Mixed numbers cannot be multiplied part by part. Convert each one to an improper fraction, do the multiplication or division, then convert back if a mixed number is wanted.
Converting to an improper fraction
3 2/5 as an improper fraction
3 x 5 = 15 15 + 2 = 17 3 2/5 = 17/5
Multiplying
2 1/2 times 1 3/4
2 1/2 = 5/2 1 3/4 = 7/4 5 7 35 - x - = --- = 4 3/8 2 4 8
Why multiplying the parts fails
A tempting shortcut is 2 x 1 = 2 and 1/2 x 3/4 = 3/8, giving 2 3/8. The true answer is 4 3/8, so the shortcut is short by two whole units. It misses the cross terms: the whole 2 also has to multiply the 3/4, and the 1/2 also has to multiply the whole 1.
The four pieces the shortcut forgets
(2 + 1/2) x (1 + 3/4) = 2x1 + 2x3/4 + 1/2x1 + 1/2x3/4 = 2 + 3/2 + 1/2 + 3/8 = 2 + 2 + 3/8 = 4 3/8
This is the same distributive structure as partial-products multiplication, which is why the area model works just as well for mixed numbers as for whole ones.
Dividing
4 1/2 divided by 1 1/2
4 1/2 = 9/2 1 1/2 = 3/2 9 3 9 2 18 - div - = - x - = -- = 3 2 2 2 3 6
Which makes sense: 1 1/2 fits into 4 1/2 exactly three times.
Estimating first
Rounding each mixed number to the nearest whole gives a quick check on the size of the answer. 2 1/2 x 1 3/4 is roughly 3 x 2 = 6, so 4 3/8 is in the right region and the shortcut answer of 2 3/8 is clearly too small.
Cancel before multiplying
Once both numbers are improper fractions, look for common factors across the diagonal before doing any multiplication. The numbers stay small and the answer often needs no simplifying at all.
2 2/3 times 1 1/2
8/3 x 3/2 the 3 on the bottom cancels the 3 on top the 8 and the 2 share a factor of 2 4/1 x 1/1 = 4
Doing it the other way gives 24/6, which simplifies to the same 4 after an extra step.
Common mistakes
Multiplying wholes and fractions separately
Always wrong except by accident. Convert first.
Converting incorrectly
3 2/5 is 17/5, not 6/5 or 32/5. Multiply the whole by the denominator, then add the numerator.
Check yourself
1 1/3 times 2 1/4
4/3 x 9/4 = 36/12 = 3.
3 3/4 divided by 1 1/2
15/4 div 3/2 = 15/4 x 2/3 = 30/12 = 5/2 = 2 1/2.
Convert 5 3/8 to an improper fraction.
5 x 8 + 3 = 43, so 43/8.
A recipe uses 2 1/4 cups and you make it twice. How much flour?
9/4 x 2 = 18/4 = 4 1/2 cups.
See also
Sources
- Mixed number multiplication and division as taught in United States middle grades mathematics. Examples on this page are our own.