Mistake Master
Turning words into equations
Most SAT word problems are a single linear equation in disguise. The math is the part you already know; the points slip away in the translation, when a relationship gets written backward or a rate gets flipped, and at the end, when the answer reported is one step short of the question. Translate carefully, then answer exactly what was asked.
§1
What this topic is about
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Most SAT word problems are a linear equation wearing a story. The work splits in two: translate the words into an equation without bending the relationship, then solve it with the usual balance moves and report the exact thing asked. The algebra is rarely the hard part. The points leak in the translation, and in stopping one step before the question does.
§2
Name the unknown
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Start by deciding what the variable stands for and writing it down in words. If the question is about a number of tickets, let $t$ be that number of tickets. A variable with a clear label keeps every later phrase honest, because you can read each sentence straight onto it.
- Name the quantity the question is actually about, not a quantity sitting near it.
- Write the label in words next to the letter, for example "$d$ is the number of defenders."
- If two things are related, name the one the other is described against, then build the second from it.
§3
Translate the relationship, in the right direction
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Each phrase maps to an operation, and the order matters. "More than" and "less than" attach to the thing named after them, and reading them backward is the most common way a word problem goes wrong.
- "$5$ more than $n$" is $n + 5$. "$5$ less than $n$" is $n - 5$, not $5 - n$.
- "$3$ times as many $A$ as $B$" means $A = 3B$: the thing named first is the larger one.
- "$4$ fewer $A$ than $B$" means $A = B - 4$: start from $B$, then take away.
Worked example. A larger number is $3$ times a smaller number. The larger number is $21$. Find the smaller number.
The larger is three times the smaller: $$\text{larger} = 3 \times \text{smaller}.$$
The larger is $21$, so $21 = 3 \times \text{smaller}$. Divide both sides by $3$: $$\text{smaller} = 7.$$ Multiplying $21$ by $3$ instead, which makes the smaller the larger, is the reversal this section is about.
§4
Rates: keep them right-side up
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A rate is one quantity per another, and which one goes on top is fixed by the units asked. "Miles per hour" is miles divided by hours. Flip it and you get hours per mile, a number that looks like an answer but is the reciprocal of the one wanted.
- Let the unit phrase set the fraction: "dollars per hour" is dollars over hours.
- A flipped rate often comes out as a small decimal or a unit fraction. If the size feels off, check the order.
- When a fixed fee rides along, take it off the total before you divide for the rate.
Worked example. A bill for a $3$-hour job was $150$ dollars, including a $30$ dollar service fee. Find the hourly rate.
The fee is not hourly, so take it off first: $150 - 30 = 120$ dollars of hourly work.
Dollars per hour means dollars over hours: $$\dfrac{120}{3} = 40.$$ Dividing hours by dollars, $\dfrac{3}{120}$, gives the rate upside down.
§5
Solve, then answer what was asked
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Once the equation is built, solving it uses the same balance moves as any linear equation: clear what is added or subtracted, then divide by the coefficient, doing each step to both whole sides. The two ways this stage costs points are stopping at a middle value, and answering for a quantity the question did not ask.
Worked example. Six more than twice a number is $20$. Find the number.
"Six more than twice a number" is $2n + 6$, so $2n + 6 = 20$. Subtract $6$ from both sides: $$2n = 14.$$
Divide both sides by $2$: $$n = 7.$$ Stopping at $2n = 14$ and reporting $14$ answers a step early. Adding the $6$ instead of subtracting it lands on $13$.
Name the unknown, translate each phrase in the direction the words set, keep rates right-side up, then solve and report the exact quantity asked.
§6
Five patterns that cost real points
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Five patterns recur on word problems. They are the same ones the diagnostic routes on.
The relationship gets written backward.
"$5$ less than $n$" becomes $5 - n$, or "$3$ times as many pens as markers" turns the pens into the smaller group. The phrase fixes the direction, and reversing it changes the answer.
Fix. Anchor on the thing named after "than" or "as," and build the other quantity from it: "$5$ less than $n$" is $n - 5$, and "$3$ times as many pens as markers" is pens $= 3 \times$ markers.
The rate gets flipped.
Miles per hour gets computed as hours over miles, or dollars per item as items over dollars. The reciprocal looks like a clean number, so it does not raise a flag.
Fix. Read the unit phrase as the fraction: the unit before "per" goes on top. "Dollars per hour" is dollars over hours.
You stop one step short.
You reach $2n = 14$ and report $14$, or you find the dollars spent on data and report that instead of the gigabytes. The middle value is correct, but it is not the answer.
Fix. Carry the solve all the way to the variable, then run the last step the question names before you commit.
You answer for the wrong quantity.
The question asks for the total but you give one group, or it asks for a count and you compute a cost, or it asks for one of two numbers and you report the other. The test writes a choice equal to each of these on purpose.
Fix. Reread the final line and confirm your number names that quantity. Ignore numbers in the problem that the question does not use.
A sign or a side slips while solving.
A constant moves across the equals sign without changing sign, or an operation lands on one side only. The setup was right, and the arithmetic of the solve undid it.
Fix. Do each step to both whole sides, and flip the sign of any term you move across the equals sign.
Ten quick checks across the patterns: translating the relationship, keeping a rate upright, the early stop, the wrong quantity, and a sign or side slip. Pick or type your answer, then check. Progress is saved.