0.3 Fractions

Learning Objectives

By the end of this section, you will be able to:

The integers from Section 0.2 count things that come in whole pieces. Most of what you measure does not. Two and a half hours of practice, three quarters of a tank of gas, a recipe scaled down to five sixths — none of those can be written with an integer, and rounding them off throws away information you may need later.

Fractions handle those in-between amounts exactly. That word matters. Section 0.4 covers decimals, which describe the same quantities in a different notation but sometimes only approximately: one third is \(0.333\dots\) forever, so any decimal you actually write down is a little bit wrong. A fraction is never a little bit wrong. That is why algebra keeps answers in fraction form and converts to decimals at the very end, if at all.

Here is the thing to watch for as you read. Most fraction mistakes are not arithmetic mistakes. They happen when a rule from one operation leaks into a different operation. Multiplication really does go straight across; addition does not, and never has. Cancelling works on factors; it does not work on terms. This section is careful to tell you not just what each rule says, but exactly where it stops.

0.3.1 Visualizing and Simplifying Fractions

Cut a pan of cornbread into 8 equal pieces and take 3 of them. You have taken \(\dfrac{3}{8}\) of the pan. The bottom number says how many equal pieces the whole was cut into, and the top number says how many of those pieces you are talking about.

The word equal is doing real work in that definition. If you hack the pan into 8 pieces of wildly different sizes and grab 3, you do not have \(\dfrac{3}{8}\) of the pan. Every part-of-a-whole reading of a fraction assumes the pieces are the same size.

Two conditions in the definition are worth pulling out on their own, because both come back later. First, the denominator can never be zero: \(\dfrac{5}{0}\) asks you to cut a whole into zero equal parts, which describes nothing at all. That is the same undefined division you met in Section 0.2, and spotting when a variable denominator could turn into zero is a job you will do over and over in Chapter 3.

Second, a fraction is also a division. The expression \(\dfrac{a}{b}\) means \(a \div b\). That is not a separate meaning bolted on; it is the same idea coming at you from the other direction. Split 3 pans of cornbread evenly among 8 people and each person gets \(3 \div 8 = \dfrac{3}{8}\) of a pan, which is exactly what the picture already showed.

The second row generalizes into a rule you will use constantly. Any nonzero number divided by itself is 1, so a fraction whose numerator and denominator match is one whole:

That small fact is the engine behind everything else in this subsection.

Definition 0.3.1 — A fraction: the denominator names the equal pieces, the numerator counts them.

Equivalent fractions

Now slice each of those 8 cornbread pieces in half. There are 16 pieces on the table, and the 3 you took have become 6. Nothing about how much cornbread you have changed — only how finely it was cut. So

Read left to right, that property builds a fraction up to a bigger denominator. Read right to left, it reduces one. Both directions are allowed for the same reason: multiplying the numerator and denominator by the same \(c\) is really multiplying by \(\dfrac{c}{c}\), which is 1, and multiplying by 1 cannot change what a number is worth.

The requirement that \(c \neq 0\) is not fussiness. Multiplying top and bottom by 0 would turn every fraction into \(\dfrac{0}{0}\), which is undefined and tells you nothing about what you started with.

Definition 0.3.2 — Equivalent Fractions Property: finer pieces, the same amount.

Simplifying to lowest terms

To simplify a fraction, divide the numerator and the denominator by their greatest common factor. If the greatest common factor does not jump out at you, divide by any common factor you do see and repeat — you land in the same place, it just takes more steps.

Definition 0.3.3 — Lowest terms: the point where no common factor is left to group by.

Two of those rows are worth a comment. The last one shows that simplifying does not always change the fraction: when the greatest common factor is 1, you are already finished. The row above it shows a fraction simplifying all the way down to a whole number, which happens whenever the denominator divides the numerator evenly.

Cancel factors, not terms

This is the most common fraction error in all of algebra, so it is worth stating exactly. Simplifying divides the whole numerator and the whole denominator by a common factor. A factor is something being multiplied. If a piece of the numerator is being added instead, it is a term, and it cannot be cancelled.

Check the bad one with actual numbers: \(\dfrac{3+5}{3+7} = \dfrac{8}{10} = \dfrac{4}{5}\), which is nowhere near \(\dfrac{5}{7}\). The two 3s look identical on the page, but in one expression the 3 is a factor and in the other it is a term.

Here is the habit that protects you. Before cancelling anything, ask whether the numerator and the denominator are each written as a single product. If either one has a \(+\) or a \(-\) sitting at the top level, nothing inside it can be cancelled until you factor it first, which is a skill Chapter 8 builds properly.

Improper fractions and mixed numbers

Neither form is more correct than the other, and they are useful in different places. Mixed numbers read the way people talk, which is what you would say at a lumber yard. Improper fractions are much easier to compute with, which is why algebra almost always converts to them first.

To go from improper to mixed, divide the numerator by the denominator. The quotient is the whole-number part. The remainder is the new numerator, and the denominator stays put. To go the other way, multiply the whole number by the denominator, then add the numerator. The denominator does not change.

That second procedure is not a recipe you have to take on faith. Since \(5\frac{3}{4}\) means \(5 + \frac{3}{4}\), and \(5 = \frac{20}{4}\), the sum is \(\frac{20}{4} + \frac{3}{4} = \frac{23}{4}\).

Definition 0.3.5 — A mixed number is the same pieces gathered into wholes plus a remainder.

One warning about mixed-number notation. Writing a whole number next to a fraction means addition, but writing a number next to a variable means multiplication, as Section 0.1 established. So \(2\frac{3}{4}\) is \(2 + \frac{3}{4}\), while \(2x\) is \(2 \cdot x\). The two notations look parallel and mean opposite things, which is one more reason algebra prefers improper fractions — \(\frac{11}{4}\) carries no such ambiguity.

Negative fractions and where the sign goes

A fraction can be negative, and the sign can be written in three different places. All three mean the same number.

The sign rules from Section 0.2 are what make that true. A fraction is a division, and a division with exactly one negative sign among its parts comes out negative. It makes no difference whether that sign is sitting on the numerator, on the denominator, or out in front.

That last row is the like-signs-give-a-positive rule from Section 0.2, and it is the fraction version of \(-(-6) = 6\).

Make moving the sign to the front an automatic habit. When you simplify \(\dfrac{-24}{36}\), for instance, the greatest common factor of 24 and 36 is 12, so the fraction reduces to \(\dfrac{-2}{3}\), and you write that as \(-\dfrac{2}{3}\).

0.3.2 Multiplying and Dividing Fractions

Multiply the numerators, multiply the denominators, then simplify. That really is the whole rule:

Here is the picture behind it. You have \(\frac{3}{7}\) of a pan of cornbread, and you want \(\frac{2}{5}\) of that. Cutting the pan into 7 strips and then cutting each strip into 5 pieces makes \(5 \cdot 7 = 35\) small pieces in the whole pan, and you are keeping \(2 \cdot 3 = 6\) of them. The denominators multiply because the two rounds of cutting compound, and the numerators multiply for the same reason. Notice also that the word "of" is signaling multiplication here, exactly as the translation table in Section 0.1 promised.

Signs work the way Section 0.2 said: like signs give a positive product, unlike signs give a negative one. A whole number joins in by writing it over 1, since \(n = \dfrac{n}{1}\).

Definition 0.3.6 — Two rounds of cutting compound, so both numerators and both denominators multiply.

Simplify before you multiply

Every product in that table needed simplifying afterward, and the numbers you had to simplify were bigger than the ones you started with. There is a better order to work in: cancel the common factors first, then multiply.

That is allowed because the Equivalent Fractions Property does not care which numerator sits above which denominator. Once the product is written as one fraction, \(\dfrac{a \cdot c}{b \cdot d}\), any factor on top can be cancelled against any factor on the bottom.

Two ways through the same product. Both columns compute \(\dfrac{14}{15} \cdot \dfrac{25}{21}\):

Same answer, but the largest product the right-hand column ever formed was \(2 \cdot 5 = 10\), against \(14 \cdot 25 = 350\) on the left. The bigger the numbers get, the more that gap matters. It matters most once the fractions contain variables: a variable factor cancels cleanly, while a variable product has to be expanded and then factored all over again.

One caution follows straight from Section 0.3.1: this cancelling is still cancelling factors. It is allowed here only because multiplication is the sole operation in sight. The moment a sum shows up in a numerator, the same-looking move becomes wrong.

Reciprocals

Two facts in that table matter later on. A reciprocal keeps the sign of the original number, since flipping a fraction over does not change whether it is positive or negative, and \(-\frac{5}{3}\) times \(-\frac{3}{5}\) is \(+1\) by the like-signs rule. And zero has no reciprocal at all, because no number times 0 gives 1. That is the same fact as "you cannot divide by zero" wearing a different hat, and it is why every division rule below comes with a nonzero condition attached.

Definition 0.3.7 — A reciprocal is the number that turns a product into 1.

Dividing by a fraction

To divide by a fraction, multiply by its reciprocal instead. Only the second fraction gets flipped — flipping both, or flipping the first one, is a common slip that throws the answer off badly enough that a rough estimate will catch it.

Definition 0.3.8 — Dividing counts how many fit inside, which is what the reciprocal does.

Why multiplying by the reciprocal works

Memorizing "flip and multiply" will get you through the exercises. Understanding it keeps you from using it in the wrong place, so here are two explanations — one you can picture, one you can prove.

The picture version is about counting. Division asks how many copies of the divisor fit inside the dividend, and when the divisor is small, a lot of them fit. The reciprocal of a small number is a large number, so multiplying by it produces the same growth that dividing by a small piece does.

The proof version is short. Write the division as a fraction, then multiply the top and the bottom by the reciprocal of the divisor. That move is legal — it is the Equivalent Fractions Property, multiplying by a well-chosen form of 1.

The reciprocal gets chosen for exactly one reason: it is the one multiplier that turns the denominator into 1 so it disappears. Nothing else about it is special. Hold onto that argument, because Section 0.3.4 uses the same move on complex fractions, and it is the reason both topics live in one section.

0.3.3 Adding and Subtracting Fractions

The denominator does not change because it is not a quantity you are adding. It is the name of the unit. Two ninths plus five ninths is seven ninths for the same reason that 2 feet plus 5 feet is 7 feet and not 7 square feet. You count the objects; you do not count the noun.

Subtraction needs no separate treatment here either. The numerators are integers, so Section 0.2 already covers them, negative results included.

Definition 0.3.9 — Like denominators add their counts; the denominator names the unit.

Why you never add the denominators

Adding the denominators instead would give \(\frac{2}{8}\), which is \(\frac{1}{4}\) — the same quarter you started with. That rule says adding a quarter to a quarter changes nothing at all, and two quarters of a pizza is plainly half a pizza. Any rule that returns a sum equal to one of the things being added is broken on its face.

The confusion is understandable, because multiplication does multiply the denominators. But multiplication is answering a different question — a part of a part — and parts of parts really are smaller. Addition combines two pieces of the same-size whole, and the size of that whole does not change just because you put pieces together.

The least common denominator

Unlike denominators cannot be added until you make them alike. Any common denominator gets the job done, but the smallest one keeps the numbers manageable.

There are two ways to find it. You can go by inspection, running through multiples of the larger denominator until you hit one that the smaller denominator also divides: for 6 and 8, try 8, then 16, then 24, and 24 works because it is \(6 \cdot 4\). Or you can use prime factorization, which keeps working when the numbers get ugly. Factor each denominator into primes, then take each prime the greatest number of times it shows up in any single factorization.

For 12 and 18, note that \(12 = 2^2 \cdot 3\) and \(18 = 2 \cdot 3^2\). Take two 2s, which is the most in either, and two 3s for the same reason, giving \(2^2 \cdot 3^2 = 36\).

Two patterns in that table are worth memorizing. When the denominators share no factor, the LCD is just their product. When one denominator divides the other, the larger one is already the LCD.

Definition 0.3.10 — The LCD is the first value both ladders of multiples land on.

Adding with unlike denominators

The procedure has four steps: find the LCD, build each fraction up to it with the Equivalent Fractions Property, add the numerators, and simplify.

Whatever you multiply a denominator by, you have to multiply its numerator by as well. Multiplying only the bottom changes the number instead of renaming it, and that mistake hides well — nothing about \(\frac{5}{36}\) looks wrong on the page, it simply is not equal to \(\frac{5}{12}\).

You are not actually required to use the least common denominator. Running that same problem over 60 gives \(\frac{42}{60} - \frac{50}{60} = \frac{-8}{60} = -\frac{2}{15}\), which is the same answer reached through bigger numbers and one extra round of simplifying. The LCD is a convenience, not a law.

Signed fractions

Nothing new is required here. Convert each fraction to the common denominator, then let the integer rules from Section 0.2 handle the numerators. Writing negative fractions with the sign up in the numerator during this step keeps the bookkeeping where you can see it.

That last row uses the rule from Section 0.2 that subtracting a negative is the same as adding, so \(-10 - (-3)\) becomes \(-10 + 3 = -7\). Rewriting the subtraction as an addition before you touch the numbers costs one extra line and prevents most of the sign errors that happen here.

Knowing which procedure you need

The two procedures are so different that the most valuable habit is checking which one the problem is asking for before you write anything down.

That fourth row is the one to watch. Cancelling across an addition is the fraction version of cancelling a term, and it is wrong for exactly the reason Section 0.3.1 gave.

0.3.4 Complex Fractions and Order of Operations

Section 0.1 listed the fraction bar among the grouping symbols, right alongside parentheses and brackets. That was not a technicality. It changes the order you have to work in.

Treat the numerator and the denominator as though each one were wrapped in invisible parentheses. Written on one line with a division sign, such an expression would need those parentheses spelled out: \((5 + 3 \cdot 4) \div (2^3 - 6)\). The bar supplies the grouping for free, which is exactly why algebra prefers it to \(\div\).

Here is a second one with signs in it. In \(\dfrac{6 - 10}{-2 - 2}\) the numerator works out to \(-4\) and the denominator also works out to \(-4\), so the value is \(\dfrac{-4}{-4} = 1\) by the like-signs rule from Section 0.2.

Complex fractions

These look intimidating and are not. That main bar is a division sign, so a complex fraction is just a division problem written vertically, and Section 0.3.2 already told you what to do with a division problem.

Definition 0.3.11 — The main bar of a complex fraction is the division that happens last.

When the numerator or the denominator has an operation inside it, the grouping rule comes first: finish the top, finish the bottom, and only then divide. Do not try to combine an addition in the numerator with the division in one move — the addition happens inside the grouping, and the division happens after. Skipping that order is the usual failure here, and it produces an answer that looks plausible but is not.

The second row makes a point about size worth repeating. Dividing 6 by a number smaller than 1 gives an answer bigger than 6. If you divide a positive number by a positive proper fraction and your answer comes out smaller than what you started with, you flipped the wrong fraction. (The size check needs both numbers positive — with negatives in play, compare magnitudes instead.)

Order of operations with fractions

The four-level order from Section 0.1 does not change at all. Fractions do not get their own rules; they just make each step take more care, the same way Section 0.2 said about negative numbers.

One fact deserves its own line before the example. An exponent on a fraction applies to the numerator and the denominator both, since \(\left(\frac{2}{5}\right)^2\) means \(\frac{2}{5} \cdot \frac{2}{5}\), which multiplies straight across to \(\frac{4}{25}\). The placement warnings from Section 0.1 still apply, so \(\left(\frac{2}{5}\right)^2\) and \(\frac{2^2}{5}\) are different numbers.

Evaluating expressions containing fractions

To evaluate an expression, substitute the given value and simplify, wrapping the substituted value in parentheses exactly as Sections 0.1 and 0.2 insisted. Those parentheses are not decoration. Without them, \(2x\) with \(x = -\frac{1}{2}\) invites the reading \(2 - \frac{1}{2}\), and the whole problem goes wrong on the first line.

A value that makes a denominator zero is not merely awkward to work with. It is not in the expression's domain at all, so the expression has no value there — the same conclusion Section 0.2 reached about \(\frac{5}{0}\).

Fractions in a real setting

A crew is repainting a hallway. Materials for the whole job cost $48, and the crew has finished \(\frac{5}{8}\) of the hallway. To find what they have used so far, take \(\frac{5}{8}\) of 48, cancelling the 8 into the 48 before multiplying:

So they have used $30 of materials, and the remaining \(\frac{3}{8}\) of the job accounts for the other $18. Notice the check built into that: \(\frac{5}{8} + \frac{3}{8} = \frac{8}{8} = 1\), so the two pieces add to one whole job.

Now flip the question. Suppose you know that $30 of materials covered \(\frac{5}{8}\) of the hallway and you want the cost of the whole job. That is a division:

Same two numbers, opposite operation, because the unknown moved from the part to the whole. Deciding which of those two setups a situation calls for is the real skill, and Section 0.5 gives you the properties that let you rearrange relationships like this one on purpose instead of by guessing.

Problem Set 0.3

Key Terms