Ratios, Rates, and Percents: Chapter Review

Vocabulary and notation

Ratio $a : b$

A comparison of two quantities by division, also written $\frac{a}{b}$. Order matters, and it gives a relationship, not a count: $2 : 3$ can describe $4$ and $6$ just as well as $20$ and $30$.

Scale factor

The number a ratio's part is multiplied by to reach a real amount. Both parts share ONE factor, so finding it from the known pair hands you the other.

Rate

A ratio of quantities in DIFFERENT units, so the units do not cancel and stay attached to the answer. The word "per" marks the denominator.

Proportion

A statement that two ratios are equal, $\frac{a}{b} = \frac{c}{d}$, read "$a$ is to $b$ as $c$ is to $d$". Like any equation it is true or false.

Percent $\%$

"Out of $100$", an instruction to divide by $100$: $25\% = \frac{25}{100} = 0.25$, never the plain $25$. $100\%$ is the whole, so above $100\%$ is more than the whole.

Base

The whole a percent is taken OF. $20\%$ of $50$ is $10$ but $20\%$ of $200$ is $40$, so a percent means nothing until its base is named. In a percent change the base is the value BEFORE it.

Formulas and theorems

• Scaling and simplifying a ratio

  $$a : b = ka : kb, \qquad a : b = \frac{a}{b}$$

  Use when $b \neq 0$ and $k \neq 0$. The factor hits BOTH parts, and it multiplies or divides, never adds. Divide by the greatest common factor for lowest terms; two ratios are equivalent exactly when their lowest terms match.

  e.g. $18 : 24$ divides by $6$ to $3 : 4$, and $0.5 : 1.5$ doubles to $1 : 3$.

• Part-to-whole fraction

  $$\text{first part of the whole} = \frac{a}{a + b}$$

  Use when The listed parts must be ALL of the whole and are positive, so the sum is nonzero. It extends to $\frac{a}{a + b + c}$. The part-to-part $\frac{a}{b}$ is a different number.

  e.g. In $3 : 5$ the first quantity is $\frac{3}{8}$ of the total, not $\frac{3}{5}$.

• Splitting a total in a ratio

  $$\text{one share} = \frac{T}{a + b + \cdots}$$

  Text description

  A bar of 400 cut into ten equal cells grouped 3, 2 and 5, so one cell is 40 and the blocks are 120, 80 and 200.

  Use when The ratio numbers must count every share, and any number of terms works. Each amount is its own ratio number times one share.

• Unit rate, and scaling it

  $$\begin{gathered} \text{unit rate} = A \div B \\ \text{total} = \text{unit rate} \times (\text{units of } B) \end{gathered}$$

  Text description

  Two aligned number lines, dollars above and pounds below, stepping together by 1.75 dollars for every one pound.

  Use when $B \neq 0$, and the quantity named after "per" is the one you divide BY. Carry the units, since a bare number does not say which way round it is; the quotient may be a decimal.

  e.g. $7$ dollars for $4$ pounds is $1.75$ per pound, so $3$ pounds cost $5.25$.

• Cross-product test

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

  Use when $b \neq 0$ and $d \neq 0$. It runs both ways, so unequal products PROVE the ratios differ.

  e.g. $\frac{5}{8}$ and $\frac{15}{24}$: $5 \times 24 = 120 = 8 \times 15$, so the proportion is true.

• Missing term of a proportion

  $$\text{blank} = \frac{\text{known diagonal product}}{\begin{array}{c} \text{number diagonally} \\ \text{opposite the blank} \end{array}}$$

  Use when Needs matching quantities in matching positions and a nonzero number diagonally opposite the blank. It works with the blank on top or underneath, and never moves a term across the equals sign.

  e.g. $\frac{9}{n} = \frac{15}{40}$: the known diagonal is $9 \times 40 = 360$, so $n = 360 \div 15 = 24$.

• Percent, fraction, and decimal

  $$p\% = \frac{p}{100} = p \div 100$$

  Use when The $\%$ sign is the division, so it must be cleared before the number enters any arithmetic. Dividing by $100$ shifts the point two places. $p$ may exceed $100$ or itself be a decimal.

  e.g. $0.045 = 4.5\%$, and $\frac{7}{20} = \frac{35}{100} = 35\%$.

• The percent relationship

  $$\frac{\text{part}}{\text{whole}} = \frac{p}{100}$$

  Use when The whole follows "of". Part $= \frac{p}{100} \times$ whole; whole $=$ part $\div \frac{p}{100}$, which needs $p \neq 0$; the percent needs a nonzero whole.

  e.g. What percent of $160$ is $36$? $36 \div 160 = 0.225 = 22.5\%$.

• Percent change

  $$\begin{gathered} \text{percent change} \\ = \frac{\text{new} - \text{original}}{\text{original}} \times 100\% \end{gathered}$$

  Use when The ORIGINAL sits underneath and must be nonzero. Positive is an increase, negative a decrease, normally reported as a positive percent with the word "decrease".

  e.g. $75$ falling to $66$: $\frac{-9}{75} = -0.12$, a $12\%$ decrease.

• The multiplier, forwards and backwards

  $$\begin{gathered} \text{new} = \left(1 \pm \frac{p}{100}\right)\text{original} \\ \text{original} = \frac{\text{new}}{1 \pm \frac{p}{100}} \end{gathered}$$

  Use when $+$ for tax, tip, or markup and $-$ for a discount, where a decrease needs $p \le 100$; reversing needs a nonzero multiplier. Build it from what REMAINS: $p\%$ off leaves $\frac{100 - p}{100}$.

  e.g. $8\%$ tax on $250$ gives $1.08 \times 250 = 270$.

• Stacked percent changes

  $$\begin{gathered} \text{final} = f_1 f_2 \times \text{original} \\ f = 1 \pm \frac{p}{100} \end{gathered}$$

  Text description

  Bars of 100, then 120 after a 20 percent rise, then 96 after a 20 percent fall, ending below the dashed line at the starting height.

  Use when Each factor acts on what the previous change produced, so the percents MULTIPLY rather than add, and the order does not matter. They add only when both are taken of the same base.

  e.g. Two $10\%$ raises give $1.10 \times 1.10 = 1.21$, a $21\%$ rise, not $20\%$.

Problem types, step by step

Exam traps

• Trap Reading a part-to-part ratio as a share of the whole, so "boys to girls is $2 : 3$" makes boys $\frac{2}{3}$ of the class.

  Fix The whole is the SUM of the parts, so boys are $\frac{2}{2 + 3} = \frac{2}{5}$. The $\frac{2}{3}$ compares boys to girls and is nobody's share of a total.

• Trap Scaling or comparing ratios by adding, so $2 : 3$ becomes $3 : 4$, or $\frac{2}{3}$ and $\frac{4}{5}$ look equal because each part grew by $2$.

  Fix Equivalence is multiplicative, so $2 : 3$ scales to $4 : 6$. Cross-multiplying settles the second case: $2 \times 5 = 10$ but $3 \times 4 = 12$.

• Trap Multiplying straight across a proportion instead of along the diagonals, or dividing by the number you just used.

  Fix Multiply the diagonal you know, then divide by the number DIAGONALLY OPPOSITE the blank. In $\frac{4}{9} = \frac{14}{n}$ that is $9 \times 14 = 126$, then $n = 126 \div 4 = 31.5$.

• Trap Building the two sides of a proportion in opposite orders, such as miles over hours on one side and hours over miles on the other.

  Fix Matching quantities belong in matching positions on both sides. A flipped side is false, yet it still cross-multiplies to a confident wrong answer.

• Trap Dividing the change by the NEW value, or reusing one percent for the reverse move.

  Fix The base is the value BEFORE the change. $40$ up to $50$ is $\frac{10}{40} = 25\%$, but $50$ down to $40$ is $\frac{10}{50} = 20\%$, not $25\%$.

• Trap Multiplying by the bare percent, so "increase $200$ by $15\%$" gives $30$ and "$30\%$ off $60$ dollars" gives $18$ dollars.

  Fix Those are only the part that changes. Use the multiplier: $1.15 \times 200 = 230$, and $0.70 \times 60 = 42$ dollars to pay. "$p\%$ more than" means $1 + \frac{p}{100}$ times the base, not $\frac{p}{100}$ times it.

• Trap Expecting a $p\%$ rise and a $p\%$ fall to cancel out.

  Fix The factors multiply to $\left(1 + \frac{p}{100}\right)\left(1 - \frac{p}{100}\right) = 1 - \left(\frac{p}{100}\right)^2$, below $1$ for every $p \neq 0$. Up $20\%$ then down $20\%$ gives $1.2 \times 0.8 = 0.96$, a $4\%$ net loss, in either order.

• Trap Undoing a change by adding the percent back onto the new value.

  Fix The percent was figured on the larger original, so divide by the multiplier: after $20\%$ off, a price of $48$ came from $48 \div 0.80 = 60$, not $48 + 0.20 \times 48 = 57.60$.