There is always another one in the gap
After the whole number 6 comes 7, with nothing between. Fractions do not work that way.
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Step 1 of 4
- 1/3 and 1/2 on a line.
- Zoom into the gap between them.
- The gap, drawn wider.
- Halfway inside it is 5/12.
Halfway is always somewhere
Give both fractions the same bottom number. 1/3 is 4/12 and 1/2 is 6/12. Halfway between 4 and 6 is (4 + 6) ÷ 2 = 5. So halfway is 5/12.
Do it again with 1/3 and 5/12. That is 8/24 and 10/24. Halfway is (8 + 10) ÷ 2 = 9, so 9/24. That is 3/8.
Adding fractions gives a fraction. Halving a fraction gives a fraction. So halfway between two fractions is another fraction. You can repeat this as often as you like.
| Pair | Halfway |
|---|---|
| 1/3, 1/2 | 5/12 |
| 1/3, 5/12 | 3/8 |
| 1/3, 3/8 | 17/48 |
For different fractions a and b, the average (a + b) / 2 lies between them. Between 3 and 4 there is no whole number. But the fraction 7/2 is between 3 and 4.