Solving for the Sum of Two Numbers

This problem asks us to find the sum of two positive numbers. We are given two pieces of information:

• The product of the two numbers is 80.
• The first number is five times the second number.

Step-by-Step Solution

Let's define the two numbers using variables to help us solve this.

1. Define Variables: Let the second positive number be $x$. According to the problem, the first number is five times the second number, so the first number is $5x$.
2. Use the Product Information: We know that the product of these two numbers is 80. We can write this as an equation: $ (5x) \times x = 80 $
3. Simplify the Equation: Multiply the terms on the left side: $ 5x^2 = 80 $
4. Solve for x: To find the value of $x$, first divide both sides by 5: $ x^2 = \frac{80}{5} $ $ x^2 = 16 $ Now, take the square root of both sides. Since we are dealing with positive numbers, we only consider the positive root: $ x = \sqrt{16} $ $ x = 4 $
5. Find the Two Numbers: Now that we know $x = 4$, we can find the values of the two numbers:
   • Second number ($x$) = 4
   • First number ($5x$) = $5 \times 4 = 20$
   So, the two positive numbers are 4 and 20.
6. Verify the Product: Let's check if their product is 80: $20 \times 4 = 80$. This is correct.
7. Calculate the Sum: The question asks for the sum of the two numbers. Add the two numbers we found: $ \text{Sum} = 20 + 4 $ $ \text{Sum} = 24 $

Conclusion

The sum of the two numbers is 24.