If a number plus its reciprocal equals 103, find the number.

Step-by-step solution

Idea: Call the number x. Its reciprocal is 1x. Clear the fractions to get a quadratic, then factorise it by splitting the middle term. A quadratic can have two answers, so check both.

1. Let the number be x (x ≠ 0, because 0 has no reciprocal). Its reciprocal is 1x.
   x + 1x = 103½ mark
2. Multiply every term by 3x to remove the fractions:
   3x² + 3 = 10x
   3x² − 10x + 3 = 0½ mark
3. Split the middle term. We need two numbers whose product is 3 × 3 = 9 and whose sum is −10: they are −9 and −1.
   3x² − 9x − x + 3 = 0
   3x(x − 3) − 1(x − 3) = 0
   (x − 3)(3x − 1) = 01 mark
4. A product is zero only when one of the factors is zero:
   x − 3 = 0 gives x = 3;   3x − 1 = 0 gives x = 13.½ mark
5. Check both: 3 + 13 = 93 + 13 = 103 ✓ and 13 + 3 = 103 ✓. So both values are correct.½ mark

Check: It makes sense that there are two answers: if 3 works, its reciprocal 13 must also work, because “number + reciprocal” gives the same total either way round.

Answer to write in the exam

Common mistakes that cost marks

• Giving only one answer (just 3). The quadratic has two roots, and 13 also satisfies the condition.
• Multiplying only some terms by 3x, e.g. writing 3x² + 1 = 10x. Every term, including 1x, must be multiplied.
• Taking the reciprocal of 103 as the answer (310). The question asks for a number whose sum with its own reciprocal is 103.

How this can come in the exam

Try one yourself

If a number plus its reciprocal equals 265, find the number.

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