The ratio of the ages of A and B seven years ago was `3:4` respectively. The ratio of their ages nine from now will be `7:8` respectively. What is B's age at present ?

Options

A. 16 years
B. 19 years
C. 28 years
D. 23 years

Explanation

To solve the problem, we will follow these steps:

### Step 1: Set up the variables
Let A's present age be \( X \) and B's present age be \( Y \).

### Step 2: Write the first equation based on the information given
According to the problem, the ratio of A's and B's ages seven years ago was \( 3:4 \). This can be expressed mathematically as:
\[ \frac{X - 7}{Y - 7} = \frac{3}{4} \]
Cross-multiplying gives us:
\[ 4(X - 7) = 3(Y - 7) \]
Expanding this, we get:
\[ 4X - 28 = 3Y - 21 \]
Rearranging it results in:
\[ 4X - 3Y = 7 \quad \text{(Equation 1)} \]

### Step 3: Write the second equation based on future ages
The problem also states that the ratio of their ages nine years from now will be \( 7:8 \). This can be expressed as:
\[ \frac{X + 9}{Y + 9} = \frac{7}{8} \]
Cross-multiplying gives us:
\[ 8(X + 9) = 7(Y + 9) \]
Expanding this, we get:
\[ 8X + 72 = 7Y + 63 \]
Rearranging it results in:
\[ 8X - 7Y = -9 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( 4X - 3Y = 7 \) (Equation 1)
2. \( 8X - 7Y = -9 \) (Equation 2)

To eliminate \( X \), we can multiply Equation 1 by 2:
\[ 8X - 6Y = 14 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (8X - 6Y) - (8X - 7Y) = 14 - (-9) \]
This simplifies to:
\[ Y = 23 \]

### Step 5: Find B's present age
From our calculations, we find that B's present age \( Y \) is:
\[ Y = 23 \]

### Final Answer
B's present age is **23 years**.

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