If the sum of two numbers is 20 and their product is 96, what are the numbers? (

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Question: If the sum of two numbers is 20 and their product is 96, what are the numbers? (2021)

Options:

Correct Answer: 12 and 8

Exam Year: 2021

Solution:

Let the numbers be x and y. We have x + y = 20 and xy = 96. Solving these equations gives x = 12 and y = 8.

If the sum of two numbers is 20 and their product is 96, what are the numbers? (2021)

If the sum of two numbers is 20 and their product is 96, what are the numbers? (2021)

Step 1: Let the two numbers be x and y.
Step 2: Write down the equations based on the problem: x + y = 20 and xy = 96.
Step 3: From the first equation (x + y = 20), we can express y in terms of x: y = 20 - x.
Step 4: Substitute y in the second equation (xy = 96) with the expression from Step 3: x(20 - x) = 96.
Step 5: Expand the equation: 20x - x^2 = 96.
Step 6: Rearrange the equation to form a standard quadratic equation: x^2 - 20x + 96 = 0.
Step 7: Factor the quadratic equation: (x - 12)(x - 8) = 0.
Step 8: Set each factor to zero: x - 12 = 0 or x - 8 = 0.
Step 9: Solve for x: x = 12 or x = 8.
Step 10: Use the values of x to find y: If x = 12, then y = 20 - 12 = 8. If x = 8, then y = 20 - 8 = 12.
Step 11: The two numbers are 12 and 8.

Simultaneous Equations – The question tests the ability to solve a system of equations involving addition and multiplication.
Quadratic Equations – The problem can be transformed into a quadratic equation, testing knowledge of factoring or using the quadratic formula.