Dividing by a fraction

This post was prompted by conversations with April Brown and Leann Miller, teachers from the Peace Wapiti School Division in Alberta whom I met at our April 2018 SNAP Math Fair Workshop at BIRS.

Invert and multiply

Is there any real-life circumstance where the invert-and-multiply rule is naturally applied? Here are two instances when I used this rule without being aware that I was using it.

The rule I am talking about is this one:

where b and c are numbers and [A] is pretty well anything.

Red Deer to Edmonton

I'm driving from Red Deer to Edmonton. That's 144 km. I know from past experience that my average speed will be 90 km per hour. How many minutes will it take me to get to Edmonton?

Did you solve it like this:

144 divided by 90 is 1.6, so it takes 1.6 hours, which is one hour and 36 minutes, which is 96 minutes.

That's how I got the answer. Putting it in mathematical notation:

I could have obtained the answer by first converting the speed from km per hour to km per minute –– 90 km per hour is 90 km per 60 minutes which is 1.5 km per minute –– which gives this solution:

To travel 144 km at 1.5 km per minute will take 144 divided by 1.5, or 96 minutes.

In mathematical notation:

The two solutions are equally valid, which means that

So, dividing by 90/60 is the same as inverting the fraction and multiplying by 60/90.

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In this next problem, the numbers work out nicely. In the actual task, the numbers were not quite so friendly.

Building a shelf

You have been asked to construct a 28 inch wide shelf using 2 × 4 lumber.

The shelf will be 8 feet long (which is convenient because two-by-fours come in that length). The shelf will have the boards snug against each other, that is, no spaces are allowed between the boards. 

How many boards will you need? 

Since 28 ÷ 4 is 7, we will require 7 boards. But there's a catch: two-by-fours are not 2 inches by 4 inches. The two-by-fours that you buy at a lumber store are milled a quarter of an inch on all sides, so that their actual dimensions are 1½ inches by 3½ inches.

So the question becomes:

How many  3½ inch wide boards are needed to make a width of 28 inches? In other words, what's 28 divided by 3½ ?

If it were me, I would shamelessly bring up the calculator on my smart phone and punch in 28 ÷ 3.5 to get the answer: eight boards.

To avoid the awkward ½ in a mental calculation, you might note that two 3½ inch wide boards are together the same as one 7 inch wide board.  Since four 7 inch boards will span 28 inches exactly, this means that eight 3½ inch boards will do the job.

If you've followed this, we've just inverted-and-multiplied again. Using the fact that 3½ can be written as ⁷/₂ , this is what we did (after a slight rearrangement of the notation):

The expression on the left is 28 divided by 3½ while the expression on the right is 28 multiplied by the reciprocal of 3½.

Some additional comments

I learned the invert-and-multiply rule in grade 8. At that time, some of us had difficulty deciding what number we should use to divide or multiply by.  In the Red Deer to Edmonton problem, should one multiply by 60/90 or by 90/60?  Should the final calculation be

To overcome difficulties in situations like this, our physics teacher gave this useful advice (which today is sometimes offered on the web):

Always check that the units cancel properly!

When the units are included, the confusion about 60/90 versus 90/60 becomes:

Which of the following should we use

or, equivalently, which of these should we use

Here's what the physics teacher meant by the units cancelling properly:

In the expression on the left, the km units cancel, leaving us with an answer in minutes –– which is what we want. The units in the expression on the right don't cancel and so we are left with km²/minute, which doesn't make sense in this problem. 

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Finally, it is interesting to note that there is a connection between the invert-and-multiply rule and the rule for the distribution of the subtraction sign.

In general, without the usual fractional notation, the invert-and-multiply rule looks like this

a ÷ (b ÷ c) = a ÷ b・c.

Here's the rule for the distribution of the subtraction sign 

a – (b – c) = a – b + c.

In both rules, the sign outside the parentheses flips the signs of the operations inside the parentheses.

There are really a lot of math connections happening here. There is a tie-in with the BEDMAS convention for the order of precedence of operations. There is the potential lead-in to additive and multiplicative inverses. There are even hints about how to remove the mystery from rules like a-negative-times-a-negative-is-a-positive.