Core Topics · High-Frequency Traps
Q 01
TRAP: "twice as many" ≠ always 2x → read direction carefully
A store sells notebooks and pens. The number of pens sold is 8 more than twice the number of notebooks sold. If the store sold a total of 62 items, how many notebooks were sold?
Mini Example
"Apples are 3 more than twice the oranges. Total = 15."Let oranges = \(x\). Then apples \(= 2x+3\).
\(x + (2x+3) = 15 \Rightarrow 3x = 12 \Rightarrow x = 4\)
Q 02
TRAP: % of ≠ % more → "30% more" means ×1.3, not ×0.3
A jacket originally costs \(\$80\). It is first discounted 25%, then the sale price is increased by 10%. What is the final price in dollars? (Round to nearest cent.)
Mini Example
\$100 → down 20% → \$80 → up 10% → \$88Key: Apply each % to the current price, not the original.
Q 03
KEY: Set up ratio as fraction → cross-multiply
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?
Mini Example
Ratio \(2:3\), group A = 10. Then \(\frac{2}{3}=\frac{10}{x} \Rightarrow x=15\). Total \(=25\).
Q 04
TRAP: Don't find both variables if asked for one → stop early!
Two friends, Ana and Ben, together have \$120. Ana has \$30 more than Ben. If each person spends \$15, how much money does Ana have left?
Mini Example
\(A + B = 120,\; A = B + 30\). Substitute → solve → apply the extra step.Many students forget the final "spends \$15" step — classic SAT trap!
Q 05
KEY: f(a+b) ≠ f(a)+f(b) → always substitute the whole expression
Let \(f(x) = 3x^2 - 2x + 1\). What is the value of \(f(x+1) - f(x)\) when \(x = 2\)?
\(f(x+1) - f(x) = [3(x+1)^2 - 2(x+1)+1] - [3x^2-2x+1]\)
Strategy
Step 1: Compute \(f(3)\). Step 2: Compute \(f(2)\). Step 3: Subtract.Do NOT expand algebraically — plug in \(x=2\) first for speed.
Q 06
ANCHOR: d = r × t → same distance means r₁t₁ = r₂t₂
Car A travels from City X to City Y at 60 mph. Car B travels the same route at 90 mph and departs 30 minutes later than Car A. How many hours after Car A departs does Car B catch up to Car A?
Mini Example
Let \(t\) = time (hrs) Car A has been driving when caught.Car B drives \((t - 0.5)\) hrs. Equal distance: \(60t = 90(t-0.5)\).
Q 07
TRAP: "no real solution" ↔ discriminant < 0, i.e. b²-4ac < 0
The equation \(x^2 - 6x + k = 0\) has exactly one real solution. What is the value of \(k\)?
Discriminant \(\Delta = b^2 - 4ac\). Exactly one solution \(\Rightarrow \Delta = 0\).
Mini Example
\(x^2 - 4x + c = 0\): one solution when \(16 - 4c = 0 \Rightarrow c = 4\).
Q 08
KEY: Similar triangles → ratios of ALL sides are equal
A tree casts a shadow of 15 feet. At the same time, a 6-foot person casts a shadow of 4 feet. How tall is the tree in feet?
Mini Example
\(\dfrac{\text{tree height}}{\text{tree shadow}} = \dfrac{\text{person height}}{\text{person shadow}}\)
Q 09
ANCHOR: Combined rate = 1/a + 1/b (work per unit time)
Printer A can print a report in 4 hours. Printer B can print the same report in 6 hours. If both printers work together, how many hours will it take to finish the report? (Express as a fraction or decimal.)
\(\dfrac{1}{t} = \dfrac{1}{4} + \dfrac{1}{6}\)
Trap
Students often average \(\frac{4+6}{2}=5\). Wrong!Rate addition gives \(t = \frac{12}{5} = 2.4\) hours.
Q 10
TRAP: Adding a value changes BOTH sum AND count → recalculate
The average (mean) of 5 test scores is 78. A sixth test is added, and the new average becomes 80. What was the score on the sixth test?
Mini Example
Old sum \(= 5 \times 78 = 390\). New sum \(= 6 \times 80 = 480\).6th score \(= 480 - 390 = 90\).
Q 11
KEY: Doubling time → multiply by 2ⁿ where n = (total time)/(doubling period)
A bacteria colony starts with 500 bacteria and doubles every 3 hours. How many bacteria will there be after 12 hours?
\(N = N_0 \cdot 2^{t/d}\) where \(d\) = doubling time
Trap
Students multiply \(500 \times 12\). Wrong!After 12 hrs: \(n = 12/3 = 4\) doublings → \(500 \times 2^4\).
Q 12
TRAP: Multiply/divide by NEGATIVE → FLIP the inequality sign!
A company's profit \(P\) (in thousands) satisfies \(-3P + 12 \geq 0\). What is the maximum value of \(P\) that still satisfies the inequality?
Trap
Dividing both sides by \(-3\) flips \(\geq\) to \(\leq\).So \(P \leq 4\). Max value = \(4\).
Q 13
KEY: P(A and B) = P(A) × P(B) only if INDEPENDENT
A bag has 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red? (Express as a simplified fraction.)
\(P = \dfrac{4}{10} \times \dfrac{3}{9}\)
Trap
"Without replacement" means the second draw denominator changes to 9!Many students use \(\frac{4}{10}\times\frac{4}{10}\) — that's WITH replacement.
Q 14
KEY: Midpoint = average of coords; Distance = Pythagorean theorem
Point \(A\) is at \((2, 5)\) and Point \(B\) is at \((8, 13)\). Point \(M\) is the midpoint of \(AB\). What is the distance from the origin \((0,0)\) to point \(M\)? (Round to nearest tenth.)
Strategy
Step 1: Find \(M = \left(\frac{2+8}{2}, \frac{5+13}{2}\right) = (5, 9)\).Step 2: Distance \(= \sqrt{5^2 + 9^2}\).
Q 15
ANCHOR: (concentration)(volume) = amount of solute → set up equation
A chemist has a 30% acid solution and a 70% acid solution. She mixes them to make 200 mL of a 50% acid solution. How many mL of the 30% solution should she use?
\(0.30x + 0.70(200-x) = 0.50 \times 200\)
Trap
Don't just average the percents. The "concentration × volume = amount" equation is the only safe approach.
Q 16
KEY: Arithmetic → add d each time; Geometric → multiply r each time
In a geometric sequence, the 3rd term is 12 and the 6th term is 96. What is the 1st term of the sequence?
\(a_n = a_1 \cdot r^{n-1}\)
Strategy
\(\dfrac{a_6}{a_3} = r^3 \Rightarrow \dfrac{96}{12} = 8 \Rightarrow r=2\).Then \(a_3 = a_1 \cdot r^2 \Rightarrow 12 = a_1 \cdot 4\).
Q 17
TRAP: Arc length uses RADIANS or degrees-to-fraction-of-circle
A circle has a radius of 10 cm. A sector of the circle has a central angle of 72°. What is the arc length of that sector? (Give answer in terms of \(\pi\).)
\(\text{Arc length} = \dfrac{\theta}{360} \times 2\pi r\)
Mini Example
90° arc of radius 5: \(\frac{90}{360}\times 2\pi(5) = \frac{1}{4}\times 10\pi = \frac{5\pi}{2}\)
Q 18
KEY: |x - a| = b → two cases: x - a = b OR x - a = -b
A machine produces bolts whose diameter must be within 0.05 mm of the target diameter of 10 mm. Which of these is the largest acceptable diameter? The acceptable range satisfies \(|d - 10| \leq 0.05\). What is the maximum value of \(d\)?
Trap
Many students only find one solution. Absolute value always gives TWO:\(d - 10 = 0.05\) or \(d - 10 = -0.05\).
Maximum \(d = 10.05\).
Q 19
TRAP: "Less than" reverses direction → "5 less than x" = x−5, NOT 5−x
Three times a number is 14 more than twice the number. Four less than the number is multiplied by 5. What is the result?
Strategy
Step 1: Find the number: \(3n = 2n + 14 \Rightarrow n = 14\).Step 2: "Four less than the number" \(= n - 4 = 10\).
Step 3: Multiply by 5 → result = \(50\).
Q 20
KEY: Compound → A = P(1 + r/n)^(nt); Simple → A = P(1 + rt)
Maria invests \$2,000 at an annual interest rate of 6%, compounded annually, for 3 years. Meanwhile, Jake invests \$2,000 at 6% simple interest for 3 years. How much more money does Maria have than Jake after 3 years? (Round to nearest cent.)
Compound: \(A = 2000(1.06)^3\) | Simple: \(A = 2000(1 + 0.06 \times 3)\)
Trap
Many students compute only one of the two. You must compare both final amounts and subtract.