🎯 SAT Math Prep

Word Problems
Trick Edition

20 deceptive SAT Math word problems — with built-in traps, memory keys, and full explanations.

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Topic 01
Algebra Word Traps
Translating English into equations — where most SAT points are lost.
🔑
Memory Keys
MORE than = ADD (+)
LESS than = subtract — FLIP order!
OF = MULTIPLY
IS / WAS = EQUALS (=)
TWICE = ×2 (not +2)
DIFFERENCE = bigger − smaller
TRICK
Linear SetupTricky
"5 less than three times a number is 22." What is the number?
⚠️ "Less than" flips word order: "5 less than 3n" = $3n - 5$, NOT $5 - 3n$!
💡 Explanation

"5 less than $3n$" $= 3n - 5$ (the phrase "less than" reverses the order)

Equation$3n - 5 = 22 \Rightarrow 3n = 27 \Rightarrow n = 9$
Wrong trap AWriting $5 - 3n = 22$ gives $n = -\frac{17}{3}$ — negative and not an integer, a clear signal something's wrong.
HARD
Rate · Distance · TimeHard
A train leaves City A at 60 mph. Two hours later, a second train leaves City A in the same direction at 90 mph. How many hours after the second train departs will it catch the first?
⚠️ Set DISTANCES equal — not speeds. The 2-hour head start creates a 120-mile gap!
💡 Explanation

Let $t$ = hours after 2nd train departs.

Train 1$d = 60(t+2)$   Train 2$d = 90t$
Set equal: $60(t+2)=90t \Rightarrow 60t+120=90t \Rightarrow 30t=120 \Rightarrow t=4$ ✓
TRICK
Consecutive Even IntegersTricky
The sum of three consecutive even integers is 78. What is the largest?
⚠️ Consecutive EVEN integers differ by 2, not 1! Use $n,\ n+2,\ n+4$.
💡 Explanation

Consecutive even: $n,\ n+2,\ n+4$

$3n + 6 = 78 \Rightarrow n = 24$. Largest $= 24+4 = 28$ ✓
Wrong trap B (27)Using $n, n+1, n+2$ gives largest $= 27$ — those are consecutive integers, not even!
HARD
Age ProblemHard
Maya is currently 3 times as old as her son. In 12 years, she will be twice as old as her son. How old is Maya now?
⚠️ "In 12 years" — add 12 to BOTH ages. Don't only add 12 to one side!
💡 Explanation

Son's age now $= s$, Maya's age now $= 3s$.

In 12 years: $3s+12 = 2(s+12) \Rightarrow 3s+12=2s+24 \Rightarrow s=12$
Maya now $= 3 \times 12 = 36$ ✓
Topic 02
Topic 02
Percent, Ratio & Proportion
Percent change, markup, discount — the SAT's favorite traps.
📊
Memory Keys
% Change = (new−old)/old × 100
+r% then −r% = net LOSS of r² % always!
After discount = original × (1−rate)
Ratio a:b = a/(a+b) of total
Mixture = weighted sum ÷ total volume
TRICK
Successive % ChangeTricky
A price increases by 20%, then decreases by 20%. The final price is what percent of the original?
⚠️ +20% then −20% does NOT return to 100%! The decrease applies to the higher price.
💡 Explanation

Net multiplier $= 1.2 \times 0.8 = 0.96 = 96\%$

Formula$(1+r)(1-r) = 1 - r^2$. Here: $1 - 0.04 = 0.96$ — always a net LOSS.
Wrong trap A (100%)"They cancel out" is the #1 SAT mistake here!
MED
Ratio PartitionMedium
The ratio of boys to girls in a school is 3:5. There are 240 students total. How many girls are there?
⚠️ Do NOT use $\frac{3}{5} \times 240$. The denominator is $3+5=8$, not 5!
💡 Explanation

Total parts $= 3+5 = 8$. Girls $= \dfrac{5}{8} \times 240 = 150$ ✓

Wrong trap C (144)$\frac{3}{5} \times 240 = 144$ — uses the ratio directly as a fraction instead of a fraction of the total.
HARD
Find Original PriceHard
After a 30% discount, a shirt costs $\$42$. What was the original price?
⚠️ Do NOT add 30% back to $42. The 30% was off the ORIGINAL — not off $42!
💡 Explanation

Original $\times (1-0.30) = 42 \Rightarrow$ Original $= \dfrac{42}{0.70} = \$60$ ✓

Wrong trap A$42 \times 1.30 = \$54.60$ — adds 30% back to the discounted price. Wrong base!
TRICK
Mixture / Weighted AverageTricky
A chemist mixes 200 mL of a 30% acid solution with 300 mL of a 50% acid solution. What is the concentration of the resulting mixture?
⚠️ You CANNOT simply average 30% and 50% = 40%. The volumes differ — use weighted average!
💡 Explanation

Acid: $200(0.30) + 300(0.50) = 60 + 150 = 210$ mL

Total volume $= 500$ mL. Concentration $= \dfrac{210}{500} = 0.42 = 42\%$ ✓
Wrong trap A (40%)Simple average of 30 and 50 — ignores that volumes are unequal!
Topic 03
Topic 03
Functions, Slope & Linear Models
Interpreting slope and y-intercept in context — the #1 most-tested SAT skill.
📈
Memory Keys
Slope = RATE of change (per unit)
y-intercept = STARTING value (when x=0)
Perpendicular = negative RECIPROCAL slope
Parallel = SAME slope
f(−a) = plug negative; (−a)² = POSITIVE!
TRICK
Slope Meaning in ContextTricky
A phone plan charges a flat fee of $\$15$/month plus $\$0.05$ per text. Monthly cost: $C = 0.05t + 15$. What does 15 represent?
⚠️ Slope = RATE (per text). y-intercept = STARTING VALUE (fee when zero texts sent).
💡 Explanation

In $C = 0.05t + 15$ (form $y = mx + b$):

b = 15y-intercept = value when $t = 0$ texts = flat monthly fee = $\$15$ ✓
m = 0.05slope = rate = cost per additional text message ($0.05 each)
HARD
Perpendicular SlopeHard
Line $\ell$ passes through $(2, 1)$ and $(6, 9)$. What is the slope of any line perpendicular to $\ell$?
⚠️ Perpendicular slope = flip AND negate. If slope $= \frac{a}{b}$, perpendicular $= -\frac{b}{a}$.
💡 Explanation

Slope of $\ell = \dfrac{9-1}{6-2} = \dfrac{8}{4} = 2$

Perpendicular slope $= -\dfrac{1}{2}$ (negative reciprocal) ✓
Check$2 \times \left(-\frac{1}{2}\right) = -1$ — perpendicular slopes always multiply to $-1$.
TRICK
Function EvaluationTricky
If $f(x) = 3x^2 - 2x + 1$, what is $f(-2)$?
⚠️ $(-2)^2 = +4$, NOT $-4$. Squaring a negative always gives a POSITIVE!
💡 Explanation

$f(-2) = 3(-2)^2 - 2(-2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = 17$ ✓

Wrong trap C (−7)Calculating $3 \times (-4) = -12$, then $-12-4+1=-15$... sign errors cascade. Always resolve $(-2)^2 = +4$ first!
Topic 04
Topic 04
Quadratics & Nonlinear Traps
Vertex, projectile motion, discriminant — context traps on every SAT.
🪁
Memory Keys
Vertex x = −b/(2a)
Max/Min = vertex y-value (plug back in!)
Hits ground = set h = 0, then solve
Exactly 1 solution = discriminant = 0
No real solution = discriminant < 0
HARD
Projectile Max HeightHard
The height (feet) of a ball thrown upward is $h(t) = -16t^2 + 48t + 4$, where $t$ is seconds. What is the maximum height?
⚠️ Max height = vertex y-value. Do NOT set $h = 0$ — that finds the landing time, not max height!
💡 Explanation

Time at max: $t = -\dfrac{b}{2a} = -\dfrac{48}{2(-16)} = \dfrac{48}{32} = 1.5$ sec

$h(1.5) = -16(1.5)^2 + 48(1.5) + 4 = -16(2.25) + 72 + 4 = -36 + 72 + 4 = 40$ feet ✓
Wrong trap C (48)Using just $b = 48$ as the max — that's a coefficient, not the height!
TRICK
Discriminant = 0Tricky
The equation $2x^2 - 4x + k = 0$ has exactly one real solution. What is $k$?
⚠️ "Exactly one solution" → discriminant $b^2 - 4ac = 0$. Do not try to solve for $x$!
💡 Explanation

Exactly one solution $\Rightarrow b^2 - 4ac = 0$. Here $a=2,\ b=-4,\ c=k$:

$(-4)^2 - 4(2)(k) = 0 \Rightarrow 16 - 8k = 0 \Rightarrow k = 2$ ✓
HARD
Maximum AreaHard
A farmer has 80 meters of fencing to enclose a rectangular garden against a straight barn wall. Only 3 sides need fencing. What is the maximum possible area of the garden?
💡 Explanation

Let width $= x$ (two sides). Long side $= 80 - 2x$.

$A = x(80-2x) = -2x^2 + 80x$. Max at $x = \dfrac{80}{2 \times 2} = 20$ m.
Length $= 80 - 40 = 40$ m. Area $= 20 \times 40 = 800\ \text{m}^2$ ✓
Topic 05
Topic 05
Systems, Statistics & Rates
Averages, overlapping groups, work rates — classic SAT traps in every test.
⚖️
Memory Keys
Mean = sum ÷ count
New mean = new sum ÷ new count
Work rate = 1/time; ADD rates together
No solution = parallel lines (same slope, diff b)
Overlapping = neither group out FIRST, then formula
Inverse ∝ = product constant: t·w = k
TRICK
Average — New ElementTricky
The average of 5 numbers is 18. A 6th number is added and the new average is 20. What is the 6th number?
⚠️ Do NOT say "new number = 20 + 2 = 22". The 6th number must raise ALL 6 positions by 2!
💡 Explanation

Old sum $= 5 \times 18 = 90$. New sum needed $= 6 \times 20 = 120$.

6th number $= 120 - 90 = 30$ ✓
Wrong trap B (22)$18 + 2 \times 2 = 22$ — this confuses average increase with the actual value needed.
HARD
Work RateHard
Pipe A fills a tank in 6 hours. Pipe B fills the same tank in 4 hours. With both pipes open together, how long does it take to fill the tank?
⚠️ The answer is NOT $(6+4)/2 = 5$ hours. ADD the work rates, not average the times!
💡 Explanation

Rate A $= \dfrac{1}{6}$ tank/hr, Rate B $= \dfrac{1}{4}$ tank/hr

Combined rate $= \dfrac{1}{6} + \dfrac{1}{4} = \dfrac{2+3}{12} = \dfrac{5}{12}$ tank/hr
Time $= \dfrac{1}{5/12} = \dfrac{12}{5} = 2.4$ hours ✓
TRICK
System — No SolutionTricky
For what value of $k$ does the system below have no solution?
$$2x + ky = 6$$ $$4x + 6y = 10$$
⚠️ No solution = parallel lines. Make coefficients proportional, but constants NOT proportional. Don't "solve" normally!
💡 Explanation

Parallel lines: coefficient ratios equal → $\dfrac{2}{4} = \dfrac{k}{6}$

$\dfrac{1}{2} = \dfrac{k}{6} \Rightarrow k = 3$
VerifyLine 1: $2x+3y=6$. Line 2: $\div 2 \Rightarrow 2x+3y=5$. Same slope, different constant → no solution ✓
TRICK
Inverse ProportionTricky
The time $t$ to complete a job varies inversely with the number of workers $w$. If 4 workers finish in 15 days, how many days for 10 workers?
⚠️ Inverse means MORE workers = FEWER days. Use $t \times w = \text{constant}$!
💡 Explanation

$t \times w = k \Rightarrow 15 \times 4 = 60$

$10 \times t = 60 \Rightarrow t = 6$ days ✓
Wrong trap B (37.5)Using direct proportion $\frac{4}{15}=\frac{10}{t}$ — completely backwards! More workers means less time.
HARD
System of EquationsHard
Adult tickets cost $\$12$ and child tickets cost $\$7$. A total of 50 tickets were sold for $\$450$. How many adult tickets were sold?
💡 Explanation

Let $a$ = adult, $c$ = child tickets.

$a + c = 50$ and $12a + 7c = 450$. Substitute $c = 50-a$:
$12a + 7(50-a) = 450 \Rightarrow 5a + 350 = 450 \Rightarrow 5a = 100 \Rightarrow a = 20$ ✓
Check$12(20) + 7(30) = 240 + 210 = 450$ ✓
TRICK ★
Overlapping GroupsTricky ★
In a survey of 100 students: 60 like math, 50 like science, and 20 like neither. How many like both?
⚠️ Subtract the "neither" group FIRST. At least one subject = 100 − 20 = 80, NOT 100. This is the most commonly missed step!
💡 Explanation

At least one subject $= 100 - 20 = 80$

$|M \cup S| = |M| + |S| - |M \cap S| \Rightarrow 80 = 60 + 50 - |M \cap S|$
$|M \cap S| = 110 - 80 = 30$ ✓
Wrong trap A (10)Using 100 instead of 80: $100 = 60+50-x \Rightarrow x=10$ — forgot to remove the "neither" group!
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