Concept Review
"Master the fundamentals, and the test becomes a familiar friend — not an adversary."
Topic 01 · Number & Operations
Fractions, Decimals & Percents
To add/subtract fractions, find a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal.
a/b + c/d = (ad + bc) / bd
(a/b) ÷ (c/d) = (a/b) × (d/c)
Percent: Part / Whole × 100
LCD
Flip & Multiply
Part/Whole × 100
(a/b) ÷ (c/d) = (a/b) × (d/c)
Percent: Part / Whole × 100
EXAMPLE ▸ 2/3 + 3/4 = 8/12 + 9/12 = 17/12
EXAMPLE ▸ 15 is what % of 60? → 15/60 × 100 = 25%
EXAMPLE ▸ 15 is what % of 60? → 15/60 × 100 = 25%
Topic 02 · Number & Operations
Ratios, Proportions & Exponents
A ratio a:b divides a total T into parts a/(a+b)·T and b/(a+b)·T. For exponents, same-base multiplication adds exponents.
Part = Total × (share / total parts)
a^m × a^n = a^(m+n)
LCM(a,b) = a×b / GCF(a,b)
Part = T × fraction
Add Exponents × Same Base
LCM × GCF = a×b
a^m × a^n = a^(m+n)
LCM(a,b) = a×b / GCF(a,b)
EXAMPLE ▸ 3:5 ratio, 40 total → boys = 40×3/8 = 15
EXAMPLE ▸ 2³ × 2⁴ = 2⁷ = 128
EXAMPLE ▸ 2³ × 2⁴ = 2⁷ = 128
Topic 03 · Algebra
Equations, Inequalities & Functions
Solve equations by isolating the variable using inverse operations. Flip inequality sign when multiplying/dividing by a negative. Functions: substitute the input value.
3x + 7 = 22 → x = (22−7)/3 = 5
2x − 3 < 7 → x < 5 (greatest integer = 4)
f(x) = 2x² − 3 → f(3) = 2(9)−3 = 15
Isolate Variable
Flip Sign ÷ by Negative
Substitute for f(x)
2x − 3 < 7 → x < 5 (greatest integer = 4)
f(x) = 2x² − 3 → f(3) = 2(9)−3 = 15
EXAMPLE ▸ x+y=10, x−y=4 → add: 2x=14 → x=7, y=3
Topic 04 · Geometry
Area, Perimeter, Volume & Angles
Key formulas to memorize. Angle sum of any triangle = 180°.
Triangle Area = ½ × base × height
Circle: C = 2πr, A = πr²
Pythagorean: a² + b² = c²
Rectangular Prism V = l × w × h
Triangle angles sum = 180°
½bh
2πr
a²+b²=c²
lwh
Circle: C = 2πr, A = πr²
Pythagorean: a² + b² = c²
Rectangular Prism V = l × w × h
Triangle angles sum = 180°
EXAMPLE ▸ Triangle: base=8, h=5 → Area = ½×8×5 = 20
EXAMPLE ▸ Legs 6,8 → hypotenuse = √(36+64) = 10
EXAMPLE ▸ Legs 6,8 → hypotenuse = √(36+64) = 10
Topic 05 · Data Analysis & Probability
Mean, Median, Mode & Probability
Mean = sum ÷ count. Median = middle value of sorted list. Mode = most frequent. Probability = favorable outcomes ÷ total outcomes.
Mean = Σx / n
Median: sort → pick middle
P(event) = favorable / total
Mean = Sum/n
Sort for Median
P = favorable/total
Median: sort → pick middle
P(event) = favorable / total
EXAMPLE ▸ {4,7,9,11,14} → Mean = 45/5 = 9, Median = 9
EXAMPLE ▸ P(blue) from 3R,5B,2G → 5/10 = 1/2
EXAMPLE ▸ P(blue) from 3R,5B,2G → 5/10 = 1/2
20 Practice Problems
Question 01
FRACTIONS
Easy
What is the value of 2/3 + 3/4?
✦ Explanation
Find the LCD of 3 and 4, which is 12. Convert each fraction:
2/3 = 8/12 | 3/4 = 9/12
8/12 + 9/12 = 17/12
8/12 + 9/12 = 17/12
The answer is 17/12. Choices A and D incorrectly add numerators and denominators. Choice B subtracts instead.
Question 02
PERCENT
Easy
15 is what percent of 60?
✦ Explanation
Use the formula: Percent = (Part / Whole) × 100
15 / 60 × 100 = 0.25 × 100 = 25%
A common mistake is computing 60/15 × 100 = 400%, or reversing the division.
Question 03
RATIO
Medium
The ratio of boys to girls in a class is 3:5. If there are 40 students total, how many are boys?
✦ Explanation
The ratio 3:5 means there are 3+5 = 8 total parts. Boys occupy 3 parts out of 8.
Boys = 40 × (3/8) = 40 × 0.375 = 15
Choice D (25) is the number of girls. Choice A (12) is incorrect — watch out for multiplying 40 × 3/10.
Question 04
EXPONENTS
Easy
What is the value of 2³ × 2⁴?
✦ Explanation
When multiplying powers with the same base, ADD the exponents.
2³ × 2⁴ = 2^(3+4) = 2⁷ = 128
Choice C (2¹²) incorrectly multiplies the exponents. Choice E subtracts. Only addition applies here.
Question 05
ALGEBRA
Easy
If 3x + 7 = 22, what is the value of x?
✦ Explanation
Isolate x by performing inverse operations on both sides:
3x + 7 = 22
3x = 22 − 7 = 15
x = 15 / 3 = 5
3x = 22 − 7 = 15
x = 15 / 3 = 5
Check: 3(5) + 7 = 15 + 7 = 22 ✓
Question 06
INEQUALITIES
Medium
What is the greatest integer value of x that satisfies the inequality 2x − 3 < 7?
✦ Explanation
Solve the inequality step by step:
2x − 3 < 7
2x < 10
x < 5
2x < 10
x < 5
x must be strictly less than 5. The greatest integer less than 5 is 4. Note: x = 5 does NOT satisfy x < 5.
Question 07
FUNCTIONS
Medium
If f(x) = 2x² − 3, what is the value of f(3)?
✦ Explanation
Substitute x = 3 into the function:
f(3) = 2(3)² − 3 = 2(9) − 3 = 18 − 3 = 15
Choice D (33) comes from computing 2(3²) as (2×3)² = 36−3 = 33 — a common order-of-operations error. Always apply the exponent before multiplying.
Question 08
WORD PROBLEM
Medium
Apples cost $0.50 each and oranges cost $0.75 each. Sam buys 4 apples and 3 oranges. What is the total cost?
✦ Explanation
Calculate each subtotal, then add:
4 apples: 4 × $0.50 = $2.00
3 oranges: 3 × $0.75 = $2.25
Total: $2.00 + $2.25 = $4.25
3 oranges: 3 × $0.75 = $2.25
Total: $2.00 + $2.25 = $4.25
Choice B ($4.00) forgets to correctly compute the orange cost. Double-check each multiplication step.
Question 09
GEOMETRY
Easy
A triangle has a base of 8 cm and a height of 5 cm. What is its area?
✦ Explanation
Use the triangle area formula:
Area = ½ × base × height = ½ × 8 × 5 = 20 cm²
Choice B (40) forgets the ½ factor — this is the most common triangle area mistake. Choice A (13) just adds base + height.
Question 10
CIRCLES
Medium
A circle has a radius of 7. What is its circumference? (Leave answer in terms of π.)
✦ Explanation
Circumference uses the formula C = 2πr:
C = 2π(7) = 14π
Choice C (49π) is the area formula πr². Choice A (7π) forgets the factor of 2. Always distinguish C = 2πr from A = πr².
Question 11
PYTHAGOREAN THEOREM
Medium
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
✦ Explanation
Apply the Pythagorean Theorem: a² + b² = c²
6² + 8² = 36 + 64 = 100
c = √100 = 10
c = √100 = 10
This is the famous 3-4-5 Pythagorean triple scaled by 2 (→ 6-8-10). Memorizing these triples saves time: (3,4,5), (5,12,13), (8,15,17).
Question 12
VOLUME
Easy
A rectangular prism has length 4, width 3, and height 5. What is its volume?
✦ Explanation
Volume of a rectangular prism = length × width × height:
V = 4 × 3 × 5 = 60
Choice C (94) is the surface area: 2(lw + lh + wh) = 2(12+20+15) = 94. Don't confuse volume with surface area.
Question 13
STATISTICS
Easy
What is the mean (average) of the numbers 4, 7, 9, 11, and 14?
✦ Explanation
Mean = Sum of all values ÷ Number of values:
(4 + 7 + 9 + 11 + 14) / 5 = 45 / 5 = 9
Note: 9 is also the median (middle value when sorted). They agree here, but mean and median are different concepts.
Question 14
MEDIAN
Medium
What is the median of the data set: 3, 7, 12, 5, 9?
✦ Explanation
ALWAYS sort the data before finding the median:
Sorted: 3, 5, 7, 9, 12
Middle value (3rd of 5) = 7
Middle value (3rd of 5) = 7
The unsorted middle value is 12 — the trap. Choice E (7.2) is the mean. Always sort first!
Question 15
PROBABILITY
Medium
A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability it is blue?
✦ Explanation
Probability = favorable outcomes / total outcomes:
Total marbles = 3 + 5 + 2 = 10
P(blue) = 5/10 = 1/2
P(blue) = 5/10 = 1/2
Choice B (5/8) uses only non-red marbles as the denominator — incorrect. Always use the total count of all items.
Question 16
NUMBER THEORY
Medium
What is the Least Common Multiple (LCM) of 12 and 18?
✦ Explanation
12 = 2² × 3, and 18 = 2 × 3². The LCM takes the highest power of each prime factor:
LCM = 2² × 3² = 4 × 9 = 36
Alternatively: LCM × GCF = 12 × 18 = 216. GCF(12,18) = 6, so LCM = 216/6 = 36. Choice A (6) is the GCF, not the LCM.
Question 17
NUMBER THEORY
Medium
What is the Greatest Common Factor (GCF) of 24 and 36?
✦ Explanation
Find prime factorizations, then take the lowest power of shared factors:
24 = 2³ × 3 | 36 = 2² × 3²
GCF = 2² × 3 = 4 × 3 = 12
GCF = 2² × 3 = 4 × 3 = 12
Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36: 1,2,3,4,6,9,12,18,36. Largest in common = 12.
Question 18
SYSTEMS OF EQUATIONS
Hard
If x + y = 10 and x − y = 4, what is the value of x?
✦ Explanation
Use the elimination method — add both equations:
(x + y) + (x − y) = 10 + 4
2x = 14
x = 7
Then y = 10 − 7 = 3
2x = 14
x = 7
Then y = 10 − 7 = 3
Verify: 7+3=10 ✓ and 7−3=4 ✓. The y terms cancel perfectly when we add, making elimination the ideal method.
Question 19
ANGLES
Easy
In a triangle, two angles measure 65° and 75°. What is the measure of the third angle?
✦ Explanation
The angles in any triangle always sum to exactly 180°:
Third angle = 180° − 65° − 75°
= 180° − 140° = 40°
= 180° − 140° = 40°
Always verify: 65 + 75 + 40 = 180 ✓. This rule applies to ALL triangles — equilateral, right, obtuse, etc.
Question 20
RATE & SPEED
Hard
A car travels 180 miles in 3 hours. If it continues at the same speed, how far will it travel in 5 hours total?
✦ Explanation
First find the speed, then calculate total distance for 5 hours:
Speed = 180 miles / 3 hours = 60 mph
Distance in 5 hours = 60 × 5 = 300 miles
Distance in 5 hours = 60 × 5 = 300 miles
Choice D (360) multiplies 180 × 2, incorrectly assuming 5 hours is double 3 hours. Choice B (270) only adds 90 more miles for 1.5 hours, not 2 more hours.
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