Fractions: Chapter Review

Vocabulary and notation

Numerator and denominator

In $\frac{a}{b}$, the denominator $b$ names the SIZE of each part (the whole was cut into $b$ EQUAL parts) and the numerator $a$ COUNTS how many of them you have.

Unit fraction $\frac{1}{b}$

One single equal part. Every fraction is a count of these: $\frac{a}{b}$ is $a$ copies of $\frac{1}{b}$.

Proper fraction

Numerator smaller than the denominator, so the value is less than $1$.

Improper fraction

Numerator at least the denominator, so the value is $1$ or more. $\frac{4}{4}$ is improper yet equals exactly $1$, and an improper answer is never a wrong form to leave.

Equivalent fractions

Two fractions naming the same number, such as $\frac{1}{2}$ and $\frac{2}{4}$. They share one lowest-terms form.

Lowest terms (simplest form)

The numerator and denominator share no common factor above $1$, so nothing is left to divide out.

Mixed number $w\tfrac{a}{b}$

A whole number beside a proper fraction, standing for their SUM: $w\tfrac{a}{b} = w + \tfrac{a}{b}$. The space means plus, not times.

Formulas and theorems

• The bar means divide

  $$\frac{a}{b} = a \div b$$

  Use when $b \neq 0$. Top divided by bottom, so $\frac{3}{4}$ is $3 \div 4$, never $4 \div 3$.

  e.g. $5 \div 8 = \frac{5}{8}$, and $\frac{12}{3} = 4$.

• Zero and one inside a fraction

  $$\begin{gathered} \frac{n}{n} = 1, \qquad \frac{n}{1} = n \\ \frac{0}{b} = 0, \qquad \frac{a}{0} \text{ is undefined} \end{gathered}$$

  Use when $n \neq 0$ in the first two, $b \neq 0$ in the third. A numerator of $0$ is fine; a denominator of $0$ is not, since no number times $0$ returns a nonzero $a$.

• Comparing at a glance

  Same denominator: the larger NUMERATOR is larger. Same numerator: the SMALLER denominator is larger, because fewer cuts make bigger parts.

  Text description

  Two equal bars each with three parts shaded: the quarters on the upper bar cover far more length than the sevenths on the lower one.

  Use when One of the two numbers must genuinely match. A shared numerator must be nonzero, since $\frac{0}{3}$ and $\frac{0}{5}$ both equal $0$.

  e.g. $\frac{5}{8} > \frac{3}{8}$, and $\frac{3}{4} > \frac{3}{7}$.

• Building rule

  $$\frac{a}{b} = \frac{a \times c}{b \times c}$$

  Text description

  Two equal bars, one cut into halves with one part shaded and one cut into quarters with two shaded, whose shadings stop at the same point.

  Use when $b \neq 0$ and $c \neq 0$. The multiplier hits BOTH numbers, and it multiplies rather than adds: $\frac{1+1}{2+1}$ is $\frac{2}{3}$, a different number from $\frac{1}{2}$.

• Simplifying rule

  $$\frac{a}{b} = \frac{a \div c}{b \div c}$$

  Use when $c$ must be a common factor, dividing BOTH $a$ and $b$ exactly, with $c \neq 0$ and $b \neq 0$. Dividing by the greatest common factor reaches lowest terms in one step.

• Cross-product test for equality

  $$\frac{a}{b} = \frac{c}{d} \quad \text{exactly when} \quad a \times d = b \times c$$

  Use when $b \neq 0$ and $d \neq 0$. It runs both ways, so differing products prove the fractions differ. Each numerator pairs with the OTHER denominator, so $a \times b$ tests nothing.

  e.g. $\frac{6}{10}$ and $\frac{9}{15}$: $6 \times 15 = 90 = 10 \times 9$, so they are equal.

• Like-denominator rule

  $$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}, \qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$

  Use when $c \neq 0$, and the SAME $c$ in both. The denominator never moves, and only the numerators combine.

  e.g. $\frac{9}{11} - \frac{4}{11} = \frac{5}{11}$.

• Least common denominator

  $$\text{LCD of } \tfrac{a}{b} \text{ and } \tfrac{c}{d} = \operatorname{lcm}(b, d)$$

  Use when The denominators are whole numbers, so the LCD is a whole NUMBER, with $b \neq 0$ and $d \neq 0$. Any common multiple works, including $b \times d$; the least one only keeps the numbers small. When one denominator divides the other, the LCD is the larger one.

  e.g. $\operatorname{lcm}(6, 8) = 24$, but $\operatorname{lcm}(6, 12) = 12$.

• Multiplication rule

  $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

  Use when $b \neq 0$ and $d \neq 0$. No common denominator is wanted or needed. A whole number joins as $n = \frac{n}{1}$.

• Cancelling before multiplying

  A factor shared by any numerator and any denominator of the product divides out of both first, taken from either fraction.

  Use when The cancel pairs a NUMERATOR with a DENOMINATOR, never a top with a top. It must divide the WHOLE numerator, never one term of a sum: in $\frac{a + c}{b}$ a factor of $a$ alone cancels nothing.

  e.g. $\frac{2}{9} \times \frac{3}{8}$: cancel $2$ with $8$ and $3$ with $9$, leaving $\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$.

• A fraction times its reciprocal

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

  Use when The reciprocal swaps numerator and denominator, so it needs $a \neq 0$ as well as $b \neq 0$: flipping $\frac{0}{b}$ would put $0$ underneath. Every number but $0$ has one.

  e.g. $\frac{3}{5} \times \frac{5}{3} = 1$, and the reciprocal of $6 = \frac{6}{1}$ is $\frac{1}{6}$.

• Division rule (flip the divisor)

  $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

  Use when $b \neq 0$, $d \neq 0$, and the divisor must be nonzero, so $c \neq 0$. ONLY the divisor flips; the first fraction is left exactly as it is.

• Mixed number and improper fraction

  $$\begin{gathered} w\tfrac{a}{b} = \frac{w \times b + a}{b} \\ \frac{n}{d} = q\tfrac{r}{d} \;\text{ where }\; n = q \times d + r \end{gathered}$$

  Text description

  Two whole bars cut into quarters plus a third bar with three of its quarters shaded, numbering eleven shaded quarters in all.

  Use when $b \neq 0$, $d \neq 0$, and $a < b$, so the fraction part is proper. The denominator never changes in either direction. A remainder is always below the divisor, so $0 \le r < d$; when $r = 0$ the value is the whole number $q$.

Problem types, step by step

Exam traps

• Trap Adding the denominators along with the numerators, so $\frac{2}{5} + \frac{1}{5}$ is written as $\frac{3}{10}$.

  Fix The answer is $\frac{3}{5}$: the denominator names the piece size, and combining groups of fifths does not shrink the pieces, so only the counts add.

• Trap Cancelling a number that sits in a SUM, such as striking the $b$'s in $\frac{w \times b + a}{b}$.

  Fix Cancelling divides a factor out of the whole numerator, and being one term of a sum does not make a number a factor of it. $2\tfrac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$, and $4$ does not divide $11$; cancelling the $4$s would claim $2 + 3 = 5$.

• Trap Cancelling a numerator against another numerator, or a denominator against another denominator.

  Fix Every cancel pairs one top with one bottom. In $\frac{3}{4} \times \frac{3}{8}$ the two $3$s are both on top, so nothing cancels and the answer is $\frac{9}{32}$, not $\frac{1}{32}$.

• Trap Flipping the first fraction in a division, or forgetting to flip and multiplying straight across.

  Fix Only the divisor turns over: $\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{5}{4}$. Straight across gives $\frac{5}{9}$, and flipping the first gives $\frac{4}{5}$, the reciprocal of the answer.

• Trap Ordering fractions with a shared numerator by their denominators, smallest denominator first.

  Fix More cuts make smaller pieces, so the order reverses: $\frac{3}{4}$, $\frac{3}{7}$, $\frac{3}{10}$ runs from greatest to least.

• Trap Calling a fraction simplified after dividing out a common factor that was not the greatest.

  Fix Halving $\frac{18}{24}$ gives $\frac{9}{12}$, which still shares a factor of $3$. Divide by the greatest common factor $6$ once to reach $\frac{3}{4}$.

• Trap Multiplying mixed numbers part by part, whole against whole and fraction against fraction.

  Fix A mixed number is a sum, and a product of sums is not the product of the parts. $2\tfrac{1}{4} \times 1\tfrac{1}{3}$ is $\frac{9}{4} \times \frac{4}{3} = 3$, not the $2\tfrac{1}{12}$ that part-by-part suggests.

• Trap Subtracting mixed numbers by taking the smaller fraction from the larger regardless of which one you started with.

  Fix Borrow instead. $5\tfrac{1}{4} - 2\tfrac{2}{3}$ is $2\tfrac{7}{12}$; taking $\tfrac{3}{12}$ from $\tfrac{8}{12}$ the wrong way round returns $3\tfrac{5}{12}$, too big by $\tfrac{5}{6}$.

• Trap Assuming a product must be larger than its factors and a quotient smaller than what it divides.

  Fix Read $\times$ as "of" and $\div$ as "how many fit". A product of two fractions each between $0$ and $1$ falls below both, as $\frac{3}{4} \times \frac{2}{5} = \frac{3}{10}$ does, while $6 \div \frac{3}{4} = 8$ grows.