Fractions

Outline — by the end of this section you will be able to

Objectives

1. Explain what a fraction means, as a part of a whole and as a division Def 0.3.1
2. Build equivalent fractions and simplify to lowest terms Def 0.3.2–0.3.3
3. Convert improper ↔ mixed, and place a negative sign correctly Def 0.3.4–0.3.5
4. Multiply and divide, and say why dividing means multiplying by the reciprocal Def 0.3.6–0.3.8
5. Add and subtract using the least common denominator Def 0.3.9–0.3.10
6. Simplify complex fractions and order operations containing them Def 0.3.11

§0.3.1 — Visualizing and simplifying fractions

Three of eight equal pieces

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

The word equal is doing real work. Hack the pan into 8 pieces of wildly different sizes and grab 3, and you do not have $\dfrac{3}{8}$ of anything.

§0.3.1 — Definition

Fraction

The denominator can never be zero: $\dfrac{5}{0}$ asks you to cut a whole into zero equal parts, which describes nothing. Same undefined division as §0.2.

§0.3.1 — Definition

Equivalent Fractions Property

Left to right it builds up to a bigger denominator; right to left it reduces. Both work because multiplying top and bottom by $c$ is multiplying by $\dfrac{c}{c}$, which is 1.

Insight — slicing does not feed more people

More pieces, not more pizza

Cutting a pizza into 16 slices instead of 8 does not give you more pizza. It gives you more pieces, each one half as big.

Equivalent fractions are the same amount described with a different piece size.

§0.3.1 — Definition

Lowest terms

§0.3.1 — Worked example

Simplifying to lowest terms

Context Pause — factors cancel, terms do not

The most common fraction error in algebra

$\dfrac{3 \cdot 5}{3 \cdot 7}$ really is $\dfrac{5}{7}$, but $\dfrac{3 + 5}{3 + 7}$ is not. In the first the 3 is being multiplied, so it is a factor. In the second it is being added, so it is a term.

Check with numbers: $\dfrac{3+5}{3+7} = \dfrac{8}{10} = \dfrac{4}{5}$, nowhere near $\dfrac{5}{7}$. The two 3s look identical on the page and are doing different jobs.

§0.3.1 — A habit that protects you

Can the 4s cancel?

Before cancelling anything, ask whether the numerator and denominator are each a single product. If either carries a $+$ or $-$ at the top level, nothing inside can be cancelled until you factor it — a Chapter 8 skill.

§0.3.1 — Definition

Proper and improper fractions

§0.3.1 — Definition

Mixed number

† A number beside a fraction means addition; a number beside a variable means multiplication. $2\tfrac{3}{4}$ is a sum, $2x$ is a product — one more reason algebra prefers $\tfrac{11}{4}$.

Insight — where the quarters are hiding

Counting the pieces inside the wholes

Each whole holds 4 quarters, so 5 wholes hold 20 of them.

That is all the "multiply by the denominator" step is doing — counting the quarter-pieces buried inside the whole numbers before adding the 3 loose ones.

§0.3.1 — Worked example

Converting both directions

§0.3.1 — Negative fractions

Three places, one number

Context Pause — keep the sign out in front

Equal in value, unequal in practice

$-\dfrac{2}{3}$, $\dfrac{-2}{3}$ and $\dfrac{2}{-3}$ are the same number, but only the first is easy to keep track of.

A minus sign buried in a denominator gets lost partway through a long problem, and once it is lost you will not find it by rereading. Make moving it to the front automatic.

§0.3.1 — Your turn

Try It Now 0.3.1

§0.3.2 — Multiplying and dividing fractions

Fraction multiplication

You have $\tfrac{3}{7}$ of a pan and want $\tfrac{2}{5}$ of that. Cutting into 7 strips then each into 5 makes $35$ pieces; you keep $6$. The word "of" signals multiplication, exactly as §0.1's table promised.

§0.3.2 — A better order to work in

Cancel first, then multiply

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.

§0.3.2 — Worked example

Cancelling before multiplying

† This is still cancelling factors, allowed only because multiplication is the sole operation in sight. The moment a sum appears in a numerator, the same-looking move becomes wrong.

§0.3.2 — Definition

Reciprocal

A reciprocal keeps the sign of the original, and zero has none — no number times 0 gives 1. That is "you cannot divide by zero" wearing a different hat, and it is why every division rule below carries a nonzero condition.

§0.3.2 — Definition

Fraction division

§0.3.2 — Worked example

Dividing by a fraction

Insight — small pieces fit many times

Why the answer gets bigger

How many quarters fit inside 3? Twelve, because each whole holds four of them. So $3 \div \tfrac{1}{4} = 12$, which is just $3 \cdot 4$.

Dividing by something smaller than 1 gives a bigger answer, and multiplying by its reciprocal is the only move that does the same.

§0.3.2 — One you can picture, one you can prove

Why multiplying by the reciprocal works

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

The reciprocal is chosen for exactly one reason: it is the multiplier that turns the denominator into 1 so it disappears. Nothing else about it is special — and §0.3.4 uses the same move on complex fractions.

§0.3.2 — Your turn

Try It Now 0.3.2

§0.3.3 — Adding and subtracting fractions

Fraction addition and subtraction

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 2 feet plus 5 feet is 7 feet and not 7 square feet.

Insight — fractions are like terms in disguise

"Ninths" is playing the role of $x$

$3x + 5x = 8x$ works because both terms count the same kind of thing. $\dfrac{2}{9} + \dfrac{5}{9} = \dfrac{7}{9}$ works for exactly the same reason.

§0.1 called those like terms. Unlike denominators are unlike terms — and that is precisely why they have to be renamed before they can combine.

§0.3.3 — Why you never add the denominators

A sum cannot equal one of its addends

Multiplication does multiply denominators — but it answers a different question. A part of a part really is smaller; addition combines pieces of the same-size whole, and that whole does not change.

§0.3.3 — Definition

Least common denominator

For 12 and 18: $12 = 2^2 \cdot 3$, $18 = 2 \cdot 3^2$, so take two 2s and two 3s — $2^2 \cdot 3^2 = 36$.

§0.3.3 — Two patterns worth memorizing

Finding the LCD

§0.3.3 — Worked example

Adding with unlike denominators

† Whatever you multiply a denominator by, multiply its numerator by too. Multiplying only the bottom changes the number instead of renaming it — and nothing about $\tfrac{5}{36}$ looks wrong on the page.

§0.3.3 — Worked example

A subtraction that comes out negative

You are not required to use the least common denominator. Over 60 the same problem gives $\tfrac{42}{60} - \tfrac{50}{60} = -\tfrac{2}{15}$ — same answer, bigger numbers, one more round of simplifying. The LCD is a convenience, not a law.

§0.3.3 — Check which one the problem is asking for

Multiplying against adding

The fourth row is the one to watch: cancelling across an addition is the fraction version of cancelling a term, wrong for exactly the reason §0.3.1 gave.

§0.3.3 — Your turn

Try It Now 0.3.3

§0.3.4 — Complex fractions and order of operations

The fraction bar is a grouping symbol

Treat each as though wrapped in invisible parentheses. On one line with $\div$ you would have to spell them out: $(5 + 3 \cdot 4) \div (2^3 - 6)$. The bar supplies the grouping for free — which is exactly why algebra prefers it.

§0.3.4 — Worked example

The bar groups the top and the bottom

† With signs in it: $\dfrac{6 - 10}{-2 - 2}$ has numerator $-4$ and denominator $-4$, so the value is $\dfrac{-4}{-4} = 1$ by the like-signs rule from §0.2.

§0.3.4 — Definition

Complex fraction

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

§0.3.4 — Worked example

Simplifying a complex fraction

§0.3.4 — Worked example

A complex fraction with a sum on top

§0.3.4 — Worked example

Order of operations with fractions

† $\left(\tfrac{2}{5}\right)^2$ and $\tfrac{2^2}{5}$ are different numbers — the §0.1 placement warning, in fraction form.

§0.3.4 — Worked example

Evaluating at a negative value

† Those parentheses are not decoration: without them, $2x$ at $x = -\tfrac{1}{2}$ invites the reading $2 - \tfrac{1}{2}$, and the whole problem goes wrong on the first line.

Context Pause — a zero denominator has no value

Every fractional expression hides a restriction

$\dfrac{2x}{x+5}$ means nothing when $x = -5$, because the denominator becomes zero.

Such a value is not merely awkward to work with — it is not in the expression's domain at all, so the expression has no value there. Finding that restriction is the first move in almost every Chapter 3 problem.

§0.3.4 — Fractions in a real setting

Same two numbers, opposite operation

† The built-in check: $\tfrac{5}{8} + \tfrac{3}{8} = \tfrac{8}{8} = 1$, so the remaining $\tfrac{3}{8}$ accounts for the other $18. Deciding which setup a situation calls for is the real skill — §0.5 gives you the properties to rearrange relationships on purpose.

§0.3.4 — Your turn

Try It Now 0.3.4

Key Terminology — the eleven words this section defines

Key terms

§0.3 — Conclusions

What to carry forward

Next: §0.4 Decimals — the same numbers written a different way, plus roots and the real number line. Back to start.