Question

Difference between two numbers is 9 and difference between their squares is 981. Lowest of the two numbers is:

• A. 40
• B. 50
• C. 55
• D. 59

Hint:

When dealing with square differences, use identity \(a^2 - b^2 = (a - b)(a + b)\). Then solve using two linear equations.

Answer 1

The Correct Option is A

Step 1: Use identity for difference of squares \[ a^2 - b^2 = (a - b)(a + b) \] Step 2: Use given data Difference between two numbers = \( a - b = 9 \)
Difference between squares = \( a^2 - b^2 = 981 \) \[ \Rightarrow (a - b)(a + b) = 981
\Rightarrow 9(a + b) = 981
\Rightarrow a + b = \frac{981}{9} = 109 \] Step 3: Solve the equations \[ a + b = 109 \quad \text{and} \quad a - b = 9 \] Add: \[ 2a = 118 \Rightarrow a = 59
\Rightarrow b = 109 - 59 = 50 \] So the smaller number is \(\boxed{50}\), but the options say 40. Wait! Let's recheck: Actually: \[ a + b = 109,\quad a - b = 9 \Rightarrow 2a = 118 \Rightarrow a = 59,\quad b = 50 \] Smaller number is \(\boxed{50}\) So correct answer is: (b) 50, not (a) % Correction Correct Answer: (b) 50