Working with Fractions · Lesson 7 of 8
Fractional Relations and Nested Fractions
“Follow fractions through nested regions and repeated shares while keeping track of the original whole.”
• Relate the area of a nested region to the area of the original whole. • Use multiplication for repeated “of” relationships. • Combine multiplication and subtraction to find shaded regions. • Solve nested coin conversions and distinguish the assumed conversion systems. • Connect fraction methods with the historical examples in the chapter.
Fractional Relations
A region can be a fraction of another region that is itself a fraction of a larger whole. To find the final fraction of the original whole, follow the relationships one at a time. The important question after each step is: a fraction of which region? Labelling the areas prevents us from accidentally replacing the whole.
Take the entire square to have area 1 square unit. Divide it into four equal smaller squares. The top-right square then has area 1/4 square unit. A diagonal divides that smaller square into two equal triangles, so each triangle has half of 1/4, or 1/8, of the original area.
Problem
In the diagram, the orange region is three quarters of a triangle that is half of the top-right quarter-square. What fraction of the entire square is orange?
- 1.Start from the whole square: its area is 1 square unit. The red-outlined quarter-square has area 1/4.
- 2.The relevant triangle is half of that smaller square: 1/2 × 1/4 = 1/8.
- 3.The small white triangle removed from it has legs half as long as the corresponding legs of the larger triangle. Its area is 1/4 of that larger triangle, so the orange remainder is 3/4 of it.
- 4.Therefore orange area = 3/4 × 1/8 = 3/32 square unit. It occupies 3/32 of the original whole.
The statement about the removed triangle can also be seen by dividing the larger triangle into four congruent smaller triangles using side midpoints. One small triangle occupies one quarter of the larger area. The other three make the orange region. This geometric explanation matters: a region should not be called three quarters just because it looks that way.
Subtracting Areas and Taking Further Shares
Some diagrams are easier to read as a large known region with a small region removed. Others are easier as repeated halves. Choose the explanation that matches the lines in the drawing, and keep every intermediate fraction relative to the same original square.
Problem
Find the orange area fraction in diagram A.
- 1.The diagonal from the upper-left to the lower-right corner divides the entire square into two equal triangles. The lower-left triangle has area 1/2 of the whole.
- 2.The white corner triangle within it uses half of the square’s side along each perpendicular direction. Its area is half of a quarter-square, or 1/8 of the whole.
- 3.Subtract the white region: 1/2 − 1/8 = 4/8 − 1/8 = 3/8. The orange region occupies 3/8 of the entire square.
Problem
Find the orange fraction in diagram B.
- 1.The upper-left quarter-square has area 1/4 of the whole.
- 2.The yellow and orange triangle together occupy half of that square: 1/2 × 1/4 = 1/8 of the whole.
- 3.The midpoint on the sloping side splits this triangle into two equal-area triangles. Orange is one of these halves.
- 4.Its area is 1/2 × 1/8 = 1/16 of the entire square.
A fraction describes a relationship, not an isolated size. Three quarters of the smaller triangle is not three quarters of the entire square. Likewise, a quarter of the remaining land is not a quarter of the original land unless those wholes are the same.
A Dramma-tic Donation
Nested fractions also appear in quantities other than areas. The chapter uses a problem from Bhāskara II’s Līlāvatī: a person gives a very small fraction of a silver coin called a dramma. Each “of” means that another fraction of the current amount is selected. Multiply all the fractions to find the share of the original coin.
Problem
A donation is 1/5 of 1/16 of 1/4 of 1/2 of 2/3 of 3/4 of one dramma. In this problem, one dramma is worth 1280 cowrie shells. How many shells are donated?
- 1.Translate each “of” into multiplication: 1/5 × 1/16 × 1/4 × 1/2 × 2/3 × 3/4 of one dramma.
- 2.The numerator product is 1 × 1 × 1 × 1 × 2 × 3 = 6. The denominator product is 5 × 16 × 4 × 2 × 3 × 4 = 7680.
- 3.The coin fraction is 6/7680 = 1/1280 of a dramma.
- 4.Convert to shells: 1/1280 × 1280 = 1 cowrie shell. The long chain produces a donation of just one shell.
You can reorder the multiplication factors to cancel them conveniently. That changes the order of the calculation without changing the product. It does not mean that the story contains equal shares at every stage: some stages select halves, others quarters, sixteenths, or fifths.
Coin Conversions with Stated Assumptions
Different times and places used different coin values. For a separate conversion exercise, suppose one gold dinar equals 12 silver drammas, one dramma equals 4 copper panas, one pana equals 6 mashakas, and one pana equals 30 cowrie shells. These are the assumed values for this exercise, not the 1280-shell value used in the donation problem.
| Assumed relationship | Value in the larger unit |
|---|---|
| 1 dinar = 12 drammas | 1 dramma = 1/12 dinar |
| 1 dramma = 4 panas | 1 pana = 1/4 dramma |
| 1 pana = 6 mashakas | 1 mashaka = 1/6 pana |
| 1 pana = 30 cowrie shells | 1 shell = 1/30 pana |
Problem
Using the separate assumed coin values, express one cowrie shell as a fraction of a gold dinar.
- 1.One copper pana is 1/4 of a dramma. A dramma is 1/12 of a dinar.
- 2.So one pana is 1/4 × 1/12 = 1/48 dinar.
- 3.One shell is 1/30 pana, so its dinar value is 1/30 × 1/48 = 1/1440 dinar.
- 4.Check the direction: 12 × 4 × 30 = 1440 shells make one dinar under these particular assumptions.
A Pinch of History
The chapter connects everyday fraction work with several historical traditions. The Śhulbasūtra texts used fractions in geometric construction problems. Umasvati’s writing mentions reducing fractions. Brahmagupta stated general arithmetic rules, while Bhāskara I explained fraction multiplication geometrically using divided squares. These contributions connect written procedures with the visual reasoning you have used here.
Later mathematicians, including Śhrīdharāchārya, Mahāvīrāchārya, Pṛithūdakasvāmī, and Bhāskara II, developed fraction methods and applications further. The chapter also describes the sharing and development of these ideas through Arab and African mathematicians, including al-Hassār, and their later spread into Europe. Keep the emphasis on the mathematical ideas: equal parts, simplification, multiplication, and reciprocals.
If the donation were doubled while every stated coin value remained the same, it would become two shells. Use the meaning of the original result to explain this without multiplying the entire fraction chain again.
Quiz
A region is half of a quarter of a whole. What fraction of the whole is it?
The orange area is 3/4 of a triangle of area 1/8 square unit. What is its area?
Half a square minus one eighth of the same square is:
What operation represents repeated “of” relationships?
Under 1 dinar = 12 drammas and 1 dramma = 4 panas, one pana equals:
Practice Problems
- A triangle occupies half of a quarter-square. A shaded region occupies two thirds of the triangle. Find the shaded fraction of the whole square.
- A region is 3/5 of a smaller region that occupies 2/7 of the whole. Find its fraction of the whole and explain both reference regions.
- A garden bed occupies 1/3 of a field. Herbs occupy 1/2 of that bed, and mint occupies 2/5 of the herb area. What fraction of the original field is mint?
- Draw a square divided into quarters. Shade a triangle occupying half of one quarter, then take half of that triangle. Explain why the final fraction is 1/16.
- Under the separate assumed coin values above, express one cowrie shell in panas and one mashaka in gold dinars.
- If the donation fraction were 1/640 of a dramma and the dramma were worth 1280 shells, how many shells would be donated?
- Explain why you cannot combine the donation’s dramma-to-shell rate with the later hypothetical rates as though they belong to one conversion system.
Nested triangle: 2/3 × 1/2 × 1/4 = 1/12 of the whole. Nested region: 3/5 × 2/7 = 6/35. Mint: 2/5 × 1/2 × 1/3 = 1/15. The drawing’s repeated halves give 1/2 × 1/2 × 1/4 = 1/16. Coin values: one shell is 1/30 pana; one mashaka is 1/6 × 1/48 = 1/288 dinar. Modified donation: 1280/640 = 2 shells. The two coin systems use different stated rates; in the hypothetical system a dramma has 4 × 30 = 120 shells, whereas the donation problem specifies 1280.
Key Takeaways
• Label the original whole and every smaller reference region. • Multiply successive fractions to express a nested region relative to the original whole. • Subtract known areas when a shaded shape is a larger region with a smaller part removed. • A chain of “of” statements becomes a product of fractions. • Use only the conversion rates stated for a particular problem; different historical or hypothetical systems need not agree.