Fractions and Decimals

Equivalent Fractions and the Number Line

A fraction $\frac{a}{b}$ says: cut one whole into $b$ equal parts and take $a$ of them. A phone battery with 10 bars and 6 of them lit is $\frac{6}{10}$ charged. Three out of five bars lit is $\frac{3}{5}$. These are the same amount of battery.

Fractions that name the same amount are equivalent fractions. You get one from another by multiplying or dividing the numerator and the denominator by the same nonzero number: $\frac{3}{5} = \frac{6}{10} = \frac{9}{15}$.

To test two fractions without drawing anything, use cross-multiplication:

A fraction is in simplest form (lowest terms) when the numerator and denominator have no common factor except 1. To get there, divide both by their GCF.

Fractions on the number line

To place $\frac{a}{b}$ on the number line, split each unit into $b$ equal pieces and count $a$ pieces from 0. Positive numbers go right of 0 and negative numbers go left. An improper fraction like $\frac{3}{2}$ is the same as the mixed number $1\tfrac{1}{2}$, so it sits halfway between 1 and 2. A decimal like $-1.5$ is the same point as $-\frac{3}{2}$.

Comparing: the number farther right is greater. To compare fractions, write them with a common denominator (or as decimals). Every negative number is less than every positive number, and among negatives the one closer to 0 is greater: $-\frac{1}{2} > -\frac{3}{2}$.

Operations with Fractions

Here is a quick review of the four operations, then the new pieces: negative exponents and fractions inside fractions.

• Add or subtract: first rewrite with a common denominator (the LCD), then add or subtract the numerators. The denominator stays the same.

• Multiply: multiply across, numerator times numerator and denominator times denominator. Simplify first if you can.

• Divide: multiply by the reciprocal of the second fraction. Division is often written $\div$ (some problems use a colon $:$). It means the same thing.

Why does division flip the fraction? Dividing by $\frac{1}{4}$ asks "how many quarters fit?" and the answer is 4 times as many, so you multiply by 4.

Order of operations works with fractions exactly as with whole numbers: parentheses, then exponents, then multiplication and division left to right, then addition and subtraction left to right. (You may know it as PEMDAS.) The most common slip is adding before multiplying.

Negative exponents

A negative exponent means "reciprocal." For any nonzero $a$:

So $3^{-1} = \frac{1}{3}$ and $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$. Why? Each time the exponent goes down by 1, you divide by the base: $2^2, 2^1, 2^0, 2^{-1}$ is $4, 2, 1, \frac{1}{2}$.

Fractions inside fractions, and square roots

A complex fraction is a fraction whose numerator or denominator contains a fraction. Work from the inside out: simplify the small pieces, then divide. For example,

A square root of a fraction is the square root of the top over the square root of the bottom: $\sqrt{\frac{4}{9}} = \frac{2}{3}$. If you are given a decimal such as $0.\overline{4}$, first write it as a fraction, as you will see in the next section.

From Fractions to Decimals and Back

Fraction to decimal

To turn a fraction into a decimal, divide the numerator by the denominator. What matters is what kind of decimal comes out:

The bar goes over the digits that repeat. Some problems write this with a hat instead, $0.\widehat{45}$. It means the same thing. Some problems call the second kind a pure repeating decimal and the third kind a mixed repeating decimal.

Why do digits repeat?

When you divide by 7, each remainder can only be 1, 2, 3, 4, 5 or 6 (if it were 0, the division would stop). So within 7 steps a remainder must come back, and from then on the digits repeat. Here is the "remainder wheel" for $1 \div 7$:

Decimal to fraction

Every terminating or repeating decimal can be written as a fraction. Here is how to find it in each case.

Terminating. Put the digits (without the point) over a 1 followed by as many zeros as there are decimal places, then simplify:

Repeating. Multiply by a power of 10 so that when you subtract, the infinite tails cancel.

Back to the start: the calculator's $0.3333333$ is written $0.\overline{3}$. To get the fraction, let $x = 0.\overline{3}$. Then $10x - x = 3.333\ldots - 0.333\ldots$, so $9x = 3$ and $x = \frac{3}{9} = \frac{1}{3}$.

Percents, Fractions and Decimals

A percent is a fraction with denominator 100: $25\%$ means $\frac{25}{100} = \frac{1}{4} = 0.25$. Fraction, decimal and percent are three names for one number.

To turn a percent into a fraction, put it over 100 and simplify. To turn a decimal into a fraction, read it as tenths, hundredths and so on. The same goes for decimals bigger than 1.

To find a percent of an amount, multiply by the decimal. Sales tax of $8\%$ on a $\$50$ jacket is $50 \times 0.08 = \$4$. To find what percent a part is, divide and convert: $12$ out of $80$ is $\frac{12}{80} = 0.15 = 15\%$.

Fractions of a Quantity

"Of" means multiply. To find $\frac{a}{b}$ of an amount, multiply the amount by $\frac{a}{b}$. Equivalently, divide the amount by $b$ to get one part, then multiply by $a$. For example, $\frac{2}{5}$ of $350$ students is $350 \div 5 = 70$ per part, times 2, so $140$.

When several people take pieces of the same whole, add their fractions to see how much is gone. The rest is $1$ minus that total, because the whole is $1 = \frac{\text{all parts}}{\text{all parts}}$.

A Fraction of What Is Left

Some problems take a fraction of the rest. The key idea: a fraction "of what is left" is a fraction of a smaller amount, so you multiply the fractions.

Working backward

Sometimes you know the final amount and need the whole. If a fraction $\frac{p}{q}$ of the total equals a known amount, then $\text{total} = \text{amount} \div \frac{p}{q}$. For example, if $\frac{3}{5}$ of a drive is $90$ GB, the drive holds $90 \div \frac{3}{5} = 90 \cdot \frac{5}{3} = 150$ GB.

Fractions in Geometry

Measurements are not always whole numbers. A side might be $\frac{9}{2}$ ft long. You still use the same formulas, rectangle perimeter $P = 2(\ell + w)$ and area $A = \ell \cdot w$, but with fractions.

What fraction of the area is shaded?

To find the fraction of a figure that is shaded, compare areas, or cut the figure into equal pieces and count. Connect the midpoints of the sides of a square. The inner square is exactly half the big one, because the four corner triangles are each $\frac{1}{8}$ of the whole.

Repeating the process, each new inner square has half the area of the previous one: $\frac{1}{2}$, then $\frac{1}{4}$, then $\frac{1}{8}$, and so on. When a figure is built "recursively," look for the repeated factor.

Rational and Irrational Numbers

Every time an operation could not be done, mathematicians invented new numbers:

• Natural numbers $\mathbb{N} = \{1, 2, 3, \ldots\}$ are the counting numbers. Add 0 to get the whole numbers. You cannot do $3 - 5$ with them.

• Integers $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ fix that. But now you cannot do $3 \div 5$.

• Rational numbers $\mathbb{Q}$ are all numbers that can be written as a fraction $\frac{a}{b}$ with integers $a$ and $b \neq 0$. This includes every integer, every terminating decimal and every repeating decimal.

• Irrational numbers are decimals that go on forever with no repeating block, like $\sqrt{2} = 1.41421356\ldots$ or $\pi = 3.14159265\ldots$

• Rational and irrational numbers together make the real numbers $\mathbb{R}$, which fill the whole number line.

To classify a number, ask for the smallest set that contains it. First simplify: $-\frac{18}{3} = -6$ is an integer (it is also rational, but $\mathbb{Z}$ is smaller). Then $4.25 = \frac{17}{4}$ is rational and not an integer. A repeating decimal is always rational.

Where is $\sqrt{2}$?

Even though its digits never end, $\sqrt{2}$ has an exact place on the number line. Draw a square with side 1. By the Pythagorean Theorem its diagonal is $\sqrt{1^2 + 1^2} = \sqrt{2}$. Swing that length down onto the line with a compass:

Finding numbers between two numbers

Between any two different real numbers there are infinitely many rationals and infinitely many irrationals. To find some, write both endpoints as decimals and compare digit by digit.

Measurement Error

No measurement is perfect. Comparing the measured value with the true value gives two numbers:

The absolute error tells you how far off you were, in the same units as the measurement. The relative error compares the mistake to the size of the thing measured, and it has no units. That is why it lets you compare measurements of different sizes. A smaller relative error means a better measurement.

Errors in scientific notation

When numbers are in scientific notation, $a \times 10^n$ with $1 \le a < 10$, first make the powers of 10 the same, then subtract the front parts and write the answer in scientific notation again.

Fractions and Powers of 2

Doubling is multiplying by 2, and fractions like $\frac{1}{64}$ are powers of 2 with a negative exponent: $\frac{1}{64} = \frac{1}{2^6} = 2^{-6}$. Each hour of doubling multiplies by $2$, which adds 1 to the exponent.

Rates and Mixtures

Work rates

If a faucet fills a pool in $6$ hours, in one hour it fills $\frac{1}{6}$ of the pool. That is its rate. When several things work together, add the rates (use a negative rate for a drain). The time to finish a job is $1 \div \text{rate}$ (or the remaining fraction $\div$ rate).

Mixtures with "parts"

"3 parts gold for every 1 part copper" means $\frac{3}{4}$ of the ingot is gold. In general, a ratio of $p : q$ means the first ingredient is $\frac{p}{p+q}$ of the total. To mix two things, let $x$ be the amount of the first. The second is then $\text{total} - x$. Then write one equation for the amount of the ingredient you care about.

Summary