Question

If the sum of two numbers is 8 and their difference is 2, then those two numbers are _____.

• A. 4, 4
• B. 7, 1
• C. 5, 3
• D. 6, 2

Hint:

If the sum and difference of two numbers are known: \[ \text{Larger Number}= \frac{\text{Sum}+\text{Difference}}{2} \] \[ \text{Smaller Number}= \frac{\text{Sum}-\text{Difference}}{2} \] This shortcut works very fast in exams.

Answer 1

The Correct Option is C

Concept: When two unknown numbers satisfy conditions involving their sum and difference, we form linear equations and solve them simultaneously.

Step 1: Assume the numbers.
Let the two numbers be: \[ x \quad \text{and} \quad y \]

Step 2: Form equations using the given conditions.
Given: \[ \text{Sum of the numbers}=8 \] Therefore, \[ x+y=8 \] Also given: \[ \text{Difference of the numbers}=2 \] Therefore, \[ x-y=2 \] Now we have the system: \[ x+y=8 \] \[ x-y=2 \]

Step 3: Add the equations.
Adding both equations: \[ (x+y)+(x-y)=8+2 \] \[ x+y+x-y=10 \] \[ 2x=10 \] \[ x=5 \]

Step 4: Find the second number.
Substitute \(x=5\) into: \[ x+y=8 \] \[ 5+y=8 \] \[ y=8-5 \] \[ y=3 \]

Step 5: Verify the answer.
Check the sum: \[ 5+3=8 \] Correct. Check the difference: \[ 5-3=2 \] Correct. Hence, the two numbers are: \[ \boxed{5 \text{ and } 3} \]