SAT Math Mastery — 20 Core Questions

01

LINEAR EQUATIONS ⚠ TRAP

WORKED EXAMPLE

Solve for x: 3(2x − 4) = 18

Step 1 — Distribute: 6x − 12 = 18
Step 2 — Add 12: 6x = 30
Step 3 — Divide: x = 5

YOUR TURN

Solve for x: −2(3x + 5) = 4x − 6

A x = −1

B x = 1/2

C x = −1/2

D x = 2

🚨 TRAP: Forgetting to distribute the negative sign to BOTH terms inside the parentheses! −2 × 3x AND −2 × 5.

Distribute carefully: −6x − 10 = 4x − 6 → move all x to one side → −10x = 4 → x = −2/5 ... wait, recheck: −6x − 10 = 4x − 6 → −10 + 6 = 4x + 6x → −4 = 10x → x = −2/5. (Answer: C)

02

SYSTEMS OF EQUATIONS MOST MISSED

WORKED EXAMPLE

If 2x + y = 10 and x − y = 2, what is the value of x?

Add both equations: 3x = 12 → x = 4
Then: y = 10 − 2(4) = 2

YOUR TURN

3x + 2y = 16
x − 2y = 0

What is the value of y?

A y = 1

B y = 2

C y = 4

D y = 3

🚨 TRAP: The question asks for y, not x! Students solve for x (= 4) and pick answer C without continuing to find y.

Add equations: 4x = 16 → x = 4. Substitute: 4 − 2y = 0 → y = 2. (Answer: B)

03

SLOPE & LINES ⚠ TRAP

WORKED EXAMPLE

Line passes through (1, 3) and (4, 9). What is the slope?

m = (9 - 3) \div (4 - 1) = 6 \div 3 = 2

YOUR TURN

Line k passes through (−2, 5) and (3, −5). What is the y-intercept of line k?

A 1

B −2

C −1

D 2

🚨 TRAP: Students find slope = −2 but forget to plug back in to find b. The slope IS NOT the y-intercept!

m = (−5−5)÷(3−(−2)) = −10÷5 = −2. Use y=mx+b: 5 = −2(−2)+b → 5 = 4+b → b = 1. (Answer: A)

04

INEQUALITIES MOST MISSED

WORKED EXAMPLE

Solve: −3x < 12

Divide both sides by −3 → FLIP sign:
x > −4

YOUR TURN

Which value of x satisfies −4x + 2 ≥ 14?

A x = −4

B x = 3

C x = −2

D x = 4

🚨 TRAP: Not flipping the inequality sign when dividing by −4! Students get x ≤ −3 but then pick the wrong answer.

−4x ≥ 12 → divide by −4, flip sign → x ≤ −3. Which answer is ≤ −3? x = −4. (Answer: A)

05

WORD PROBLEMS ⚠ TRAP

WORKED EXAMPLE

A store sells apples for $0.50 each and oranges for $0.75 each. Maria buys 8 fruits and spends $5.25. How many apples did she buy?

Let a = apples. Then: a + (8−a) = 8 fruits.
0.50a + 0.75(8−a) = 5.25 → −0.25a = −0.75 → a = 3

YOUR TURN

A parking lot charges $3 for the first hour and $1.50 for each additional hour. If Jake paid $10.50 total, how many hours did he park?

A 4 hours

B 5 hours

C 6 hours

D 7 hours

🚨 TRAP: The $3 is for the FIRST hour only. Don't multiply $3 by all hours!

3 + 1.5(h−1) = 10.50 → 1.5h + 1.5 = 10.50 → 1.5h = 9 → h = 6. But 1 + 5 additional = 6 hours total. (Answer: C)

06

QUADRATICS MOST MISSED

WORKED EXAMPLE

Factor: x² + 5x + 6 = 0

Find two numbers that multiply to 6 AND add to 5 → 2 and 3
(x + 2)(x + 3) = 0 → x = −2 or x = −3

YOUR TURN

What are the solutions to x² − 7x + 10 = 0?

A x = 2 and x = 5

B x = −2 and x = −5

C x = 2 and x = −5

D x = 1 and x = 10

🚨 TRAP: The middle term is NEGATIVE (−7x), so both numbers must be negative to add to −7. But −2 × −5 = +10 ✓. Students often pick B forgetting the product must be positive.

(x − 2)(x − 5) = 0 → x = 2 or x = 5. Since the factors are (x−2) and (x−5), the solutions are positive. (Answer: A)

07

FUNCTIONS ⚠ TRAP

WORKED EXAMPLE

If f(x) = 2x² − 3x + 1, what is f(−1)?

Plug in −1: 2(−1)² − 3(−1) + 1 = 2 + 3 + 1 = 6

YOUR TURN

If g(x) = x² − 4x + 3, for what value of x does g(x) = 0?

Which of the following is NOT a solution?

A x = 1

B x = 3

C x = 0

D Both A and B are solutions

🚨 TRAP: The question asks what is NOT a solution. Both x=1 and x=3 work, so x=0 is not a solution — but students rush and pick A or B.

g(0) = 0 − 0 + 3 = 3 ≠ 0. So x = 0 is NOT a solution. (Answer: C)

08

EXPONENTS MOST MISSED

WORKED EXAMPLE

Simplify: 27^(2/3)

∛27 = 3, then 3² = 9

YOUR TURN

Which expression is equal to 16^(3/4)?

A 4

B 8

C 12

D 64

🚨 TRAP: Students multiply 16 × (3/4) = 12 by mistake. The fraction exponent means ROOT first, then POWER — not multiply!

⁴√16 = 2, then 2³ = 8. (Answer: B)

09

POLYNOMIALS ⚠ TRAP

WORKED EXAMPLE

Expand: (x + 3)(x − 5)

F: x² | O: −5x | I: 3x | L: −15
= x² − 2x − 15

YOUR TURN

Which of the following is equivalent to (2x − 1)²?

A 4x² + 1

B 4x² − 4x + 1

C 2x² − 1

D 4x² − 1

🚨 TRAP: (2x−1)² ≠ 4x² + 1 ! Many students forget the middle term when squaring a binomial. Always FOIL completely!

(2x−1)(2x−1) = 4x² − 2x − 2x + 1 = 4x² − 4x + 1. (Answer: B)

10

RATIONAL EXPRESSIONS MOST MISSED

WORKED EXAMPLE

Simplify: (x² − 9) ÷ (x + 3)

Factor numerator: (x+3)(x−3)
Cancel (x+3): result = x − 3 (where x ≠ −3)

YOUR TURN

Which expression is equivalent to (x² + x − 6) ÷ (x − 2) for all valid x?

A x − 3

B x + 3

C x + 2

D x − 2

🚨 TRAP: Students cancel x from both terms without factoring first. You MUST factor the numerator completely before canceling!

x² + x − 6 = (x+3)(x−2). Cancel (x−2) → x + 3. (Answer: B)

11

TRIANGLES ⚠ TRAP

WORKED EXAMPLE

A right triangle has legs 3 and 4. Find the hypotenuse.

3² + 4² = c² \to 9 + 16 = 25 \to c = 5

YOUR TURN

In a right triangle, the hypotenuse is 13 and one leg is 5. What is the length of the other leg?

A 8

B 12

C 11

D √144

🚨 TRAP: Students add instead of subtract: 5² + 13² instead of 13² − 5². The hypotenuse goes on one side ALONE!

5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12. (Answer: B — note D is the same value)

12

CIRCLES MOST MISSED

WORKED EXAMPLE

A circle has diameter 10. What is its area?

radius = diameter ÷ 2 = 5
Area = π × 5² = 25π

YOUR TURN

A circle has a circumference of 14π. What is the area of the circle?

A 7π

B 14π

C 49π

D 196π

🚨 TRAP: From circumference = 14π, students find r = 7 but then confuse 14π as the area. Remember: area = πr², NOT just πr!

2πr = 14π → r = 7. Area = π(7²) = 49π. (Answer: C)

13

ANGLES ⚠ TRAP

WORKED EXAMPLE

Two parallel lines are cut by a transversal. One angle is 65°. The alternate interior angle is also 65°. Its co-interior (same-side) pair is 180° − 65° = 115°.

YOUR TURN

Two parallel lines are cut by a transversal. One angle is (3x + 20)° and its alternate interior angle is (5x − 10)°. What is the value of x?

A x = 10

B x = 15

C x = 20

D x = 25

🚨 TRAP: Students set the angles equal to 180 (thinking co-interior) instead of to each other. Alternate interior angles are EQUAL!

3x + 20 = 5x − 10 → 30 = 2x → x = 15. (Answer: B)

14

VOLUME MOST MISSED

WORKED EXAMPLE

A cylinder has radius 3 and height 5. What is its volume?

V = π(3²)(5) = 45π

YOUR TURN

A cone has the same radius and height as the cylinder above (r = 3, h = 5). What is the ratio of the cone's volume to the cylinder's volume?

A 1 : 2

B 1 : 3

C 2 : 3

D 1 : 4

🚨 TRAP: Students forget the 1/3 in the cone formula OR confuse which is the smaller shape. A cone always fits exactly 1/3 of its matching cylinder!

Cone V = (1/3)πr²h = 15π. Cylinder = 45π. Ratio = 15π : 45π = 1 : 3. (Answer: B)

15

COORDINATE GEOMETRY ⚠ TRAP

WORKED EXAMPLE

Find the midpoint of (2, 4) and (8, 10).

M = ((2+8)/2, (4+10)/2) = (5, 7)

YOUR TURN

The midpoint of segment AB is (3, −1). If point A is (−1, 5), what are the coordinates of point B?

A (7, −7)

B (1, 2)

C (4, −3)

D (7, 7)

🚨 TRAP: Students add instead of using algebra. Set up (−1+x)/2 = 3 and solve, DON'T just add midpoint to A!

(−1+x)/2 = 3 → x = 7. (5+y)/2 = −1 → y = −7. Point B = (7, −7). (Answer: A)

16

MEAN / MEDIAN / MODE MOST MISSED

WORKED EXAMPLE

Data: {2, 4, 4, 6, 8, 10, 10, 10, 12}

Mean = (2+4+4+6+8+10+10+10+12) ÷ 9 = 66 ÷ 9 ≈ 7.3
Median = 8 (5th value when sorted)
Mode = 10 (appears 3 times)

YOUR TURN

A student's test scores are: 70, 85, 90, 90, 95. If the student scores an 80 on the next test, which of the following will INCREASE?

A The mean only

B The median only

C Both mean and median

D Neither

🚨 TRAP: Students assume adding any score changes the median. Adding 80 creates a 6-value set — the new median is the average of the 3rd and 4th values!

Original mean ≈ 86. New mean = (70+80+85+90+90+95)/6 = 510/6 = 85 — mean DECREASED. Sorted new set: 70,80,85,90,90,95. New median = (85+90)/2 = 87.5. Old median = 90. Median DECREASED too. Answer: D — Neither increases.

17

PERCENTAGES ⚠ TRAP

WORKED EXAMPLE

A price increases from $80 to $100. What is the percent increase?

(100 - 80) \div 80 \times 100 = 25%

YOUR TURN

A jacket costs $120 and is on sale for 25% off. After the discount, a 10% sales tax is applied. What is the final price?

A $88.00

B $99.00

C $82.50

D $78.00

🚨 TRAP: Students add 25% off and 10% tax to get "only" 15% off, giving $102. But the tax is applied AFTER the discount, on the discounted price!

Discounted: 120 × 0.75 = $90. With tax: 90 × 1.10 = $99.00. (Answer: B)

18

PROBABILITY MOST MISSED

WORKED EXAMPLE

A bag has 3 red and 7 blue marbles. P(picking red) = 3/10 = 30%.

YOUR TURN

A class has 12 boys and 18 girls. Two students are selected at random without replacement. What is the probability that both are girls?

A 18/30 × 18/30

B 18/30 × 17/29

C 17/30 × 16/29

D 18/29 × 17/28

🚨 TRAP: Without replacement means the denominator CHANGES after the first pick! Don't use 30/30 twice.

P(1st girl) = 18/30. After selecting one girl, 17 girls remain out of 29 total. P(both) = 18/30 × 17/29 = 306/870 ≈ 35.2%. (Answer: B)

19

RATIOS & PROPORTIONS ⚠ TRAP

WORKED EXAMPLE

If 3 pencils cost $1.50, how much do 8 pencils cost?

3/1.50 = 8/x → x = (8 × 1.50)/3 = $4.00

YOUR TURN

On a map, 2 cm represents 50 km. Two cities are 7 cm apart on the map. If a car travels at 80 km/h, how many minutes will the trip take?

A 131 min

B 175 min

C 218 min

D 262 min

🚨 TRAP: The answer must be in MINUTES not hours! Students find 2.9 hours and choose 131... but 2.9 × 60 ≈ 175 minutes.

2cm/50km = 7cm/x → x = 175km. Time = 175/80 = 2.1875 hours = 2.1875 × 60 = 131.25 ≈ 131 min. (Answer: A)

20

DATA INTERPRETATION MOST MISSED

WORKED EXAMPLE

The table below shows test score distribution:

Score Range	Students
60–69	4
70–79	10
80–89	14
90–99	12

Total students = 4+10+14+12 = 40. Students scoring 80+ = 14+12 = 26.

YOUR TURN

Using the same table above, what percent of students scored below 80?

A 25%

B 35%

C 40%

D 65%

🚨 TRAP: "Below 80" means ONLY the 60–69 and 70–79 groups (4 + 10 = 14 students). The 80–89 range starts AT 80, so it does NOT count!

Below 80: 4 + 10 = 14 students out of 40 total. 14/40 = 0.35 = 35%. (Answer: B)