Proportional Reasoning and Percentages

Five topics on proportional reasoning, the first half of the SAT's Problem-Solving and Data Analysis domain. Ratios and rates, unit conversion, percentages, proportional relationships, and mixtures and weighted averages. The errors here are setup errors: a part treated as a whole, a rate flipped upside down, a percent taken of the wrong base.

Topics

3.1

Ratios and Rates

Unit Conversion

Percentages

Proportional Relationships

Mixtures and Weighted Averages

Lesson

Diagnostic

after diagnostic

Reference The ratio, rate, percent, and proportion rules behind every problem in this unit

Ratio as a fraction (order matters)

$a:b = \dfrac{a}{b}$, so $a:b = 3:5$ means $\dfrac{a}{a+b} = \dfrac{3}{8}$

Unit rate (per ONE of the denominator)

$\text{rate} = \dfrac{\text{amount}}{\text{time or count}}$

Unit conversion (cancel the unit you leave)

$12\ \text{in} \cdot \dfrac{2.54\ \text{cm}}{1\ \text{in}} = 30.48\ \text{cm}$

Percent of a number

$p\%\ \text{of}\ x = \dfrac{p}{100}\,x$

Percent change (divide by the ORIGINAL)

$\%\ \text{change} = \dfrac{\text{new}-\text{original}}{\text{original}} \cdot 100$

Increase or decrease as one multiplier

$x \to x(1 \pm r)$; two changes multiply, they never add

Proportion (cross-multiply)

$\dfrac{a}{b} = \dfrac{c}{d} \iff ad = bc$

Direct proportion (constant ratio, through the origin)

$y = kx,\quad k = \dfrac{y}{x}$

Weighted average (weights sum to the total)

$\bar{x} = \dfrac{n_{1}\bar{x}_{1} + n_{2}\bar{x}_{2}}{n_{1}+n_{2}}$

Unit 3 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Mixed items pulled from across the unit's topics, with no labels showing which is which. It surfaces the misconceptions that still bite when you cannot tell what a question is testing.