Equivalent Fractions and Reciprocals

Two fractions are equivalent when they are the same number, so they name the same amount of the same whole. Take a strip of paper that is 24 cm long. Cut it into 4 equal pieces and each piece is 6 cm long, so 3 pieces, or $\frac{3}{4}$ of the strip, measure 18 cm. Now cut every piece in half. The strip is in 8 equal pieces of 3 cm each, and that same 18 cm is 6 pieces, or $\frac{6}{8}$ of the strip. The pieces are half as long and there are twice as many, so the amount has not changed: $\frac{3}{4} = \frac{6}{8}$. The comparison works because both fractions describe the same whole, the one strip.

Why does this work for every fraction? Multiplying any number by 1 leaves it unchanged, and $\frac{n}{n} = 1$ for every nonzero integer $n$. So when you multiply the numerator and the denominator by the same nonzero integer $n$, you multiply the fraction by 1 in disguise:

$\frac{3}{4} = \frac{3}{4} \cdot \frac{2}{2} = \frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8}$

The integer $n$ cannot be zero. That would put a zero in the denominator, and division by zero is undefined. In general:

Read that rule from right to left and it tells you how to simplify. A common factor of two integers is a positive integer that divides both with no remainder, so it is never zero. When the numerator and the denominator share a common factor $n$ greater than 1, dividing both by $n$ takes out a hidden $\frac{n}{n} = 1$. The new fraction is equivalent to the old one, just written more simply.

A fraction is in simplest form when the only common factor of its numerator and denominator is 1. Dividing both by the greatest common factor (GCF), the largest common factor, gets you there in one step. You can also divide out common factors greater than 1 one at a time until the only one left is 1. A negative fraction stays negative when you simplify, because taking out a factor of 1 never changes a sign.

Now put 1 to work a second way. What do you multiply 4 by to get exactly 1? Four copies of $\frac{1}{4}$ make one whole, so $4 \cdot \frac{1}{4} = 1$. What about $\frac{2}{3}$? Multiplying by 3 gives $\frac{2}{3} \cdot 3 = 2$, and then multiplying by $\frac{1}{2}$ gives 1. Together, those two steps multiply by $3 \cdot \frac{1}{2} = \frac{3}{2}$, so $\frac{2}{3} \cdot \frac{3}{2} = 1$.

The reciprocal of a nonzero number is the number you multiply it by to get a product of exactly 1. Its other name is the multiplicative inverse, because multiplying by it undoes a multiplication: $12 \cdot \frac{3}{4} = 9$, and $9 \cdot \frac{4}{3} = 12$ again. Multiplying by a nonzero number and then by its reciprocal is the same as multiplying by 1.

For a fraction $\frac{a}{b}$ with $a \neq 0$ and $b \neq 0$, the reciprocal is $\frac{b}{a}$: the numerator and the denominator trade places. The swap works because the product has the same nonzero number on top and bottom, and a nonzero number divided by itself is 1:

$\frac{a}{b} \cdot \frac{b}{a} = \frac{a \cdot b}{b \cdot a} = 1$

Both conditions matter. You need $b \neq 0$ for $\frac{a}{b}$ to be a number at all, and $a \neq 0$ so that $\frac{b}{a}$ does not have a zero denominator. That is why zero has no reciprocal: $0$ times any number is $0$, never $1$, and swapping $\frac{0}{1}$ would give $\frac{1}{0}$, which is undefined.

Nonzero integers and mixed numbers (a whole number plus a fraction, such as $1\frac{1}{2}$) can be written as fractions, so the same swap finds their reciprocals. You have relied on this idea already: dividing by a nonzero number gives the same result as multiplying by its reciprocal.

Keep this one idea: the number 1 does both jobs. Multiplying or dividing a numerator and a denominator by the same nonzero integer only puts in or takes out a form of 1, so the new fraction is equivalent to the old one, and dividing out the greatest common factor gives simplest form. A nonzero number and its reciprocal multiply to exactly 1 and have the same sign, while zero has no reciprocal. Next, you will convert between decimals, fractions, mixed numbers and percents.