Fractions · Lesson 8 of 9
Fractions Through History and Puzzles
“Explore how fractions have been named and written, then solve puzzles with different unit fractions.”
• Connect historical fraction names with the idea of parts of a whole. • Recognise how fraction notation and common-denominator methods developed. • Express a fraction as a sum of distinct unit fractions. • Explain why two different proper unit fractions cannot sum to one. • Reason systematically about three- and four-unit-fraction sums.
People have long needed to describe parts
Sharing food and measuring lengths did not begin with the symbols we use today. People in different places developed names, written forms, and calculation methods for fractions. The Sanskrit word bhinna means “broken”; bhaga and ansha refer to a part or piece. These names connect directly to dividing a whole into smaller amounts.
Indian mathematical traditions include early work with fractions in the Sulba-sutras and later arithmetic texts. The Bakhshali manuscript shows a stacked form, with one number above another. Aryabhata, Brahmagupta, Sridharacharya, and Mahaviracharya belong to the long tradition of developing and explaining calculations. Brahmagupta’s work of 628 CE described the common-denominator approach that you used for addition and subtraction.
The horizontal bar became part of fraction notation over time; its use is associated with the Moroccan mathematician Al-Hassar around the twelfth century. Mathematical ideas travelled through Arabic-language scholarship into Europe and became more widely used in later centuries. The notation developed across a long history, while the underlying need remained the same: describe a part accurately and calculate with it.
| Historical feature | Connection with your work |
|---|---|
| Bhinna, bhaga, ansha | A fraction describes a broken-off part or share |
| Stacked numerator and denominator | The two numbers have different jobs |
| Brahmagupta’s common-denominator method | Make the units match before adding or subtracting |
| The horizontal fraction bar | Separates the count from the number of equal parts |
| Egyptian unit-fraction descriptions | Combine several unit fractions to name an amount |
One amount can be a sum of unit fractions
Different cultures used different ways to write fractional amounts. Egyptian calculations often expressed a quantity through sums of unit fractions, while Babylonian calculations used their own place-value notation. A sum of different unit fractions is now commonly called an Egyptian-fraction representation. Here we explore the unit-fraction idea using notation you already know.
Unit fractions that are all different, such as 1/2, 1/6, and 1/8. In the puzzles here, each is positive and less than one, so its denominator is a whole number greater than 1.
Problem
Check that 19/24 = 1/2 + 1/6 + 1/8.
- 1.Use a common denominator of 24.
- 2.Convert 1/2 to 12/24, 1/6 to 4/24, and 1/8 to 3/24.
- 3.Their sum is (12 + 4 + 3)/24 = 19/24.
- 4.The three addends are distinct unit fractions. They are not three equal shares; their different sizes combine to make the required amount.
If repeated units are allowed, making one whole is easy: two halves, three thirds, four quarters, or n copies of 1/n give one. Requiring every unit fraction to differ changes the problem. We must balance different sizes rather than simply repeat one convenient part.
Why two different unit fractions are insufficient
Among the unit fractions smaller than one, 1/2 is the largest. The next largest is 1/3, then 1/4, and so on. To make the biggest possible sum of two different unit fractions, choose the two largest different ones. If even that sum is less than one, no smaller pair can reach one.
Problem
Can two distinct unit fractions smaller than one add to one?
- 1.Each fraction is at most 1/2.
- 2.Two halves sum to one, but they are the same fraction and so are not allowed as a distinct pair.
- 3.For a distinct pair, at least one fraction must be smaller than 1/2. Even keeping the other at 1/2 gives a sum smaller than one.
- 4.For example, the largest distinct pair is 1/2 + 1/3 = 5/6. Therefore no distinct pair can sum to one.
1/2 + 1/2 = 1 is a correct equation, but it does not answer a question requiring different unit fractions. A valid solution must satisfy both the total and the distinctness condition.
Find the unique three-term solution
With three different unit fractions there is a solution. Instead of guessing indefinitely, ask what the largest piece must be. If it were smaller than a half, every piece would be at most a third. Three thirds give one only by repetition; making them different reduces their total below one. Thus a half is necessary.
Problem
Why must a three-term solution contain both 1/2 and 1/3?
- 1.If no half is used, every addend is at most 1/3. For distinct addends, at least one is smaller, so their sum is less than 1.
- 2.Therefore one addend must be 1/2. The other two together must equal 1/2.
- 3.If neither of those two is 1/3, both are at most 1/4. Two quarters equal 1/2 only when repeated; distinct choices give less.
- 4.Hence a third is also necessary. Only the final unit fraction remains to be found.
Problem
Find the third unit fraction that completes 1/2 + 1/3 to one.
- 1.First combine the known pieces: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- 2.Subtract from the whole: 1 − 5/6 = 6/6 − 5/6 = 1/6.
- 3.Check: 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1.
- 4.The fractions are all different. The earlier reasoning forced the half and third, so the sixth is forced too. Changing their order gives the same solution.
You can also follow the balancing idea. Begin with three thirds, replace one third by the larger half, and then adjust the remaining pieces to compensate. Or start with a half and two quarters and replace one quarter by a third; the other must shrink to a sixth. Both approaches lead to the same total, but the forced-choice argument explains why no other distinct three-term solution exists.
Create four different pieces
Once a valid three-term sum is known, try splitting one of its pieces into two unequal unit fractions. The total remains one, but there are now four pieces. Check that the new pieces do not repeat either of the pieces you kept. This is a constructive strategy: use a known solution to make a new one.
Problem
Construct four distinct unit fractions that sum to one.
- 1.Keep 1/2 and 1/3 from the three-term solution.
- 2.Split the remaining sixth as 1/7 + 1/42: with denominator 42, this is 6/42 + 1/42 = 7/42 = 1/6.
- 3.Therefore 1/2 + 1/3 + 1/7 + 1/42 = 1.
- 4.The denominators 2, 3, 7, and 42 differ, so all four unit fractions are distinct.
Try finding more solutions before opening the list below. A useful plan is to choose a large first piece, then calculate the amount still required. Choose the next piece only if the leftover amount stays positive. Finally check that the last remainder is a unit fraction and that it differs from the pieces already chosen. There are six four-term solutions when order is ignored.
1/2 + 1/3 + 1/7 + 1/42 = 1; 1/2 + 1/3 + 1/8 + 1/24 = 1; 1/2 + 1/3 + 1/9 + 1/18 = 1; 1/2 + 1/3 + 1/10 + 1/15 = 1; 1/2 + 1/4 + 1/5 + 1/20 = 1; 1/2 + 1/4 + 1/6 + 1/12 = 1. Check each sum with a common denominator. Within each sum, all four denominators are different. Reordering an existing sum does not create a new solution.
Quiz
Which name describes one equal part of a whole?
Which method is associated with Brahmagupta in this chapter?
Which sum equals 19/24?
Why does 1/2 + 1/2 fail the distinct-unit-fraction puzzle?
What completes 1/2 + 1/3 + ? = 1?
Which is a valid sum of four distinct unit fractions equal to one?
Practice Problems
- Explain how the words broken, part, and piece connect to fractions. Find a familiar fraction name in a language spoken at home.
- Describe how stacked numerator-denominator notation and a separating bar help us distinguish the two jobs of a fraction’s numbers.
- Verify 19/24 = 1/2 + 1/6 + 1/8 by using a common denominator.
- Explain why no pair of distinct unit fractions smaller than one can sum to one. Your reasoning should rule out every pair, not just a few trials.
- Without using a memorised answer, explain why a three-term distinct-unit-fraction sum of one must contain 1/2 and 1/3. Find the remaining term.
- Find at least one four-term solution before opening the answer list. Then verify all six sums and check distinctness in each.
- Draw a circle or strip to represent a four-term solution. Explain which pieces combine to make a half, a third, or a sixth.
- Would writing the terms of a solution in a different order make a new solution? Explain using the meaning of combining amounts.
Key Takeaways
• Fraction names and notation reflect a long history of measuring and sharing. • Common-denominator methods connect historical arithmetic with present-day calculations. • Distinct unit fractions have different denominators and different sizes. • Two distinct proper unit fractions cannot sum to one. • The unique three-term sum is 1/2 + 1/3 + 1/6 = 1; splitting a piece can produce a four-term solution.