PART 01
Algebra 1
Word Problems
QUICK MEMORY POINT
Word problems → always DEFINE your variable first!
"Let x = ___" before writing any equation. Then TRANSLATE words → math symbols.
Key words: more than = +, less than = −, times = ×, of = ×, is / are / equals = =
"Let x = ___" before writing any equation. Then TRANSLATE words → math symbols.
Key words: more than = +, less than = −, times = ×, of = ×, is / are / equals = =
Q01
The Pocket Money Problem
Emma has three times as much money as her brother Jake. After Emma gives Jake $12, they both have the same amount. How much money did Emma start with?
STEP-BY-STEP SOLUTION
Let Jake's money = x. Then Emma's money = 3x.
After the transfer: Emma has \(3x - 12\), Jake has \(x + 12\).
Set equal: \(3x - 12 = x + 12\)
\(2x = 24 \Rightarrow x = 12\)
Emma started with \(3 \times 12 = \)$36. ✗ Wait—re-read: they end up equal, so Emma = Jake = $24. Emma started with $36+$12=$48. Correct answer: D) $48.
Common mistake: forgetting that after giving, Emma's amount decreases AND Jake's increases.
After the transfer: Emma has \(3x - 12\), Jake has \(x + 12\).
Set equal: \(3x - 12 = x + 12\)
\(2x = 24 \Rightarrow x = 12\)
Emma started with \(3 \times 12 = \)$36. ✗ Wait—re-read: they end up equal, so Emma = Jake = $24. Emma started with $36+$12=$48. Correct answer: D) $48.
Common mistake: forgetting that after giving, Emma's amount decreases AND Jake's increases.
Q02
The Train Problem
Two trains leave the same station at the same time, traveling in opposite directions. One travels at 60 mph, the other at 80 mph. After how many hours are they 280 miles apart?
STEP-BY-STEP SOLUTION
Key Concept: Opposite directions → add the speeds.
Combined speed = 60 + 80 = 140 mph
Time = Distance ÷ Speed = \(\dfrac{280}{140} = \)2 hours
Common mistake: using only one train's speed. Always add speeds for opposite directions!
Combined speed = 60 + 80 = 140 mph
Time = Distance ÷ Speed = \(\dfrac{280}{140} = \)2 hours
Common mistake: using only one train's speed. Always add speeds for opposite directions!
Q03
The Sum of Integers
The sum of three consecutive even integers is 78. What is the largest of the three integers?
STEP-BY-STEP SOLUTION
Consecutive even integers differ by 2. Let them be \(n,\ n+2,\ n+4\).
\(n + (n+2) + (n+4) = 78\)
\(3n + 6 = 78 \Rightarrow 3n = 72 \Rightarrow n = 24\)
The three integers: 24, 26, 28. Largest = C) 28.
Tricky part: "consecutive even" uses +2, not +1. Don't use n, n+1, n+2!
\(n + (n+2) + (n+4) = 78\)
\(3n + 6 = 78 \Rightarrow 3n = 72 \Rightarrow n = 24\)
The three integers: 24, 26, 28. Largest = C) 28.
Tricky part: "consecutive even" uses +2, not +1. Don't use n, n+1, n+2!
Q04
The Coffee Blend
A café mixes coffee worth $8/lb with coffee worth $12/lb to make 20 lb of a blend worth $9.50/lb. How many pounds of the $8 coffee should be used?
STEP-BY-STEP SOLUTION
Let \(x\) = lb of $8 coffee. Then \((20-x)\) = lb of $12 coffee.
\(-4x = -50 \Rightarrow x = 12.5\)
Use 12.5 lb of the $8 coffee → D)
Setup trick: (value × amount) for each type = total value of blend.
\(8x + 12(20-x) = 9.50 \times 20\)
\(8x + 240 - 12x = 190\)\(-4x = -50 \Rightarrow x = 12.5\)
Use 12.5 lb of the $8 coffee → D)
Setup trick: (value × amount) for each type = total value of blend.
Q05
The Discount & Tax Trap
A jacket costs $80. It is discounted 25%, then taxed 10%. What is the final price?
STEP-BY-STEP SOLUTION
After 25% discount: \(80 \times 0.75 = \$60\)
After 10% tax: \(60 \times 1.10 = \$66\)
Answer: B) $66.00
Common trap: Some students compute 25% − 10% = 15% off. Wrong! Apply each percentage step by step to the current price.
After 10% tax: \(60 \times 1.10 = \$66\)
Answer: B) $66.00
Common trap: Some students compute 25% − 10% = 15% off. Wrong! Apply each percentage step by step to the current price.
Q06
Tickets at the Fair
Adult tickets cost $7 and child tickets cost $4. A family bought 9 tickets total and paid $48. How many adult tickets did they buy?
STEP-BY-STEP SOLUTION
Let \(a\) = adult tickets, \(c\) = child tickets.
\(a + c = 9\) and \(7a + 4c = 48\)
From first: \(c = 9 - a\). Substitute:
\(7a + 4(9-a) = 48 \Rightarrow 7a + 36 - 4a = 48 \Rightarrow 3a = 12 \Rightarrow a = 4\)
Answer: A) 4 adult tickets
\(a + c = 9\) and \(7a + 4c = 48\)
From first: \(c = 9 - a\). Substitute:
\(7a + 4(9-a) = 48 \Rightarrow 7a + 36 - 4a = 48 \Rightarrow 3a = 12 \Rightarrow a = 4\)
Answer: A) 4 adult tickets
Q07
The Age Puzzle
Right now, a mother is 4 times as old as her daughter. In 6 years, she will be only 3 times as old. How old is the daughter now?
STEP-BY-STEP SOLUTION
Let daughter's age now = \(d\). Mother's age now = \(4d\).
In 6 years: daughter = \(d+6\), mother = \(4d+6\).
\(4d + 6 = 3(d+6)\)
\(4d + 6 = 3d + 18 \Rightarrow d = 12\)
Answer: B) 12 years old
Key: Add the same number to BOTH ages for future problems. Don't just add to the parent's age!
In 6 years: daughter = \(d+6\), mother = \(4d+6\).
\(4d + 6 = 3(d+6)\)
\(4d + 6 = 3d + 18 \Rightarrow d = 12\)
Answer: B) 12 years old
Key: Add the same number to BOTH ages for future problems. Don't just add to the parent's age!
Q08
The Recipe Scale-Up
A recipe for 4 people uses \(2\frac{1}{2}\) cups of flour. If you want to make it for 10 people, how many cups of flour do you need?
STEP-BY-STEP SOLUTION
Set up a proportion: \(\dfrac{2.5}{4} = \dfrac{x}{10}\)
Cross multiply: \(4x = 25 \Rightarrow x = 6.25 = 6\frac{1}{4}\)
Answer: C) \(6\frac{1}{4}\) cups
Cross multiply: \(4x = 25 \Rightarrow x = 6.25 = 6\frac{1}{4}\)
Answer: C) \(6\frac{1}{4}\) cups
Q09
The Grade Requirement
Maya scored 72, 85, and 90 on three tests. What is the minimum score she needs on her fourth test to have an average of at least 82?
STEP-BY-STEP SOLUTION
Let \(x\) = fourth test score.
\(\dfrac{72 + 85 + 90 + x}{4} \geq 82\)
\(247 + x \geq 328\)
\(x \geq 81\)
Answer: B) 81
\(\dfrac{72 + 85 + 90 + x}{4} \geq 82\)
\(247 + x \geq 328\)
\(x \geq 81\)
Answer: B) 81
Q10
The Coin Jar
A jar has only dimes and quarters. There are 24 coins total, worth $3.75. How many quarters are there?
STEP-BY-STEP SOLUTION
Let \(q\) = quarters, \(d\) = dimes.
\(q + d = 24\) and \(0.25q + 0.10d = 3.75\)
Multiply second equation by 100: \(25q + 10d = 375\)
From first: \(d = 24 - q\). Substitute:
\(25q + 10(24-q) = 375 \Rightarrow 25q + 240 - 10q = 375 \Rightarrow 15q = 135 \Rightarrow q = 9\)
Answer: C) 9 quarters
\(q + d = 24\) and \(0.25q + 0.10d = 3.75\)
Multiply second equation by 100: \(25q + 10d = 375\)
From first: \(d = 24 - q\). Substitute:
\(25q + 10(24-q) = 375 \Rightarrow 25q + 240 - 10q = 375 \Rightarrow 15q = 135 \Rightarrow q = 9\)
Answer: C) 9 quarters
PART 02
Geometry
Shapes · Angles · Proofs
QUICK MEMORY POINT
SOHCAHTOA for triangles · Angles in triangle = 180° · Angles on line = 180° · Full turn = 360°
Area formulas: Triangle = ½bh · Circle = πr² · Rectangle = lw
Pythagorean theorem: a² + b² = c² (c = hypotenuse, always longest side)
Area formulas: Triangle = ½bh · Circle = πr² · Rectangle = lw
Pythagorean theorem: a² + b² = c² (c = hypotenuse, always longest side)
Q11
The Ladder Problem
A 13-foot ladder leans against a wall. The bottom of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
STEP-BY-STEP SOLUTION
Use \(a^2 + b^2 = c^2\) where \(c = 13\) (hypotenuse = ladder).
\(5^2 + h^2 = 13^2\)
\(25 + h^2 = 169\)
\(h^2 = 144 \Rightarrow h = 12\)
Answer: C) 12 feet
This is the 5-12-13 Pythagorean triple — memorize it!
\(5^2 + h^2 = 13^2\)
\(25 + h^2 = 169\)
\(h^2 = 144 \Rightarrow h = 12\)
Answer: C) 12 feet
This is the 5-12-13 Pythagorean triple — memorize it!
Q12
The Angle Chase
Two angles are supplementary. One angle is 40° more than the other. What is the smaller angle?
STEP-BY-STEP SOLUTION
Supplementary = two angles that add to 180°.
Let smaller angle = \(x\). Larger = \(x + 40\).
\(x + (x+40) = 180\)
\(2x = 140 \Rightarrow x = 70°\)
Answer: B) 70°
Memory: SUPplementary = Sum of 180°. COMplementary = Combined 90°.
Let smaller angle = \(x\). Larger = \(x + 40\).
\(x + (x+40) = 180\)
\(2x = 140 \Rightarrow x = 70°\)
Answer: B) 70°
Memory: SUPplementary = Sum of 180°. COMplementary = Combined 90°.
Q13
The Diameter Trap
A circle has a diameter of 10 cm. What is its area? (Use \(\pi \approx 3.14\))
STEP-BY-STEP SOLUTION
Diameter = 10, so radius = 5 (divide by 2!).
\(A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5\) cm²
Answer: B) 78.5 cm²
#1 mistake: Using the diameter instead of radius in the formula. Always halve it!
\(A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5\) cm²
Answer: B) 78.5 cm²
#1 mistake: Using the diameter instead of radius in the formula. Always halve it!
Q14
The Missing Angle
In a triangle, two angles measure 55° and 72°. What is the third angle?
STEP-BY-STEP SOLUTION
Sum of angles in any triangle = 180°.
Third angle = \(180 - 55 - 72 = 53°\)
Answer: C) 53°
Third angle = \(180 - 55 - 72 = 53°\)
Answer: C) 53°
Q15
The Garden Fence
A rectangular garden has a perimeter of 56 meters. The length is 4 meters more than twice the width. What is the width?
STEP-BY-STEP SOLUTION
Let width = \(w\), length = \(2w + 4\).
Perimeter = \(2(l + w) = 56\), so \(l + w = 28\).
\((2w+4) + w = 28 \Rightarrow 3w = 24 \Rightarrow w = 8\)
Answer: B) 8 meters
Perimeter = \(2(l + w) = 56\), so \(l + w = 28\).
\((2w+4) + w = 28 \Rightarrow 3w = 24 \Rightarrow w = 8\)
Answer: B) 8 meters
Q16
Parallel Lines Cut by a Transversal
Two parallel lines are cut by a transversal. One of the co-interior angles (same-side interior) measures 115°. What does the other co-interior angle measure?
STEP-BY-STEP SOLUTION
Co-interior angles (also called consecutive interior angles) are supplementary → they add to 180°.
\(180 - 115 = 65°\)
Answer: B) 65°
Memory trick: Co-interior angles are on the SAME side → they're SUPPLEMENTARY (180°). Alternate interior angles are on OPPOSITE sides → they're EQUAL.
\(180 - 115 = 65°\)
Answer: B) 65°
Memory trick: Co-interior angles are on the SAME side → they're SUPPLEMENTARY (180°). Alternate interior angles are on OPPOSITE sides → they're EQUAL.
Q17
The Triangle in the Rectangle
A rectangle is 12 cm long and 8 cm wide. A diagonal is drawn, creating two triangles. What is the area of one triangle?
STEP-BY-STEP SOLUTION
Rectangle area = \(12 \times 8 = 96\) cm².
A diagonal splits it into two equal triangles.
Each triangle = \(\dfrac{96}{2} = 48\) cm²
(Or directly: \(\frac{1}{2} \times 12 \times 8 = 48\) cm²)
Answer: B) 48 cm²
A diagonal splits it into two equal triangles.
Each triangle = \(\dfrac{96}{2} = 48\) cm²
(Or directly: \(\frac{1}{2} \times 12 \times 8 = 48\) cm²)
Answer: B) 48 cm²
Q18
Filling the Box
A rectangular box has a length of 6 cm, width of 4 cm, and height of 5 cm. If you double only the height, the new volume is how many times the original?
STEP-BY-STEP SOLUTION
Original volume = \(6 \times 4 \times 5 = 120\) cm³
New volume = \(6 \times 4 \times 10 = 240\) cm³
Ratio = \(\dfrac{240}{120} = 2\)
Answer: B) 2 times
When only ONE dimension is multiplied by k, the volume is multiplied by k. Doubling one side → 2× volume.
New volume = \(6 \times 4 \times 10 = 240\) cm³
Ratio = \(\dfrac{240}{120} = 2\)
Answer: B) 2 times
When only ONE dimension is multiplied by k, the volume is multiplied by k. Doubling one side → 2× volume.
Q19
The Shadow Problem
A 6-foot person casts a 4-foot shadow. At the same time, a nearby tree casts a 20-foot shadow. How tall is the tree?
STEP-BY-STEP SOLUTION
Similar triangles → corresponding sides are proportional.
\(\dfrac{\text{person height}}{\text{person shadow}} = \dfrac{\text{tree height}}{\text{tree shadow}}\)
\(\dfrac{6}{4} = \dfrac{h}{20}\)
\(h = \dfrac{6 \times 20}{4} = 30\) ft
Answer: C) 30 feet
\(\dfrac{\text{person height}}{\text{person shadow}} = \dfrac{\text{tree height}}{\text{tree shadow}}\)
\(\dfrac{6}{4} = \dfrac{h}{20}\)
\(h = \dfrac{6 \times 20}{4} = 30\) ft
Answer: C) 30 feet
Q20
Wheels & Distance
A bicycle wheel has a radius of 35 cm. How far does the bike travel in 10 full rotations of the wheel? (Use \(\pi \approx 3.14\))
STEP-BY-STEP SOLUTION
Circumference = \(2\pi r = 2 \times 3.14 \times 35 = 219.8\) cm per rotation.
10 rotations = \(219.8 \times 10 = 2{,}198\) cm
Answer: C) 2,198 cm
Distance per rotation = circumference (one full circle around the wheel).
10 rotations = \(219.8 \times 10 = 2{,}198\) cm
Answer: C) 2,198 cm
Distance per rotation = circumference (one full circle around the wheel).