Mixed Numbers

What a mixed number means

A mixed number is a whole number written directly beside a proper fraction, like $2\tfrac{3}{4}$, and it stands for their sum. By a long-standing convention, a whole number set beside a fraction means add:

So $2\tfrac{3}{4}$ is read “two and three fourths,” and the word “and” is doing real work: it is the plus sign you do not write. The whole-number part counts complete wholes, and the fraction part counts the leftover that has not reached another whole. That fraction part is always a proper fraction, with a numerator smaller than its denominator. Picture three identical bars, each cut into fourths. Fill two of them completely and shade three fourths of the third, and the total shaded amount is $2\tfrac{3}{4}$.

This is genuinely the same number as an improper fraction. Every one of those bars is cut into fourths, so count all the shaded fourths. The two full bars give $4 + 4 = 8$ fourths, and the partial bar gives $3$ more, for $11$ fourths in total. So the picture you just read as $2\tfrac{3}{4}$ is the very same shaded amount as $\frac{11}{4}$:

Recall that $\frac{1}{2}$ and $\frac{2}{4}$ were two names for one value in the equivalent-fractions lesson. A mixed number and an improper fraction are therefore two names for one value in exactly the same way. A mixed number is easier to picture and to compare at a glance, since you can see immediately that $2\tfrac{3}{4}$ sits between $2$ and $3$. An improper fraction is easier to calculate with. The reason is that the rules for multiplying, dividing, adding, and subtracting fractions all expect a single numerator over a single denominator.

Why “side by side” means add, not multiply

In algebra, and even in plain arithmetic once a fraction is involved, two numbers written next to each other normally means multiply. Yet $2\tfrac{3}{4}$ means $2 + \tfrac34$, an addition. Why the exception?

The honest answer is that it is a convention. It was settled long before algebra fixed the “side by side means multiply” rule, and kept because it is so convenient for everyday measuring. Reading $2\tfrac{3}{4}$ as a product would give $2 \times \tfrac34 = \tfrac{3}{2}$, which is less than $2$, an odd thing to write a "$2$" in front of. Reading it as a sum gives a value between $2$ and $3$, which matches what the leading $2$ promises. That is not special to $2\tfrac{3}{4}$. A mixed number always pairs a whole number with a proper fraction, a fraction whose value is less than $1$. If the two were meant to be multiplied, the result would be smaller than the whole number. This is because multiplying by a proper fraction shrinks a quantity.

From an improper fraction to a mixed number

Sometimes an answer comes out as a top-heavy fraction and you want to see its size. Going from $\frac{11}{4}$ to $2\tfrac{3}{4}$ is the conversion you reach for whenever that happens. The tool is division with remainder, the same idea that the very first fractions lesson used to read the fraction bar as a division.

Recall what the bar means: $\frac{11}{4}$ is $11 \div 4$. So ask the plain division question, how many whole groups of $4$ fit inside $11$, and what is left over? Four goes into $11$ two times, using up $8$, and leaves a remainder of $3$:

The quotient $2$ counts the complete groups of four fourths, and four fourths is one whole. So $2$ is the number of whole units, the whole-number part. The remainder $3$ is the count of fourths that did not make it into a complete whole. So it is $3$ leftover fourths, the fraction part $\tfrac34$. Putting the whole part and the leftover fraction together gives the mixed number:

The leftover is always a proper fraction, and that is not luck. A remainder is by definition smaller than the number you divided by, so the remainder is always less than the denominator. That means the fraction part always has a numerator smaller than its denominator.

To rewrite an improper fraction as a mixed number, divide the numerator by the denominator. The quotient is the whole-number part, the remainder is the new numerator, and the denominator stays the same. Then simplify the fraction part if it is not already in lowest terms. When the division comes out even, with remainder $0$, the fraction is really a whole number. Then there is no fraction part at all, for example $\frac{12}{4} = 3$.

The grouping is worth doing with your hands. In the figure below, each dot is one of the equal parts. The row width is how many of those parts it takes to fill one whole. Set the dots to $11$ and the row width to $4$ to lay out eleven fourths.

Read the mixed number straight off the rows. Two complete rows are two whole units, and the hollow dots are the fourths that could not finish a third row. Those leftover fourths are the remainder that sits over the denominator, so $\frac{11}{4} = 2\tfrac{3}{4}$. Step the count down and the leftover shrinks, $3$ then $2$ then $1$. It shrinks until at $8$ dots the rows come out even, and $\frac{8}{4}$ is the whole number $2$ with no fraction part at all.

The row width is the denominator, so changing it changes what the dots are made of. Put the count back to $11$ and set the row width to $3$. The same eleven dots regroup into $3$ full rows with $2$ left over, which is the conversion $\frac{11}{3} = 3\tfrac{2}{3}$ done by eye.

From a mixed number to an improper fraction

The reverse conversion is the one you do most often. That is because to multiply, divide, add, or subtract mixed numbers you almost always rewrite them as improper fractions first.

A mixed number is a sum, $2\tfrac34 = 2 + \tfrac34$, so the task is to add the whole number to the fraction. The adding and subtracting lesson in this chapter showed how to add a whole number to a fraction. Write the whole number over the common denominator. Each whole is $\frac{4}{4}$, because four fourths make one whole, so $2$ wholes is $2 \times 4 = 8$ fourths:

Now both terms are fourths, so add the numerators:

Look at what the arithmetic actually did: it multiplied the whole number by the denominator and then added the numerator, all over the original denominator.

Multiply the whole number by the denominator, add the numerator, and write the result over the same denominator. For $3\tfrac{2}{5}$ that is $3 \times 5 + 2 = 17$ over $5$, giving $\frac{17}{5}$, which matches the checkpoint above run in reverse. The denominator never changes, because rewriting a whole number in fifths does not change the size of the pieces. The rewriting changes only how many of those pieces you have.

Multiplying and dividing mixed numbers

Multiplication and division both use one strategy: turn each mixed number into an improper fraction, then apply the fraction rule you already know. Take $1\tfrac12 \times 2\tfrac13$. Rewrite each factor as an improper fraction, $1\tfrac12 = \frac{3}{2}$ and $2\tfrac13 = \frac{7}{3}$, then multiply across and cancel where you can:

The last steps simplify $\frac{21}{6}$ by its greatest common factor $3$ to $\frac{7}{2}$. They then convert back to the mixed number $3\tfrac12$ by dividing $7 \div 2 = 3$ remainder $1$.

For multiplication and division, converting first is the only sensible route. A tempting shortcut, multiplying the whole parts and the fraction parts separately, simply does not work. That is because a mixed number is a sum, and a product of sums is not the product of the parts. Division works the same way: convert, then flip and multiply, using the reciprocal of the divisor. Convert the answer back to a mixed number if the problem asks for that form.

Adding and subtracting mixed numbers

Addition and subtraction can also be done by converting to improper fractions, and that always works. But there is a second method that is often quicker and matches how you would do it in your head. That method handles the whole parts and the fraction parts separately.

Take $2\tfrac14 + 1\tfrac12$. Add the whole numbers, $2 + 1 = 3$, and separately add the fractions, $\tfrac14 + \tfrac12$. The fractions need a common denominator: $\tfrac12 = \tfrac24$, so $\tfrac14 + \tfrac24 = \tfrac34$. Combine the whole part and the fraction part:

This separate-parts method is allowed because adding is free to regroup: $2 + \tfrac14 + 1 + \tfrac12$ can be rearranged as $(2 + 1) + (\tfrac14 + \tfrac12)$ without changing the total. When the fraction parts add up to $1$ or more, the extra whole has to be carried over to the whole-number part. For instance $\tfrac34 + \tfrac34 = \tfrac64 = 1\tfrac12$, so that whole $1$ joins the others. Subtraction has the mirror-image wrinkle: the fraction you are subtracting may be bigger than the fraction you have. In that case you must borrow one whole and rewrite it as a fraction first.

Common mistakes