Multiplication and division of fractions need no common denominator. That makes them mechanically easier than addition, and it is worth saying so, because students often expect the opposite.
Multiplying
Multiply the numerators together and the denominators together, then simplify.
2/3 times 4/5
2 4 2 x 4 8 - x - = ------- = -- 3 5 3 x 5 15
Cancelling before multiplying keeps the numbers small. In 3/4 x 8/9, the 3 and the 9 share a factor of 3, and the 4 and the 8 share a factor of 4, leaving 1/1 x 2/3 = 2/3.
What of means
In fraction problems the word of means multiply. Two thirds of a half hour is 2/3 x 30 = 20 minutes. Once that translation is automatic, a large group of word problems becomes routine.
Why the product can be smaller than both factors
Multiplying by a number less than one makes things smaller. Half of a half is a quarter, which is smaller than either. This contradicts the habit built up with whole numbers, where multiplying always made things bigger, and it is worth confronting directly rather than hoping it settles.
Dividing
To divide by a fraction, multiply by its reciprocal, which is the fraction turned upside down.
3/4 divided by 2/5
3 2 3 5 15 - div - = - x - = -- 4 5 4 2 8 which is 1 7/8
Why flipping works. Dividing asks how many of the second fraction fit inside the first. Fitting 2/5 pieces into 3/4 is the same as asking how many fifths fit, then halving the count, and that is exactly what multiplying by 5/2 does.
Common mistakes
Flipping the wrong fraction
Only the divisor, the one after the division sign, gets flipped. Flipping the first one gives the reciprocal of the true answer.
Looking for a common denominator
Needed for addition and subtraction. Never needed for multiplication or division, and the extra work invites mistakes.
Check yourself
3/5 times 5/9
Cancel the fives: 3/9 = 1/3.
What is 3/4 of 20?
3/4 x 20 = 15.
5/6 divided by 1/3
5/6 x 3/1 = 15/6 = 5/2 = 2 1/2.
Why is 1/2 divided by 1/8 equal to 4?
Because the question is how many eighths fit inside a half, and a half is four eighths.
See also
Sources
- Fraction multiplication and division as taught in United States middle grades mathematics. Examples on this page are our own.