what multiplies to 24 and adds to

Natashya

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22-12-2024

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What Two Numbers Multiply to 24 and Add to 11? Finding Factor Pairs

This article will guide you through the process of finding two numbers that satisfy the conditions: their product is 24, and their sum is 11. We'll explore different methods to solve this type of mathematical problem, which is common in algebra and number theory.

Understanding the Problem

The problem presents two key pieces of information:

1. Product: The two numbers multiply to 24 (24 is the product).
2. Sum: The two numbers add up to 11 (11 is the sum).

Method 1: Listing Factors

The most straightforward approach involves listing the factor pairs of 24:

• 1 x 24
• 2 x 12
• 3 x 8
• 4 x 6

Now, let's check the sum of each pair:

• 1 + 24 = 25
• 2 + 12 = 14
• 3 + 8 = 11
• 4 + 6 = 10

The pair 3 and 8 satisfies both conditions: 3 x 8 = 24 and 3 + 8 = 11.

Therefore, the two numbers are 3 and 8.

Method 2: Algebraic Approach

We can solve this using a system of two equations:

• Let 'x' and 'y' represent the two numbers.
• Equation 1: x * y = 24 (Product)
• Equation 2: x + y = 11 (Sum)

Solving this system of equations requires substituting one equation into the other. Let's solve Equation 2 for x:

• x = 11 - y

Now, substitute this value of 'x' into Equation 1:

• (11 - y) * y = 24

Expand and rearrange the equation into a quadratic equation:

• 11y - y² = 24
• y² - 11y + 24 = 0

This quadratic equation can be factored:

• (y - 3)(y - 8) = 0

This gives us two possible solutions for 'y': y = 3 or y = 8.

If y = 3, then x = 11 - 3 = 8. If y = 8, then x = 11 - 8 = 3.

Again, we find that the two numbers are 3 and 8.

Method 3: Trial and Error (For smaller numbers)

With smaller numbers, a trial-and-error approach can be efficient. You can mentally test different factor pairs of 24 until you find one that adds up to 11. This method is less systematic than the others but can be quicker for simple problems.

Expanding the Concept

This problem illustrates a common type of mathematical puzzle. You can adapt these methods to solve similar problems with different product and sum values. Just remember to systematically list factors or use algebraic techniques to find the solution.

Conclusion

We've explored three different methods to determine the two numbers that multiply to 24 and add to 11. Whether you prefer listing factors, using algebra, or trial and error, understanding these techniques will help you tackle similar mathematical problems effectively. The solution, as we've shown using various methods, is definitively 3 and 8.