( \frac{1}{3} \) of B's age)

### Step 2: Express Present Ages
Now, we can express their present ages:
- A's present age = \( x + 5 \)
- B's present age = \( 3x + 5 \)

### Step 3: Set Up the Second Condition
According to the problem, the ratio of B's age 6 years hence to A's age 12 years hence is \( 7:4 \).
- B's age 6 years hence = \( (3x + 5) + 6 = 3x + 11 \)
- A's age 12 years hence = \( (x + 5) + 12 = x + 17 \)

We can set up the equation based on the ratio:
\[ \frac{3x + 11}{x + 17} = \frac{7}{4} \]

### Step 4: Cross-Multiply and Solve for \( x \)
Cross-multiplying gives us:
\[ 4(3x + 11) = 7(x + 17) \]
Expanding both sides:
\[ 12x + 44 = 7x + 119 \]
Now, rearranging gives:
\[ 12x - 7x = 119 - 44 \]
\[ 5x = 75 \]
\[ x = 15 \]

### Step 5: Calculate Present Ages
Now that we have \( x \):
- A's present age = \( 15 + 5 = 20 \)
- B's present age = \( 3(15) + 5 = 45 + 5 = 50 \)

### Step 6: Find Ages 3 Years Ago and 3 Years Hence
- A's age 3 years ago = \( 20 - 3 = 17 \)
- B's age 3 years hence = \( 50 + 3 = 53 \)

### Step 7: Calculate the Ratio
Now we can find the ratio of A's age 3 years ago to B's age 3 years hence:
\[ \text{Ratio} = \frac{A's \, age \, 3 \, years \, ago}{B's \, age \, 3 \, years \, hence} = \frac{17}{53} \]

### Final Answer
The ratio between A's age three years ago and B's age three years hence is \( \frac{17}{53} \).

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