Multiplying Large Mixed Numbers by Fractions
The standard method of multiplying a mixed number by a fraction is to convert the mixed number to an improper fraction, multiply straight across and then convert it back to a mixed number. That’s fine if the whole in the mixed number is small, such as 2, or 4 but becomes rather unwieldy if the whole number is large, such as 23.
Let’s look at an example. How about (23 2/3)(5/7)
The normal way would give us (71/3)(5/7) = 355/21.
Nope. Not doing that. I’ll invent a better way rather than work that hard.
I’ve recently figured out a way to do this multiplication more easily.
Like many of my math hacks, it involves the distributive property. We break up the mixed number into pieces and multiply the fractions times those pieces.
Since we have a 7 in the denominator of the fraction, we look for factors of 7 in the whole part of the mixed number. 7 X 3 = 21. So, we split the 23 into 21 + 2 2/3.
Now we have (21)(5)/7. If we divide first, that gives us 3 X 5 = 15.
Looking at the remaining bit, we have (2 2/3)(5/7). This we can do the normal way. 2 2/3 = 8/3, so we have (8/3)(5/7) = 40/21 = 1 19/21.
Then we just add them together. 15 + 1 19/21 = 16 19/21.
All the steps are easy enough to do in your head.
Let’s see another example: (57 1/3)(3/4)
56 = 4 X 14
So we have (56)(3/4) + (1 1/3)(3/4)
We already know that 56/4 = 14 so all we need to do there is multiply by 3. 14 X 3 = 42
Now the other part.
1 1/3 = 4/3