Ratios, Percents, and Proportion: Chapter Review

Vocabulary and notation

Common multiplier $k$

The size of one equal part: a ratio $a : b : c$ describes amounts $ak$, $bk$, $ck$, sharing one $k$. A ratio hides exactly this one unknown.

Part-to-part and part-to-whole

A ratio $a : b$ compares the parts to each other. The first quantity is $\frac{a}{a + b}$ of the total, not $\frac{a}{b}$: the whole is the sum of the parts.

Conversion factor

A fraction whose top and bottom name equal amounts, like $\frac{12 \text{ in}}{1 \text{ ft}}$, so it equals $1$. Multiplying by it changes units, not the amount.

Dimensional analysis

Arranging conversion factors so unwanted units cancel. The surviving units are the check: a stray $\text{ft}^2/\text{in}$ means a factor is upside down.

Rate $r$ (a percent as a decimal)

The percent divided by $100$, so $30\%$ gives $r = 0.30$. Percent formulas take this decimal, never the whole number $30$.

Percent change factor

The single multiplier that performs a change: $1 + r$ for an increase, $1 - r$ for a decrease, each applied to the original amount.

Constant of proportionality $k$

The one fixed number governing a variation relationship. A single complete set of matching values determines it, and it answers every other question.

Direct variation

$y$ varies directly with $x$ when $y = kx$, so the ratio $y/x$ stays constant. Doubling $x$ doubles $y$, and the graph is a line through the origin.

Inverse variation

$y$ varies inversely with $x$ when $y = \frac{k}{x}$, so the product $xy$ stays constant. Doubling $x$ halves $y$.

Work rate

The fraction of a job finished per unit of time: a job taking $T$ alone runs at rate $\frac{1}{T}$, so rate and time are reciprocals. A drain is a negative rate.

Formulas and theorems

• Finding the common multiplier $k$

  A ratio $a : b : c$ describes $ak$, $bk$, $ck$. Total: $(a + b + c)k = T$. Difference of two terms: $(a - b)k = D$, with $a > b$. One known amount: $ak = A$.

  Use when Every term shares the SAME $k$, never one letter per term. The total form needs the listed terms to make up all of $T$; each form is one linear equation in the single unknown $k$.

  e.g. $7 : 4$ with a difference of $15$: $3k = 15$, so $k = 5$ and the amounts are $35$ and $20$.

• A conversion factor equals $1$

  $$\begin{gathered} 1 \text{ ft} = 12 \text{ in} \\ \implies \frac{12 \text{ in}}{1 \text{ ft}} = \frac{1 \text{ ft}}{12 \text{ in}} = 1 \end{gathered}$$

  Use when The two sides must name genuinely equal amounts; an invented ratio like $\frac{10 \text{ in}}{1 \text{ ft}}$ is not $1$. Both orientations exist: use the one putting the unwanted unit opposite itself.

• Squared and cubed units

  $$\begin{gathered} \left(\frac{12 \text{ in}}{1 \text{ ft}}\right)^{n} = 1 \\ 1 \text{ ft}^2 = 144 \text{ in}^2 \\ 1 \text{ ft}^3 = 1728 \text{ in}^3 \end{gathered}$$

  Text description

  On the left, a 12 inch square ruled into a 12 by 12 grid of 144 unit squares, with one corner square shaded. On the right, a 12 inch cube drawn as a plain outline in three quarter view, with no interior lines: its front face carries one shaded square a twelfth of an edge wide, so a reader can compare one cubic inch with the whole cubic foot. The count of 1728 is stated in the caption rather than drawn, since 12 layers of 12 rows of 12 would not read at this size.

  Use when A unit with the $n$th power hides $n$ lengths, so raise the length factor to the $n$th power. It is still $1$, so the amount is untouched.

  e.g. $2 \text{ yd}^3 \times \left(\frac{3 \text{ ft}}{1 \text{ yd}}\right)^3 = 2 \times 27 = 54 \text{ ft}^3$.

• The percent relationship

  $$p = r\,w, \qquad w = \frac{p}{r}, \qquad r = \frac{p}{w}$$

  Use when $r$ is the percent as a decimal, $w$ the whole (the number after "of"; for a commission, the sales). Dividing needs $r \neq 0$ for the whole, $w \neq 0$ for the rate.

  e.g. $27$ out of $45$: $r = \frac{27}{45} = 0.6 = 60\%$.

• Percent change and its reversal

  $$\text{new} = (1 \pm r)\,w, \qquad w = \frac{\text{new}}{1 \pm r}$$

  Use when $1 + r$ for tax, tip, or markup (a markup is figured on the cost); $1 - r$ for a discount, with $r < 1$ so the reversal can divide. The rate is measured against the original $w$, so undoing divides by that same factor.

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

• Successive percent changes

  $$\begin{gathered} \text{final} = f_1 f_2 \, w \\ (1 + r_1)(1 + r_2) \\ = 1 + r_1 + r_2 + r_1 r_2 \end{gathered}$$

  Use when Each $f$ is $1 + r$ or $1 - r$, acting on the amount the previous change produced. Rates add only when taken of the same base.

  e.g. $1.10 \times 1.10 = 1.21$, so two $10\%$ raises make a $21\%$ raise, not $20\%$.

• Simple interest

  $$I = P r t, \qquad A = P(1 + rt)$$

  Use when $r$ is a decimal rate per period and $t$ counts those same periods (an annual rate needs $t$ in years). The principal $P$ stays fixed, which is what makes it simple, not compound, so the balance grows in a straight line.

  e.g. $1200$ at $4\%$ for $5$ years: $I = 1200 \times 0.04 \times 5 = 240$, so $A = 1440$.

• Direct and inverse variation

  $$\begin{gathered} y = kx, \quad \text{equivalently} \quad \frac{y}{x} = k \\ y = \frac{k}{x}, \quad \text{equivalently} \quad xy = k \end{gathered}$$

  Text description

  Side by side, direct variation graphs as a straight line through the origin while inverse variation graphs as a falling curve that never touches either axis.

  Use when Direct needs the ratio $y/x$ steady across every matching pair, inverse needs the product $xy$ steady. Both need $x \neq 0$ to divide.

  e.g. $(2, 3)$ and $(4, 6)$ share $y/x = 1.5$, so direct; $(2, 6)$ and $(4, 3)$ share $xy = 12$, so inverse.

• Joint and combined variation

  $$y = kxz, \qquad y = \frac{kx}{z}$$

  Use when A quantity varied with directly belongs on top, one varied with inversely on the bottom, and $z \neq 0$. One complete set of matching values fixes $k$.

• Work rates add

  $$\frac{1}{T_1} + \frac{1}{T_2} = \frac{1}{T}$$

  Use when Steady paces on the same whole job, all times in one unit, nobody in the way. An opposing agent enters with a minus sign, and the job finishes only if the net rate is positive.

  e.g. $3$ hours and $6$ hours alone: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$, so together $T = 2$ hours.

• Distance, rate, and time

  $$d = rt, \qquad r = \frac{d}{t}, \qquad t = \frac{d}{r}$$

  Use when The speed must be steady across the stretch measured, and the units must agree (mph with hours). A rate needs $t \neq 0$, a time needs $r \neq 0$.

• Average speed

  $$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$

  Use when The one steady speed that would cover the same distance in the same total time, for any mix of speeds. On a two-leg trip it equals the mean of the two speeds only when the speeds are equal or the two TIMES are equal; over equal distances at two different speeds it lands below the mean.

Problem types, step by step

Exam traps

• Trap Treating a percent increase and an equal percent decrease as cancelling out.

  Fix Factors multiply: $(1 + r)(1 - r) = 1 - r^2$, below $1$ for every $r > 0$. Up $20\%$ then down $20\%$ gives $0.96$, a $4\%$ net loss, in either order.

• Trap Reading "$20\%$ more than $50$" as "$20\%$ of $50$".

  Fix "Of" is the part alone, $0.20 \times 50 = 10$; "more than" is the whole plus the part, $1.20 \times 50 = 60$.

• Trap Reversing a change by taking the percent off the new amount.

  Fix The percent was figured on the original, so divide by the factor: $92$ after $15\%$ tax undoes to $\frac{92}{1.15} = 80$, not to $92 - 0.15 \times 92 = 78.20$.

• Trap Averaging the two speeds of a there-and-back trip.

  Fix Use total distance over total time. Over equal distances at two different speeds the true average always falls below the mean, since more time is spent on the slow leg.

• Trap Averaging or adding the workers' solo times, or answering with the combined rate.

  Fix Only rates add. Sum $\frac{1}{T_1} + \frac{1}{T_2}$, then flip it: $\frac{5}{12}$ of the job per hour means $\frac{12}{5}$ hours.

• Trap Handling an inverse relationship by adding or subtracting the change, or by reaching for $y = kx$.

  Fix Inverse variation scales: $4$ workers taking $9$ days give $k = 36$, so $6$ workers take $\frac{36}{6} = 6$ days, not $9 - 2 = 7$.

• Trap Converting a squared or cubed unit with the length factor used once.

  Fix Area hides two lengths and volume three, so square or cube the whole factor: $12^2 = 144$ and $12^3 = 1728$, never $12$.

• Trap Multiplying by the bottom unit's factor when converting a rate.

  Fix A unit in the denominator has to leave the denominator, so its factor goes in flipped: $90$ km/h needs $\times \frac{1 \text{ h}}{3600 \text{ s}}$, a division by $3600$, not a multiplication.