The sum of two numbers is 18 and the sum of their reciprocals is 9/40. Find the numbers.

MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 27

From the official CBSE sample question paper for 2024-25, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerQuadratic Equations

The sum of two numbers is $18$ and the sum of their reciprocals is $\frac{9}{40}$ . Find the numbers.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

Open one hint at a time and try again before the next.

Hint 1

Call one number $x$ . What is the other number?

Hint 2

Write $\frac{1}{x} + \frac{1}{18 - x} = \frac{9}{40}$ , clear the fractions and solve the quadratic.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let the numbers be $x$ and $18 - x$ .
½ mark
Given $\frac{1}{x} + \frac{1}{18 - x} = \frac{9}{40}$ , so $\frac{18 - x + x}{x ( 18 - x )} = \frac{9}{40}$ , that is, $\frac{18}{x ( 18 - x )} = \frac{9}{40}$ .
1 mark
Cross-multiplying: $720 = 9 x (18 - x) \Rightarrow 80 = 18 x - x^{2} \Rightarrow x^{2} - 18 x + 80 = 0$ .
Factorising: $(x - 10) (x - 8) = 0 \Rightarrow x = 10$ or $x = 8$ .
1 mark
If $x = 10$ , the other number is $8$ ; if $x = 8$ , the other is $10$ . The numbers are $8$ and $10$ .
Check: $\frac{1}{8} + \frac{1}{10} = \frac{5 + 4}{40} = \frac{9}{40}$ ✓
½ mark

Answer: 8 and 10

Where the marks are

Takes the numbers as $x$ and $18 - x$ (½ mark)
Forms the equation from the sum of reciprocals (1 mark)
Reduces it to $x^{2} - 18 x + 80 = 0$ and solves (1 mark)
States the numbers 8 and 10 (½ mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 27

Hints

Step-by-step solution

Revise the chapter

More questions from Quadratic Equations