The ratio between the ages of A and B is 3 : 5 and the sum of their ages is 56 Yr. The ratio between their ages 7 yr ago was

Options

A. `3 :4`
B. `3 : 5`
C. `1 : 2`
D. `2 : 3`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the ages of A and B
Let the age of A be \(3x\) and the age of B be \(5x\), where \(x\) is a common multiplier.

### Step 2: Set up the equation for the sum of their ages
According to the problem, the sum of their ages is 56 years. Therefore, we can write the equation:
\[ 3x + 5x = 56 \]

### Step 3: Solve for \(x\)
Combine like terms:
\[ 8x = 56 \]
Now, divide both sides by 8:
\[ x = \frac{56}{8} = 7 \]

### Step 4: Find the actual ages of A and B
Now that we have \(x\), we can find the ages:
- Age of A = \(3x = 3 \times 7 = 21\) years
- Age of B = \(5x = 5 \times 7 = 35\) years

### Step 5: Calculate their ages 7 years ago
Now, we need to find their ages 7 years ago:
- Age of A 7 years ago = \(21 - 7 = 14\) years
- Age of B 7 years ago = \(35 - 7 = 28\) years

### Step 6: Find the ratio of their ages 7 years ago
Now we can find the ratio of their ages 7 years ago:
\[ \text{Ratio} = \frac{\text{Age of A 7 years ago}}{\text{Age of B 7 years ago}} = \frac{14}{28} \]
This simplifies to:
\[ \frac{14}{28} = \frac{1}{2} \]

### Final Answer
The ratio between the ages of A and B 7 years ago is \(1 : 2\).

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