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Sum equals product

3

and

1.5

are a special pair of numbers, as

3 + 1.5 = 4.5

and

3 \times 1.5 = 4.5

so

3 + 1.5 = 3 \times 1.5

.

Given a number

a

, can you find a number

b

such that

a + b = a \times b

?

Show answer & extension

Hide answer & extension

If

a + b = a \times b

, then:

a b - b = a

b (a - 1) = a

b = \frac{a}{a - 1}

This will work for any

a \neq 1

(

a = 1

will not work as this will mean division by zero).

Extension

(i) Given a number

a

, can you find a number

b

such that

b - a = \frac{b}{a}

?

(ii) Given a number

a

, can you find a number

b

such that

b - a = \frac{a}{b}

?

(iii) Given a number

a

, can you find a number

b

such that

a - b = \frac{b}{a}

?

(iv) Given a number

a

, can you find a number

b

such that

a - b = \frac{a}{b}

?

If you enjoyed this puzzle, check out Sunday Afternoon Maths XXI,
puzzles about numbers, or a random puzzle.

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