The difference between two positive numbers is 10 and the ratio between them is `5 : 3`. Find the product of the two numbers.

Options

A. 375
B. 175
C. 275
D. 125

Answer

To solve the problem, we need to find two positive numbers based on the information given: their difference is 10 and their ratio is 5:3. We will denote the two numbers as \( a \) and \( b \).

### Step-by-Step Solution:

1. **Set Up the Variables**:
- Let the first number be \( a = 5x \).
- Let the second number be \( b = 3x \).
- Here, \( x \) is a common multiplier.

**Hint**: When given a ratio, you can express the numbers in terms of a variable (like \( x \)).

2. **Use the Information About the Difference**:
- According to the problem, the difference between the two numbers is 10:
\[ a - b = 10 \]
- Substituting the expressions for \( a \) and \( b \):
\[ 5x - 3x = 10 \]

**Hint**: Always substitute the expressions you have set up into the equation provided by the problem.

3. **Simplify the Equation**:
- Simplifying the left side gives:
\[ 2x = 10 \]

**Hint**: Combine like terms to simplify your equations.

4. **Solve for \( x \)**:
- Divide both sides by 2:
\[ x = 5 \]

**Hint**: Isolate the variable by performing the same operation on both sides of the equation.

5. **Find the Two Numbers**:
- Substitute \( x \) back into the expressions for \( a \) and \( b \):
\[ a = 5x = 5 \times 5 = 25 \]
\[ b = 3x = 3 \times 5 = 15 \]

**Hint**: Once you find \( x \), plug it back into your expressions to find the actual numbers.

6. **Calculate the Product of the Two Numbers**:
- Now, find the product of \( a \) and \( b \):
\[ a \times b = 25 \times 15 \]
- Calculate the product:
\[ 25 \times 15 = 375 \]

**Hint**: To find the product, simply multiply the two numbers you have found.

### Final Answer:
The product of the two numbers is **375**.