The sum of two numbers is 15. If the sum of their reciprocals is 3/10, find the two numbers.

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 29(A)

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, section C, question 29(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(A) · 3 marks · Short answerQuadratic Equations

The sum of two numbers is 15. If the sum of their reciprocals is $\frac{3}{10}$ , find the two numbers.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

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

Hint 1

If one number is x, express the other using the sum 15.

Hint 2

Add the reciprocals, cross-multiply and solve the quadratic by factorisation.

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 one number be $x$ ; then the other is $15 - x$ .
½ mark
Given: $\frac{1}{x} + \frac{1}{15 - x} = \frac{3}{10}$ .
1 mark
$\frac{15}{x ( 15 - x )} = \frac{3}{10} \Rightarrow 150 = 45 x - 3 x^{2} \Rightarrow 3 x^{2} - 45 x + 150 = 0 \Rightarrow x^{2} - 15 x + 50 = 0$ .
½ mark
$(x - 10) (x - 5) = 0 \Rightarrow x = 10$ or $x = 5$ .
½ mark
If $x = 10$ the other number is 5, and if $x = 5$ it is 10. So the numbers are 10 and 5.
½ mark

Answer: The numbers are 10 and 5

Where the marks are

Takes the numbers as x and 15 − x (½ mark)
Forms the reciprocal equation (1 mark)
Reduces it to $x^{2} - 15 x + 50 = 0$ (½ mark)
Solves to get x = 10, 5 (½ mark)
States the two numbers (½ mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/6/1 (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 2023: question 29(A)

Hints

Step-by-step solution

Revise the chapter

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