Question

Sum of squares of two numbers is 3341 and difference is 891. Find numbers.

• A. 35 and 46
• B. 35 and 50
• C. 40 and 55
• D. 45 and 60

Hint:

When both sum and difference of squares are given, add the equations first to eliminate one variable quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The sum of the squares of two numbers is 3341 and the difference of their squares is 891. We need to find the two numbers.

Step 2: Key Formula or Approach:
Let the numbers be \(x\) and \(y\). Given: \[ x^2+y^2=3341 \] \[ x^2-y^2=891 \] Using addition of equations: \[ (x^2+y^2)+(x^2-y^2)=3341+891 \]

Step 3: Detailed Explanation:
Add the two equations: \[ 2x^2=4232 \] \[ x^2=2116 \] \[ x=\sqrt{2116}=46 \] Now substitute into: \[ x^2+y^2=3341 \] \[ 2116+y^2=3341 \] \[ y^2=3341-2116 \] \[ y^2=1225 \] \[ y=\sqrt{1225}=35 \] Therefore, the two numbers are: \[ 35 \text{ and } 46 \]

Step 4: Final Answer:
Hence, the correct option is: \[ \boxed{\text{(A) 35 and 46}} \]

Answer 2

Since the two conditions given are about the squares of two numbers, one direct way to confirm the right pair is to square each candidate pair and check both the sum and the difference of their squares.

1. 35 and 46: Squaring gives \( 35^2 = 1225 \) and \( 46^2 = 2116 \). Their sum is \( 1225 + 2116 = 3341 \), matching the given sum, and their difference is \( 2116 - 1225 = 891 \), matching the given difference as well.
2. 35 and 50: Squaring gives \( 35^2 = 1225 \) and \( 50^2 = 2500 \). Their sum is \( 1225 + 2500 = 3725 \), which does not match 3341, so this pair is incorrect.
3. 40 and 55: Squaring gives \( 40^2 = 1600 \) and \( 55^2 = 3025 \). Their sum is \( 1600 + 3025 = 4625 \), far from 3341, so this pair does not fit.
4. 45 and 60: Squaring gives \( 45^2 = 2025 \) and \( 60^2 = 3600 \). Their sum is \( 2025 + 3600 = 5625 \), well above 3341, so this pair is also incorrect.

Only 35 and 46 satisfy both the sum of squares being 3341 and the difference of squares being 891.

Therefore, the correct answer is 35 and 46.