Question

The sum of two integers is 10 and the sum of their reciprocals is $\frac{5}{12}$. Then the larger of these integers is:

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

Use substitution when given a sum and a condition on reciprocals — convert the equation to a rational form and solve the resulting quadratic.

Answer 1

The Correct Option is C

Step 1: Let integers be $x$ and $10 - x$ Sum of integers = 10 $\Rightarrow$ second number is $10 - x$ Step 2: Use reciprocal condition \[ \frac{1}{x} + \frac{1}{10 - x} = \frac{5}{12} \] Step 3: Solve the equation \[ \frac{10 - x + x}{x(10 - x)} = \frac{5}{12} \Rightarrow \frac{10}{x(10 - x)} = \frac{5}{12} \] Cross-multiplying: \[ 12 \cdot 10 = 5x(10 - x) \Rightarrow 120 = 5x(10 - x) \Rightarrow 120 = 50x - 5x^2 \Rightarrow 5x^2 - 50x + 120 = 0 \] Divide by 5: \[ x^2 - 10x + 24 = 0 \Rightarrow (x - 6)(x - 4) = 0 \Rightarrow x = 6 \text{ or } 4 \] Step 4: Find larger integer The two numbers are 6 and 4. Hence the larger one is: \[ \boxed{6} \]