A fraction of a fraction
A barfi tray has 3 columns and 4 rows. Shade 2/3 of the columns. Then shade 3/4 of the rows. How much of the tray is shaded twice?
- A tray, 3 columns by 4 rows.
- Shade 2/3 of the columns.
- Shade 3/4 of the rows over it.
- The overlap is 6/12 = 1/2.
Multiply tops, multiply bottoms
The cells shaded twice are 3/4 of 2/3 of the tray. That is 1/2 of the tray.
To multiply two fractions, multiply the tops together. Multiply the bottoms together.
The tray has 3 × 4 = 12 cells. The shaded columns are 2 and the shaded rows are 3. The two shadings overlap on 2 × 3 = 6 cells. So 3/4 of 2/3 is 6/12, which is 1/2.
Try 1/2 × 1/3. Tops: 1 × 1 = 1. Bottoms: 2 × 3 = 6. The answer is 1/6.
Watch out!
Renaming first turns 2/3 and 3/4 into 8/12 and 9/12. Multiplying those tops and keeping 12 gives 72/12, or 6 whole trays. Too much. Renaming belongs to adding, not to multiplying.
Both fractions are less than one whole. So the answer is smaller than each. Half of a third is smaller than a half. Half of a third is also smaller than a third.
Shade columns, then rows, in the lab. The overlap, the cells shaded twice, is the answer.
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