Can you think of other way(s) to find a rational number between any two rational numbers?

Step-by-step solution

Idea: Any method works if it gives a fraction with integer numerator and non-zero integer denominator that is bigger than the smaller number and smaller than the bigger one. Test each on 13 and 12.

1. Equivalent fractions: 13 = 26 and 12 = 36 have no whole number between 2 and 3, so multiply top and bottom by 10: 2060 and 3060. Now 2160, 2260, …, 2960 all lie between.1 mark
2. Decimals: 13 = 0.333… and 12 = 0.5, so 0.4 = 25 (or 0.45, 0.35, …) lies between.½ mark
3. Part of the way: a + b − an moves 1/n of the gap from a towards b. With n = 3: 13 + 13 × 16 = 13 + 118 = 718. (n = 2 gives the average.)1 mark
4. Mediant (positive fractions): add tops and add bottoms: 1 + 13 + 2 = 25, and 13 < 25 < 12 (0.33 < 0.4 < 0.5). This works whenever both denominators are positive.½ mark

Check: 718 ≈ 0.389, 25 = 0.4, 2560 ≈ 0.417 all lie between 0.333 and 0.5 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Thinking 13 and 12 have nothing between them because 2 and 3 are consecutive numerators over 6. Scaling up the denominator makes room.
• Using the mediant with a negative denominator, e.g. 1−2; keep denominators positive first.
• Describing a method without testing that the result really lies between the two numbers.

How this can come in the exam

Try one yourself

Use the mediant to find a rational number between 49 and 12, and check your answer.

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