The sum of two numbers is 15. If the sum of their reciprocals is $\frac{3}{10}$, find the two numbers.
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding rag
Model Answer
Let the two numbers be $x$ and $15 - x$.
Given: sum of reciprocals = $\dfrac{3}{10}$
$$\frac{1}{x} + \frac{1}{15-x} = \frac{3}{10}$$
$$\frac{(15-x)+x}{x(15-x)} = \frac{3}{10}$$
$$\frac{15}{x(15-x)} = \frac{3}{10}$$
$$3x(15-x) = 150$$
$$x(15-x) = 50$$
$$x^2 - 15x + 50 = 0$$
Factorising: $x^2 - 10x - 5x + 50 = 0$
$$x(x-10) - 5(x-10) = 0$$
$$(x-5)(x-10) = 0$$
So $x = 5$ or $x = 10$.
The two numbers are 5 and 10.
Source: Chapter 4, Section 4.3 (Factorisation method)
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Explanation
- Key approach: Let one number be $x$, the other is $15 - x$ (using sum = 15). Form the equation from the reciprocal condition, simplify to a quadratic, then factorise.
- Common mistake: Students forget to cross-multiply carefully or make sign errors when rearranging. Always write the quadratic in standard form $ax^2 + bx + c = 0$ before factorising.
- Verification tip (worth mentioning if time allows): $5 + 10 = 15$ ✓ and $\frac{1}{5} + \frac{1}{10} = \frac{3}{10}$ ✓