Try to explain why the average of two rational numbers a and b, which equals (a + b)2, is always a rational number between a and b.

Step-by-step solution

Idea: Two separate claims: (1) the average is rational (closure), (2) it is bigger than the smaller number and smaller than the bigger one. For (2), compare by subtraction.

1. Rational: a + b is rational (rational numbers are closed under addition), and dividing a rational number by 2 (non-zero) gives a rational number. So a + b2 is rational.1 mark
2. Between: suppose a < b (if a = b there is nothing between them). Then b − a > 0.½ mark
3. a + b2 − a = a + b − 2a2 = b − a2 > 0, so a + b2 > a.½ mark
4. b − a + b2 = 2b − a − b2 = b − a2 > 0, so a + b2 < b.½ mark
5. Hence a < a + b2 < b. In fact the average is the midpoint: it is the same distance b − a2 from both. For example, between 1 and 32 it gives 54.½ mark

Check: a = −13, b = 12: average = 12 × 16 = 112, and −412 < 112 < 612 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Checking one example and calling it a proof. An example only illustrates; the subtraction argument works for all a, b.
• Proving only one side (that the average is bigger than a) and forgetting to show it is smaller than b.
• Forgetting to say why the average is rational; the question asks for that too.

How this can come in the exam

Try one yourself

Use averages to find two rational numbers between 15 and 13.

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