Mistake Master
Parts, wholes, and rates
A ratio problem hands you two or three small numbers and hides one big decision: what does each number count? The arithmetic is short division and multiplication. The points slip away when a part gets treated as a whole, when a rate goes into the setup upside down, and at the end, when the right split gets reported as the wrong quantity. Read what each number counts, then answer exactly what was asked.
§1
What this topic is about
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A ratio compares quantities; a rate is a ratio whose two sides carry different units. Every question in this topic reduces to three moves: decide what each ratio number counts (a part or the whole), scale in the direction the question needs, and report the quantity the question names. Each move has its own classic failure, and the SAT writes a wrong answer for every one of them.
§2
Read what each number counts
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The ratio $3:5$ can compare part to part (dogs to cats) or part to whole (red marbles to all marbles), and nothing about the notation tells you which. The sentence does. Getting this read wrong poisons everything after it, because the whole is $8$ parts in one reading and $5$ in the other.
- "The ratio of $A$ to $B$" compares the two named things directly: both numbers are parts, and the whole is their sum.
- "$3$ of every $8$" and "ratio of $A$ to total" are part-to-whole: the second number already contains the first.
- To convert part-to-whole into part-to-part, subtract: $3$ of every $8$ means $3$ against the other $5$.
§3
Split a whole with a ratio
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When both ratio numbers are parts and you know the whole, work through the size of one part. Add the ratio numbers to count the parts, divide the whole by that count, then multiply by the number the question asks about.
- Parts first: $a:b$ makes $a + b$ parts. A three-way ratio $a:b:c$ makes $a + b + c$.
- One part's size is the whole divided by the number of parts, never by one of the ratio numbers.
- Each quantity is its ratio number times the part size.
Worked example. The ratio of dogs to cats at a shelter is $3:5$, and the shelter has $24$ dogs. How many animals are there in all?
The $24$ dogs fill $3$ parts, so one part is $$24 \div 3 = 8 \text{ animals}.$$
Dogs and cats together fill $3 + 5 = 8$ parts: $$8 \cdot 8 = 64 \text{ animals}.$$ Scaling $24$ by $\dfrac{5}{3}$ gives the cats, $40$; treating the $5$ as the whole gives nonsense. Every number in this problem comes from the part size.
§4
Rates and proportions: keep them right-side up
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A rate scales one quantity into another, and its orientation is fixed by the units of the answer. Write the units next to every number: the unit you want must survive, and the unit you are leaving must cancel. An upside-down rate makes the answer shrink when it should grow, which is the fastest self-check there is.
- To go from $B$-units to $A$-units, multiply by $\dfrac{A}{B}$, the rate with $B$ on the bottom.
- In a proportion, matching units sit in matching positions: $\dfrac{\text{mi}}{\text{in}} = \dfrac{\text{mi}}{\text{in}}$, never diagonal.
- More of the input should mean more of the output. If your answer moved the wrong way, the rate was flipped.
Worked example. On a map, $2$ inches represents $5$ miles. How many miles does $7$ inches represent?
The answer is in miles, so miles goes on top: $$\dfrac{5 \text{ mi}}{2 \text{ in}}.$$
$$7 \text{ in} \cdot \dfrac{5 \text{ mi}}{2 \text{ in}} = 17.5 \text{ mi}.$$ The flipped setup, $7 \cdot \dfrac{2}{5} = 2.8$, shrinks the distance below the $5$-mile mark that just $2$ inches already earns, which is the wrong direction.
§5
Report the quantity that was asked
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A ratio problem produces a family of correct numbers: both parts, the whole, the difference, the part size, the unit rate. The test asks for exactly one of them and writes answer choices equal to the others. The last move of every problem in this topic is matching your number to the name in the question.
Worked example. The ratio of boys to girls in a class is $2:3$, and the class has $30$ students. How many girls are there?
$2 + 3 = 5$ parts, each worth $30 \div 5 = 6$ students: $$12 \text{ boys}, \quad 18 \text{ girls}.$$
The question says girls, so the answer is $$18.$$ The $12$, the $30$, and the gap of $6$ are all sitting in the choices, each one correct arithmetic attached to the wrong name.
Decide what each ratio number counts, size one part from the whole, keep every rate right-side up by its units, then report the exact quantity the question names.
§6
Three patterns that cost real points
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Three patterns recur on ratio and rate questions. They are the same ones the diagnostic routes on.
A part gets treated as a whole.
The ratio $3:5$ with $24$ animals becomes $24 \cdot \dfrac{3}{5}$, as if the $24$ were all cats, or the whole gets divided by $5$ alone. The other direction happens too: a part-to-whole ratio like "$3$ of every $8$" gets read as two separate parts.
Fix. Before touching the numbers, say what each one counts. Two parts make a whole of their sum; a part-to-whole pair converts to part-to-part by subtraction.
The rate goes in upside down.
Cups per muffin gets used as muffins per cup, or a proportion is cross-multiplied with miles matched against inches. The flipped answer is a clean-looking number, so nothing feels wrong.
Fix. Write units next to every number and orient the rate so the unwanted unit cancels. Then sense-check the direction: more minutes must mean more pages.
The right split, the wrong quantity.
The work is flawless: $12$ boys, $18$ girls. The question asks for girls and the answer sheet gets $12$, or it asks for one part and gets the whole, or the difference. Nothing in the arithmetic can catch this one.
Fix. Reread the final line of the question, name the quantity it wants out loud, and match your number to that name before you commit.
Ten quick checks across the patterns: reading what each ratio number counts, splitting a whole, keeping rates and proportions right-side up, and reporting the quantity asked. Pick or type your answer, then check. Progress is saved.