SAT Math — Word Problems and Strategy

Chapter Overview & Weightage

Word problems are the dominant format in SAT Math. Of the 44 questions in the SAT Math section, roughly 30-35 involve some form of contextual or word problem. The math itself is rarely harder than Class 10-12 level — the challenge is extracting the right equation from the problem’s context.

Weightage by domain: Heart of Algebra (linear equations, systems) — about 33%. Problem Solving and Data Analysis — about 29%. Passport to Advanced Math (quadratics, functions) — about 28%. Additional Topics (geometry, trig) — about 10%.

Word problems appear across all domains. Mastering the translation skill — converting English sentences to math equations — is the single highest-leverage skill for SAT Math.

Key Concepts You Must Know

1. Setting up equations from word problems

Identify the unknown → assign a variable
Identify the relationships → write equations
Check units → ensure both sides match

2. System of equations (two unknowns, two relationships)

Substitution method for simple systems
Elimination method for coefficient-heavy systems

3. Percentage and ratio problems

Percentage change: $\frac{\text{new} - \text{old}}{\text{old}} \times 100$
Percentage of a quantity: $\frac{x}{100} \times \text{total}$

4. Rate, distance, time

$d = rt$ (distance = rate × time)
Average speed problems often trap students

5. Function notation

$f(a) = b$ means when input is $a$ , output is $b$
$f(g(x))$ means apply $g$ first, then $f$

6. Data interpretation

Reading graphs, tables, and scatter plots
Understanding slope as rate of change in context

Important Formulas

Model: $y = mx + b$

$m$ = rate of change (slope) — what changes per unit

$b$ = initial value (y-intercept) — starting amount

Common setups: cost per item + fixed fee, speed × time, interest problems

Percentage increase: $\text{New} = \text{Old} \times (1 + \frac{r}{100})$

Percentage decrease: $\text{New} = \text{Old} \times (1 - \frac{r}{100})$

Finding the percentage: $\% = \frac{\text{part}}{\text{whole}} \times 100$

Rate of work = $\frac{1}{\text{time to complete alone}}$

Combined rate = Sum of individual rates

Time together $= \frac{1}{R_1 + R_2}$

Solved Previous Year Questions

SAT Question Type 1 — Linear model (Medium)

A plumber charges $75 for the first hour and$ 45 for each additional hour. If a customer’s bill was $255, how many hours did the plumber work?

Let $h$ = additional hours after the first.

75 + 45h = 255

45h = 180

h = 4

Total hours = $1 + 4 = \mathbf{5}$ hours.

Always check whether the variable represents “additional hours” or “total hours.” In this problem, if you let $h$ = total hours: $75 + 45(h-1) = 255 \Rightarrow 30 + 45h = 255 \Rightarrow h = 5$ . Same answer — both setups work if you’re careful.

SAT Question Type 2 — System of equations (Hard)

A jar contains red and blue marbles. There are 24 marbles total. Red marbles are worth 3 points each and blue ones are worth 5 points each. Total points = 92. How many red marbles are there?

Let $r$ = red, $b$ = blue.

r + b = 24 \quad (1)

3r + 5b = 92 \quad (2)

From (1): $b = 24 - r$ . Substitute in (2):

3r + 5(24 - r) = 92

3r + 120 - 5r = 92

-2r = -28

r = 14

14 red marbles. (Verify: $b = 10$ , points $= 42 + 50 = 92$ ✓)

SAT Question Type 3 — Function in context (Hard)

The function $P(t) = 12000(1.05)^t$ models the population of a town $t$ years after 2010. Which statement best describes the model?

$P(0) = 12000$ → population in 2010 was 12,000
Base $1.05 = 1 + 0.05$ → 5% growth per year
This is exponential growth at 5% per year

The answer would be: “The population was 12,000 in 2010 and grows by 5% each year.”

Difficulty Distribution

Level	%	Characteristics
Easy	30%	One-step translation, direct substitution
Medium	45%	Two equations, percentage with context
Hard	25%	Function notation, multi-step percentage chains

Expert Strategy

Strategy 1 — Read the question first. Before reading the full problem, look at the question being asked. This helps you know what variable to solve for and avoids spending time understanding irrelevant parts.

Strategy 2 — Name your variables explicitly. Write “Let $x$ = number of apples sold” rather than just ” $x$ .” This prevents confusion mid-solution and helps if you need to recheck.

Strategy 3 — Units check. If the answer should be in dollars and your variable is “number of items,” multiplying by price gives dollars. If units don’t work out, the equation is wrong.

Strategy 4 — Plug answers back in. For multiple-choice questions, check your answer by substituting back. This catches arithmetic errors quickly.

Strategy 5 — Don’t over-calculate. SAT word problems rarely require heavy computation. If you’re grinding through long arithmetic, you’ve probably set up the equation inefficiently.

For rate problems: if two people work together, their rates add. If a tap fills a tank in 6 hours, its rate is $\frac{1}{6}$ tank/hour. Two taps with rates $\frac{1}{6}$ and $\frac{1}{4}$ together fill at $\frac{1}{6} + \frac{1}{4} = \frac{5}{12}$ tank/hour, completing it in $\frac{12}{5} = 2.4$ hours.

Common Traps

Trap 1: “Percent more than” vs “percent of.” “A is 20% more than B” means $A = 1.2B$ . “A is 20% of B” means $A = 0.2B$ . These are completely different.

Trap 2: Average speed ≠ average of speeds. If you drive 60 km/h for 1 hour and 120 km/h for 1 hour, average speed = $\frac{180 \text{ km}}{2 \text{ hr}} = 90$ km/h. If you drive 60 km/h for 60 km and 120 km/h for 60 km, average speed = $\frac{120}{\frac{60}{60} + \frac{60}{120}} = \frac{120}{1.5} = 80$ km/h.

Trap 3: In percentage decrease problems, a 20% decrease followed by a 20% increase does NOT return to the original. $100 \to 80 \to 96$ . Net change = $-4\%$ .

Trap 4: In function problems, $f(2x)$ does not mean $2f(x)$ in general. Always substitute the entire argument into the function definition.