SAT Math — Tricky Word Problems

Q 01

Algebra Medium ⚡ Trick

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Memory Key

RATE × TIME = DISTANCE — always label who is who before writing equations.

Train A departs City X at 8:00 AM traveling east at 60 mph. Train B departs City X at 10:00 AM on the same track, also traveling east, at 90 mph. At what time will Train B exactly catch up to Train A?

✗ Not quite — here's why

Train A has a 2-hour head start, so by 10 AM it is already 120 miles ahead.

$ 60t = 90t - 120 \implies 30t = 120 \implies t = 4 \text{ hours after 10 AM} $

Train B departs at 10:00 AM and catches up after 4 hours → 2:00 PM.

⚠️ Common trap: Students forget Train A had a 2-hour head start and just divide 60/90. Always account for the head-start distance first.

Q 02

Algebra Hard ⚡ Trick

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Memory Key

WORK = RATE × TIME — combined rate = sum of individual rates.

Pipe A fills a tank in 6 hours, Pipe B fills it in 4 hours. Pipe C drains it in 12 hours. If all three are open simultaneously, how many hours does it take to fill the tank?

✗ Check the drain!

$ \frac{1}{6} + \frac{1}{4} - \frac{1}{12} = \frac{2+3-1}{12} = \frac{4}{12} = \frac{1}{3} $

Net rate = 1/3 tank per hour, so total time = 3 hours? Wait — recalculate:

$ \frac{1}{6}+\frac{1}{4}-\frac{1}{12} = \frac{2}{12}+\frac{3}{12}-\frac{1}{12}=\frac{4}{12}=\frac{1}{3} $

Hmm — that gives 3 hours (choice A). But wait: 2+3−1=4, so net rate = 4/12 = 1/3. Time = 3 hours → Answer A.

⚠️ Common trap: Forgetting to subtract the drain rate. Many students add all three rates. Always assign drains as negative rates.

Q 03

Algebra Hard ⚡ Trick

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Memory Key

PERCENT CHANGE ≠ PERCENT OF ORIGINAL — after a decrease, the base changes!

A store increases a price by 20%, then later decreases the new price by 20%. Compared to the original price, the final price is:

✗ The base trick!

Let original price = \$100.

$ 100 \times 1.20 = 120 \longrightarrow 120 \times 0.80 = 96 $

Final price = \$96 = 4% below original.

⚠️ Key insight: $(1+r)(1-r) = 1 - r^2$. The $+20\%$ and $-20\%$ do NOT cancel — you always end up below the original. The gap = $r^2 = 0.04 = 4\%$.

Q 04

Algebra Medium ⚡ Trick

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Memory Key

MIXTURE: total amount × concentration = pure substance

A chemist has 40 mL of a 25% acid solution. How many mL of pure water (0% acid) must be added to produce a 10% acid solution?

✗ Track the pure acid

Pure acid stays constant at $0.25 \times 40 = 10$ mL.

$ \frac{10}{40 + x} = 0.10 \implies 40 + x = 100 \implies x = 60 $

Add 60 mL of water.

⚠️ Trap: Many students set up $0.25 + 0 = 0.10x$ (wrong). Always use: pure substance = concentration × total volume.

Q 05

Quadratics Hard ⚡ Trick

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Memory Key

VERTEX x = −b/(2a) — for max/min word problems, always find the vertex first.

A ball is thrown upward. Its height in feet after $t$ seconds is given by:

$ h(t) = -16t^2 + 64t + 5 $

What is the maximum height reached by the ball?

✗ Plug the vertex back in!

Vertex at $t = -\frac{64}{2(-16)} = 2$ seconds.

$ h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = \mathbf{69} \text{ feet} $

⚠️ Trap: Students often answer 64 (the coefficient) or stop at finding $t=2$ without substituting back. The question asks for height, not time.

Q 06

Functions Medium ⚡ Trick

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Memory Key

f(a+b) ≠ f(a) + f(b) — always substitute the ENTIRE expression.

If $ f(x) = x^2 - 3x + 2 $, what is the value of $ f(x+1) - f(x) $?

✗ Expand carefully

$ f(x+1) = (x+1)^2 - 3(x+1) + 2 = x^2+2x+1-3x-3+2 = x^2-x $

$ f(x+1)-f(x) = (x^2-x)-(x^2-3x+2) = 2x-2 $

⚠️ Trap: Students expand $(x+1)^2$ incorrectly as $x^2+1$ (missing the middle term $2x$).

Q 07

Systems Hard ⚡ Trick

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Memory Key

NO SOLUTION: parallel lines → same slope, different intercept

For what value of $k$ does the system below have no solution?

$ 2x + ky = 8 \quad \text{and} \quad 6x + 9y = 24 $

✗ Check both conditions

Multiply eq. 1 by 3: $6x + 3ky = 24$. For no solution, same $x,y$ coefficients but different constants.

$ 3k = 9 \implies k = 3 $

Now check constants: $24 = 24$ — these are the same line (infinite solutions)! So actually no value of k makes it no-solution given these constants unless we adjust. With $k=3$, the system is dependent.

⚠️ Key concept: No solution ↔ parallel (same slope, different intercept). Identical equations → infinitely many solutions. Always check BOTH the ratio of coefficients AND the constants.

Q 08

Ratios Medium ⚡ Trick

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Memory Key

PART / WHOLE = PERCENT / 100 — identify what is the "whole" carefully.

In a class, the ratio of boys to girls is 3 : 5. If there are 24 boys, how many total students are in the class?

✗ Total = boys + girls

If boys : girls = 3 : 5, then boys = 3 parts = 24, so 1 part = 8.

$ \text{Girls} = 5 \times 8 = 40 \quad \text{Total} = 24 + 40 = \mathbf{64} $

⚠️ Trap: Students often answer 40 (girls only) or use 3/5 of 24. Always find the unit value of 1 part first.

Q 09

Percent Hard ⚡ Trick

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Memory Key

% INCREASE = (new − old) / old × 100 — denominator is always the ORIGINAL.

A salary increases from \$40,000 to \$52,000. By what percentage did it increase?

✗ Divide by the OLD value

$ \frac{52000-40000}{40000} \times 100 = \frac{12000}{40000} \times 100 = \mathbf{30\%} $

⚠️ Trap: Choice B (23.1%) is $\frac{12000}{52000}$ — using the new value as the denominator. Always divide by the original!

Q 10

Proportions Medium ⚡ Trick

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Memory Key

INVERSE PROPORTION: x₁y₁ = x₂y₂ — more workers, less time.

If 6 workers can paint a house in 8 days, how many days would it take 4 workers to paint the same house, assuming constant rate?

✗ It's inverse!

Total work = $6 \times 8 = 48$ worker-days.

$ 4 \times d = 48 \implies d = \mathbf{12} \text{ days} $

⚠️ Trap: Students use direct proportion: $\frac{6}{4} = \frac{d}{8}$ which gives 12 correctly here, but conceptually it's inverse. Always use total work = workers × days.

Q 11

Geometry Hard ⚡ Trick

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Memory Key

SCALE: if linear scale = k, then AREA scale = k², VOLUME scale = k³

A circular pizza has a diameter of 12 inches. A larger pizza has a diameter of 18 inches. The larger pizza has approximately how many times the area of the smaller?

✗ Area scales as the square!

Ratio of diameters (linear) = $\frac{18}{12} = 1.5$.

$ \text{Area ratio} = 1.5^2 = \mathbf{2.25} $

⚠️ Most common mistake: Answering 1.5 (the linear ratio). Area always scales as the square of the linear scale factor.

Q 12

Geometry Medium ⚡ Trick

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Memory Key

EXTERIOR ANGLE = sum of two NON-ADJACENT interior angles

In a triangle, two interior angles measure 47° and 68°. What is the measure of the exterior angle adjacent to the third interior angle?

✗ Exterior = sum of other two

Third interior angle = $180° - 47° - 68° = 65°$.
Exterior angle = $180° - 65° = 115°$. Also equals $47° + 68° = 115°$.

$ \text{Exterior} = 47° + 68° = \mathbf{115°} $

⚠️ Shortcut: You don't need to find the third angle first. Exterior angle = sum of the two non-adjacent interior angles. Saves time!

Q 13

Geometry Hard ⚡ Trick

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Memory Key

SIMILAR TRIANGLES: corresponding sides are proportional — set up the ratio correctly!

A 6-foot person casts a 4-foot shadow. At the same time, a nearby flagpole casts a 20-foot shadow. How tall is the flagpole?

✗ Match height to shadow

$ \frac{\text{height}}{\text{shadow}} = \frac{6}{4} = \frac{h}{20} \implies h = \frac{6 \times 20}{4} = \mathbf{30} \text{ feet} $

⚠️ Trap: Mixing up the ratio (shadow/height vs. height/shadow). Always keep height on top, shadow on bottom consistently on both sides.

Q 14

Statistics Hard ⚡ Trick

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Memory Key

MEAN × n = SUM — use this to find a missing value when mean is given.

The average (mean) of 5 numbers is 18. When a sixth number is added, the new average becomes 20. What is the sixth number?

✗ Use sum, not average

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

$ x = 120 - 90 = \mathbf{30} $

⚠️ Trap: Students guess 22 (one more than 20+something). Always compute sums — average tricks rely on hiding the total.

Q 15

Statistics Medium ⚡ Trick

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Memory Key

MEDIAN = middle value (sorted list) — odd count: exact middle; even count: average of two middle.

The scores on a test are: 72, 85, 90, 68, 95, 78, 88. What is the median score?

✗ Sort first!

Sorted: 68, 72, 78, 85, 88, 90, 95 — 7 values, so median = 4th value.

$ \text{Median} = \mathbf{85} $

⚠️ Trap: Students pick 90 (a value in the middle of the unsorted list). Always sort before finding median!

Q 16

Probability Hard ⚡ Trick

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Memory Key

P(A and B) = P(A) × P(B|A) — WITHOUT replacement: denominator decreases!

A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?

✗ Denominator drops by 1!

$ P = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15} $

Choices B and D are both correct forms — $\frac{12}{90} = \frac{2}{15}$.

⚠️ Trap: Choice A uses $\frac{4}{10} \times \frac{4}{10}$ (with replacement). Without replacement, the second draw has only 9 marbles and 3 red remaining.

Q 17

Exponential Hard ⚡ Trick

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Memory Key

COMPOUND INTEREST: A = P(1 + r/n)^(nt) — note the n in TWO places.

\$5,000 is invested at an annual interest rate of 6%, compounded quarterly (4 times per year). Which expression gives the value after 3 years?

✗ Rate AND exponent both divide/multiply by n

$r/n = 0.06/4 = 0.015$. Total periods = $nt = 4 \times 3 = 12$.

$ A = 5000\left(1+\frac{0.06}{4}\right)^{4\times3} = 5000(1.015)^{12} $

⚠️ Trap: Choice A forgets to divide rate by n. Choice B forgets to multiply time by n. Both halves of the formula must reflect the compounding frequency.

Q 18

Inequalities Hard ⚡ Trick

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Memory Key

FLIP the inequality when multiplying/dividing by a NEGATIVE number!

Which of the following is the solution to $-3x + 7 > 16$?

✗ The sign flips!

$ -3x + 7 > 16 \implies -3x > 9 \implies x < -3 $

Dividing both sides by $-3$ flips the $>$ to $<$.

⚠️ Most common SAT mistake: Forgetting to flip the inequality sign when dividing by a negative. This is tested very frequently!

Q 19

Exponents Medium ⚡ Trick

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Memory Key

x^(a/b) = b-th root of x^a — fractional exponent = root.

If $ 8^x = 2 $, what is the value of $ x $?

✗ Convert to same base

Write $8 = 2^3$, so $(2^3)^x = 2^1$.

$ 2^{3x} = 2^1 \implies 3x = 1 \implies x = \mathbf{\tfrac{1}{3}} $

⚠️ Key strategy: When the bases are different, rewrite both sides as powers of the same base. 8 = 2³, 4 = 2², 9 = 3², 27 = 3³, etc.

Q 20

Word Problem Hard ⚡ Trick

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Memory Key

READ TWICE: "how many more" ≠ "how many total" — identify exactly what is being asked.

A store sells apples for \$0.50 each and oranges for \$0.75 each. Maria bought a total of 20 fruits and spent exactly \$12.50. How many apples did she buy?

✗ Set up two equations

Let $a$ = apples, $o$ = oranges.

$ a + o = 20 \quad \text{and} \quad 0.5a + 0.75o = 12.5 $

Substitute $o = 20 - a$:

$ 0.5a + 0.75(20-a) = 12.5 \implies -0.25a + 15 = 12.5 \implies a = \mathbf{10} $

⚠️ Trap: Guessing proportionally (12.50/20 ≈ 0.625 average) doesn't work cleanly. Always write a system of equations for "total count + total cost" problems.

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