Fractions that fight back
Tonight's idea
A division question is really a "how many fit?" question. \(12 \div 3\) asks how many 3s fit into 12. So \(\tfrac32 \div \tfrac14\) asks how many quarters fit into one and a half pizzas. Count them: 6.
Now the rule makes sense. Dividing by \(\tfrac14\) is multiplying by 4, because every whole contains four quarters. Dividing by \(\tfrac34\): count the quarters (×4), then bundle them in threes (÷3). That is ×\(\tfrac43\). The flip means "count the little pieces, then bundle them."
Second idea: "of" means multiply. Two thirds of three fifths is \(\tfrac23 \times \tfrac35 = \tfrac{6}{15} = \tfrac25\). Half of a half is a quarter.
Third idea, from 4,000 years ago. Egyptian scribes wrote nearly every fraction with a 1 on top: \(\tfrac12, \tfrac13, \tfrac14, \ldots\) To write \(\tfrac34\) they wrote \(\tfrac12 + \tfrac14\). Every fraction can be written this way using different unit fractions, and the hunt for them is a good puzzle.
Watch one
A ribbon is \(\tfrac72\) metres long. How many pieces of \(\tfrac38\) metre can you cut, and what is left over?
\(\tfrac72 \div \tfrac38 = \tfrac72 \times \tfrac83 = \tfrac{56}{6} = \tfrac{28}{3} = 9\tfrac13\). So 9 whole pieces. The \(\tfrac13\) is a third of a piece, not a third of a metre: \(\tfrac13 \times \tfrac38 = \tfrac18\) metre is left over.
Egyptian version of \(\tfrac25\): take the biggest unit fraction that fits, \(\tfrac13\). What is left? \(\tfrac25 - \tfrac13 = \tfrac{6-5}{15} = \tfrac{1}{15}\). So \(\tfrac25 = \tfrac13 + \tfrac1{15}\).
Your turn
- Work these out, and for the first one draw the picture that shows why: \(2 \div \tfrac13\), \(\tfrac34 \div \tfrac18\), \(\tfrac56 \div \tfrac23\).
- A water tank is \(\tfrac23\) full. You use \(\tfrac34\) of the water in it. What fraction of the whole tank is left?
- Find \(\tfrac23\) of \(\tfrac35\) of 10. Then find \(\tfrac35\) of \(\tfrac23\) of 10. Explain in a sentence why the answers had to be the same.
- Explain, without using the word "flip", why dividing by \(\tfrac12\) doubles a number.
- Write each as a sum of different unit fractions, Egyptian style: \(\tfrac34\), \(\tfrac23\), \(\tfrac56\), \(\tfrac78\).
- Which is bigger, \(\tfrac{5}{8} \div \tfrac{3}{4}\) or \(\tfrac{3}{4} \div \tfrac{5}{8}\)? Decide before computing, then compute both to check. What do you notice about the two answers?
Stretch: the seventeen camels
An old man leaves 17 camels to his three children: half to the eldest, a third to the middle child, a ninth to the youngest. Camels do not divide nicely. A passing traveller lends them her camel, making 18. The eldest takes 9, the middle child takes 6, the youngest takes 2. That is 17 camels. The traveller takes her own camel back and rides off.
Everyone got more than their share, and the traveller lost nothing. How is that possible? Add up \(\tfrac12 + \tfrac13 + \tfrac19\). The shares were never a whole: they leave \(\tfrac1{18}\) unclaimed, and the borrowed camel is what makes that leftover share out evenly. Then find the only way to write 1 as a sum of three different unit fractions, and at least two ways using four.
Answers
- \(2 \div \tfrac13 = 6\) (six thirds in two wholes). \(\tfrac34 \div \tfrac18 = 6\) (a quarter is two eighths; three quarters is six eighths). \(\tfrac56 \div \tfrac23 = \tfrac56 \times \tfrac32 = \tfrac{15}{12} = \tfrac54\).
- You use \(\tfrac34\) of \(\tfrac23\), which is \(\tfrac12\) of the tank. Left: \(\tfrac23 - \tfrac12 = \tfrac16\). Or directly: a quarter of \(\tfrac23\) remains, \(\tfrac14 \times \tfrac23 = \tfrac16\).
- Both are 4. Multiplication can be done in any order, and "of" is multiplication: \(\tfrac23 \times \tfrac35 \times 10 = \tfrac35 \times \tfrac23 \times 10\).
- Dividing by \(\tfrac12\) asks how many halves fit. Every whole contains two halves, so a number contains twice that many halves.
- \(\tfrac34 = \tfrac12 + \tfrac14\). \(\tfrac23 = \tfrac12 + \tfrac16\). \(\tfrac56 = \tfrac12 + \tfrac13\). \(\tfrac78 = \tfrac12 + \tfrac14 + \tfrac18\). (Other correct splittings exist.)
- \(\tfrac58 \div \tfrac34 = \tfrac58 \times \tfrac43 = \tfrac{20}{24} = \tfrac56\), which is less than 1 because \(\tfrac58\) is smaller than \(\tfrac34\). The other way round gives \(\tfrac65\). The two answers are reciprocals: multiply them and you get 1. That always happens when you swap the order of a division.
Stretch. \(\tfrac12 + \tfrac13 + \tfrac19 = \tfrac{9+6+2}{18} = \tfrac{17}{18}\). The shares do not add up to a whole, so a careful division of 17 camels would leave \(\tfrac{17}{18}\) of a camel unclaimed. The traveller's camel just makes the fractions come out whole; the "extra" everyone received is that leftover eighteenth, shared out. Three different unit fractions summing to 1: only \(\tfrac12 + \tfrac13 + \tfrac16\). Four: \(\tfrac12 + \tfrac14 + \tfrac16 + \tfrac1{12}\), \(\tfrac12 + \tfrac13 + \tfrac1{10} + \tfrac1{15}\), \(\tfrac12 + \tfrac13 + \tfrac17 + \tfrac1{42}\), \(\tfrac12 + \tfrac13 + \tfrac18 + \tfrac1{24}\), \(\tfrac12 + \tfrac13 + \tfrac19 + \tfrac1{18}\), \(\tfrac12 + \tfrac14 + \tfrac15 + \tfrac1{20}\).