Difference of two numbers is 1600. If `7.5%` of one number is `12.5%` of the other number, find the two numbers?

Options

A. A)`2490, 4000`
B. B)`2400, 4150`
C. C)`2490, 4150`
D. D)`2400, 4000`

Answer

To solve the problem step by step, we will denote the two numbers as \( A \) and \( B \).

### Step 1: Set up the equations based on the problem statement.
We know from the problem that the difference between the two numbers is 1600. Therefore, we can write the first equation as:
\[ A - B = 1600 \quad \text{(1)} \]

### Step 2: Set up the second equation using the percentage condition.
The problem states that \( 7.5\% \) of one number is equal to \( 12.5\% \) of the other number. We can express this mathematically as:
\[ 0.075A = 0.125B \quad \text{(2)} \]

### Step 3: Simplify the second equation.
To eliminate the decimals, we can multiply both sides of equation (2) by 1000:
\[ 75A = 125B \]
Now, we can simplify this equation by dividing both sides by 25:
\[ 3A = 5B \quad \text{(3)} \]

### Step 4: Express one variable in terms of the other.
From equation (3), we can express \( A \) in terms of \( B \):
\[ A = \frac{5}{3}B \quad \text{(4)} \]

### Step 5: Substitute equation (4) into equation (1).
Now, we can substitute equation (4) into equation (1):
\[ \frac{5}{3}B - B = 1600 \]
To combine the terms, we can express \( B \) as \( \frac{3}{3}B \):
\[ \frac{5}{3}B - \frac{3}{3}B = 1600 \]
This simplifies to:
\[ \frac{2}{3}B = 1600 \]

### Step 6: Solve for \( B \).
To find \( B \), multiply both sides by \( \frac{3}{2} \):
\[ B = 1600 \times \frac{3}{2} = 2400 \]

### Step 7: Substitute \( B \) back to find \( A \).
Now that we have \( B \), we can substitute it back into equation (4) to find \( A \):
\[ A = \frac{5}{3} \times 2400 = 4000 \]

### Conclusion:
The two numbers are:
\[ A = 4000 \quad \text{and} \quad B = 2400 \]