The sum of two numbers is 405 and their ratio is `8:7`. Find the numbers.

Answer

To solve the problem, we need to find two numbers based on the information provided: their sum is 405, and their ratio is 8:7. We will follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the two numbers be \( a \) and \( b \).

2. **Set Up the Equations**:
We know from the problem statement that:
- The sum of the two numbers is 405:
\[ a + b = 405 \quad \text{(Equation 1)} \]
- The ratio of the two numbers is 8:7. This can be expressed as:
\[ \frac{a}{b} = \frac{8}{7} \quad \text{(Equation 2)} \]

3. **Express One Variable in Terms of the Other**:
From Equation 2, we can express \( a \) in terms of \( b \):
\[ a = \frac{8}{7}b \]

4. **Substitute into the First Equation**:
Substitute the expression for \( a \) from Equation 2 into Equation 1:
\[ \frac{8}{7}b + b = 405 \]
To combine the terms, we can express \( b \) as \( \frac{7}{7}b \):
\[ \frac{8}{7}b + \frac{7}{7}b = 405 \]
This simplifies to:
\[ \frac{15}{7}b = 405 \]

5. **Solve for \( b \)**:
Multiply both sides by 7 to eliminate the fraction:
\[ 15b = 405 \times 7 \]
Calculate \( 405 \times 7 \):
\[ 15b = 2835 \]
Now, divide both sides by 15:
\[ b = \frac{2835}{15} = 189 \]

6. **Find \( a \)**:
Now that we have \( b \), we can find \( a \) using the expression we derived earlier:
\[ a = \frac{8}{7}b = \frac{8}{7} \times 189 \]
Calculate \( a \):
\[ a = \frac{8 \times 189}{7} = 216 \]

7. **Final Answer**:
The two numbers are:
\[ a = 216 \quad \text{and} \quad b = 189 \]

### Verification:
- Check the sum:
\[ 216 + 189 = 405 \quad \text{(Correct)} \]
- Check the ratio:
\[ \frac{216}{189} = \frac{8}{7} \quad \text{(Correct)} \]