Theorem

A product with a fixed sum is largest when the factors are equal

Algebra • Expressions • Maximum and Minimum Values

Formula

x y \leq (\frac{x + y}{2})^{2}

Among pairs of non-negative numbers with a fixed sum

s

, the product

x y

is greatest when

x = y = \frac{s}{2}

and falls away as the two move apart; over the integers with

s

odd the maximum is attained at the two nearest integers

\frac{s - 1}{2}

and

\frac{s + 1}{2}

, which tie.

How to use it

1Check first that the two factors really do have a constant sum, for instance $k$ and $s - k$ .
2For integers with an odd sum, expect a tie between the two middle values rather than a single winner.
3Read backwards, the same statement gives the least sum for a fixed product.

Worked example

Two positive integers add to

15

. What is their greatest possible product?

Step-by-step

1
The product is $k (15 - k)$ , a pair with fixed sum $15$ .
Why: Write the pair in terms of a single variable.
2
$15$ is odd, so the nearest available pair is $7$ and $8$ .
Why: The exact maximum at $7.5 \times 7.5$ is not open to integers.
3
$7 \times 8 = 56$

Answer

56

Watch out

It applies only when the sum is fixed; with a fixed difference the product grows without bound. When the integer sum is odd the maximum is a genuine tie at two values, and reporting only one of them is the usual error.

Also known as

AM-GM for two numbers, maximise a product with fixed sum, product largest when factors equal

Appears in past papers

Formula

Worked example

Appears in past papers

2019 (II)

2018 (II)

2017 (I)

2016 (II)