It is a common misconception that mixed numbers always need to be converted into top-heavy fractions (improper fractions) before we can work with them.
For multiplication and division, converting them first will normally be the sensible approach. However, when adding or subtracting mixed numbers, we can often leave them as mixed numbers and deal with the whole-number and fractional parts separately.
This can be considerably quicker and can also reduce the chance of making unnecessary arithmetic mistakes.
A mixed number is simply a whole number plus a fraction.
For example:
158 7/8 = 158 + 7/8
So if we want to add:
158 7/8 + 23 23/24
we are really calculating:
158 + 7/8 + 23 + 23/24
There is therefore no reason why we cannot add the whole-number parts together and then deal with the fractional parts separately.
Consider:
158 7/8 + 23 23/24
There are two useful ways of looking at this calculation, both of which avoid converting the original mixed numbers into top-heavy fractions.
First add the whole numbers:
158 + 23 = 181
Now add the fractions:
7/8 + 23/24
The lowest common denominator is 24:
7/8 = 21/24
so:
21/24 + 23/24 = 44/24
This simplifies to:
11/6 = 1 5/6
We therefore have:
181 + 1 5/6 = 182 5/6
So:
158 7/8 + 23 23/24 = 182 5/6
We have obtained the answer without ever converting either of the original mixed numbers into a top-heavy fraction.
We can also use exactly the same calculation but deal with the fractional part slightly differently.
As before:
158 + 23 = 181
and:
7/8 + 23/24 = 21/24 + 23/24 = 44/24
Now notice that 44/24 contains one complete whole, because:
44/24 = 24/24 + 20/24
The 24/24 is simply 1, so we can transfer that whole into the whole-number part:
181 + 1 + 20/24
which gives:
182 20/24
and simplifying the fraction:
182 5/6
So again:
158 7/8 + 23 23/24 = 182 5/6
This second method is useful because it shows exactly what is happening when the two fractional parts add up to more than one whole.
We simply take one whole out of the fractional part and add it to the whole-number part.
Nothing about the value of the number has changed. We have simply regrouped it.
Imagine converting:
158 7/8
into an improper fraction before starting.
We would immediately be dealing with a very large numerator, despite the fact that there is absolutely no need to do so.
We would then have to do the same with:
23 23/24
before finding a common denominator and carrying out the addition.
That creates:
In an examination, that can waste valuable time.
If the whole-number parts can simply be added immediately, why make the calculation harder?
Subtraction works on exactly the same principle.
However, occasionally the fractional part of the number being subtracted is larger than the fractional part we start with.
This does not mean that the method fails.
It simply means that we need to regroup in the opposite direction.
With addition, we can move one whole out of the fractional part and into the whole-number part.
With subtraction, we may need to take one whole from the whole-number part and turn it into a fraction.
Consider:
68 1/4 − 25 1/3
This means:
(68 + 1/4) − (25 + 1/3)
which can be written as:
68 + 1/4 − 25 − 1/3
Subtracting the whole numbers gives:
43
and the fractional part is:
1/4 − 1/3
Using a denominator of 12:
1/4 = 3/12
and:
1/3 = 4/12
so we have:
43 + (3 − 4)/12
which is:
43 − 1/12
At this point, nothing has gone wrong.
We simply rewrite 43 as:
42 + 1
and express that 1 as twelfths:
42 + 12/12
We then have:
42 + 12/12 − 1/12
which gives:
42 11/12
Therefore:
68 1/4 − 25 1/3 = 42 11/12
Again, we have subtracted the whole-number and fractional parts separately.
The only additional step was that, because the fractional part became negative, we took one whole from the whole-number part and rewrote it as 12/12.
The overall value did not change.
This is one of the most useful things to notice.
In the addition example:
44/24 = 24/24 + 20/24
so we moved one whole from the fractional part into the whole-number part.
In the subtraction example:
43 = 42 + 12/12
so we moved one whole from the whole-number part into the fractional part.
They are really opposite forms of the same regrouping process.
That helps make the method much more logical.
We are not using a mysterious shortcut. We are simply changing the way in which the same number is written.
Pupils are already familiar with this kind of idea from ordinary subtraction.
If we cannot subtract the units immediately, we borrow from the tens.
With mixed numbers, we are doing something similar.
If we are working in twelfths, then one whole can be written as:
12/12
If we are working in twenty-fourths, then one whole can be written as:
24/24
So we can move one whole backwards or forwards between the whole-number part and the fractional part whenever that makes the calculation easier.
The mathematics is therefore not really a new trick at all.
It is simply another application of place value, equivalent fractions and regrouping.
It is important not to turn this into another rule that pupils apply automatically.
For addition and subtraction, keeping mixed numbers in mixed-number form can often be the quickest and clearest approach.
For multiplication and division, however, converting them to improper fractions is generally much more useful because we cannot simply multiply or divide the whole-number and fractional parts independently.
So the real lesson is not:
“Never convert mixed numbers.”
Nor should it be:
“Always convert mixed numbers.”
The better question is:
“Which form makes this particular calculation easiest?”
This is a good example of why I prefer pupils to understand what a mixed number actually represents rather than simply memorising a procedure.
If a pupil sees:
158 7/8
as nothing more than a special type of fraction that must immediately be converted, they may carry out several unnecessary steps.
If they understand that it simply means:
158 + 7/8
then a much more efficient method becomes obvious.
The same applies to regrouping.
If the fractional part becomes greater than one whole during addition, we can transfer a whole into the whole-number part.
If the fractional part is not large enough during subtraction, we can transfer a whole in the opposite direction.
Once pupils understand this, mixed numbers stop looking like a separate collection of rules and start to behave like numbers that can be reorganised sensibly.
Mathematics is often easier when we understand the structure of what we are doing rather than automatically following a memorised sequence of steps.
In an exam especially, an efficient method can save time, reduce the risk of mistakes and make a complicated-looking calculation considerably simpler.