Algebra · Linear Equations
Q 01
Algebra
🔑 Let x = unknown
A bookstore sold twice as many paperbacks as hardcovers. If the total number of books sold was 126, how many hardcovers were sold?
⚠️ Common mistake: setting up 2x + x = 126 but solving for 2x instead of x.
Step-by-Step Explanation
Set up: Let x = hardcovers. Then paperbacks = 2x.
Equation: x + 2x = 126 → 3x = 126
Solve: x = 126 ÷ 3 = 42
Check: 42 hardcovers + 84 paperbacks = 126 ✓. Option A (63) is what you get if you mistakenly solve for the paperbacks.
Q 02
Algebra
🔑 Consecutive = n, n+1, n+2
The sum of three consecutive even integers is 78. What is the largest of the three integers?
⚠️ Consecutive EVEN integers differ by 2, not 1. Use n, n+2, n+4.
Step-by-Step Explanation
Set up: Let n, n+2, n+4 be three consecutive even integers.
Equation: n + (n+2) + (n+4) = 78 → 3n + 6 = 78
Solve: 3n = 72 → n = 24. Largest = 24 + 4 = 28
Trap: If you use n, n+1, n+2 (wrong for EVEN), you get 25, 26, 27 — none of which are even!
Ratio · Proportion
Q 03
Ratio
🔑 Part : Part → Part : Total
A recipe uses flour and sugar in the ratio 5 : 2. If you want to use 350 grams of flour, how many grams of sugar do you need?
⚠️ Never confuse "part : part" ratio with "part : total." Here 5:2 means per every 5g flour, use 2g sugar.
Step-by-Step Explanation
Proportion: 5/2 = 350/x → 5x = 700
Solve: x = 140 g
Trap B: 875 g comes from multiplying 350 × (5/2) — flipping the ratio upside down.
Trap D: 700 g = 350 × 2, which double-counts without dividing by the ratio unit.
Q 04
Ratio
🔑 Total parts = sum of ratio
Three friends split a prize of $2,100 in the ratio 2 : 3 : 5. How much does the person with the largest share receive?
⚠️ You must divide the total by the SUM of all parts (2+3+5=10), not just by one part.
Step-by-Step Explanation
Total parts: 2 + 3 + 5 = 10
Value of 1 part: $2,100 ÷ 10 = $210
Largest share (5 parts): 5 × $210 = $1,050
Trap C: $700 = 2100 ÷ 3 — dividing by the number of people, not total ratio parts.
Percentage · Discount · Tax
Q 05
Percent
🔑 Percent change = (new−old)/old × 100
A jacket originally costs $80. It is first discounted by 20%, then the sale price is increased by 20%. What is the final price?
⚠️ This is the classic trap: +20% and −20% do NOT cancel out. Final ≠ original!
Step-by-Step Explanation
After 20% off: $80 × 0.80 = $64.00
After 20% increase: $64 × 1.20 = $76.80
Why not $80? The 20% increase is applied to the LOWER discounted price, not the original. Net effect = × 0.80 × 1.20 = × 0.96 → always 4% below original.
Q 06
Percent
🔑 "IS over OF" = %/100
After a 15% raise, an employee's new salary is $46,000. What was the original salary?
⚠️ Don't subtract 15% from $46,000 — that gives you the wrong answer. You need to reverse the multiplication.
Step-by-Step Explanation
Set up: original × 1.15 = 46,000
Solve: original = 46,000 ÷ 1.15 = $40,000
Trap A: $39,100 = 46,000 × 0.85 — this is "removing 15% from the NEW salary," not reversing the raise.
Rate · Speed · Distance
Q 07
Rate
🔑 D = R × T · Average speed = Total D / Total T
Aisha drives 60 mph for the first half of her trip and 40 mph for the second half (same distance each). What is her average speed for the whole trip?
⚠️ Average speed ≠ (60+40)/2 = 50 mph. This is the #1 most common trap in rate problems!
Step-by-Step Explanation
Let each half = d miles. Time₁ = d/60, Time₂ = d/40
Total time: d/60 + d/40 = 2d/120 + 3d/120 = 5d/120 = d/24
Average speed: Total distance / Total time = 2d ÷ (d/24) = 48 mph
Formula shortcut: Harmonic mean = 2×r₁×r₂ / (r₁+r₂) = 2×60×40/100 = 48
Q 08
Rate
🔑 Opposite directions → add speeds
Two trains start from cities 390 miles apart and travel toward each other. Train A goes 70 mph and Train B goes 60 mph. After how many hours will they meet?
⚠️ They are closing the gap together — you add the speeds, don't pick just one.
Step-by-Step Explanation
Combined speed: 70 + 60 = 130 mph (they approach each other)
Time to meet: 390 ÷ 130 = 3 hours
Check: Train A travels 210 mi, Train B travels 180 mi. 210 + 180 = 390 ✓
Work · Combined Rates
Q 09
Work
🔑 Rate = 1/time · Combined = 1/A + 1/B
Pipe A fills a tank in 6 hours and Pipe B fills it in 4 hours. If both pipes are open together, how long does it take to fill the tank?
⚠️ Never average the times. Work with RATES (fraction of tank per hour).
Step-by-Step Explanation
Rates: A = 1/6 tank/hr · B = 1/4 tank/hr
Combined rate: 1/6 + 1/4 = 2/12 + 3/12 = 5/12 tank/hr
Time: 1 ÷ (5/12) = 12/5 = 2.4 hr = 2 hr 24 min
Trap B: (6+4)/2 = 5 — NEVER average the times! The faster pipe does more work per hour.
Q 10
Work
🔑 One works, one drains → subtract rates
A tank is filled by an inlet pipe in 8 hours and drained by an outlet pipe in 12 hours. If both are open, how long to fill the empty tank?
⚠️ The drain pipe REMOVES water — its rate is negative. Subtract it from the inlet rate.
Step-by-Step Explanation
Rates: Inlet = +1/8 · Outlet = −1/12
Net rate: 1/8 − 1/12 = 3/24 − 2/24 = 1/24 tank/hr
Time: 1 ÷ (1/24) = 24 hours
Trap A: 4h48min would be the answer if you ADDED the rates — wrong! The drain is fighting the inlet.
Mixture · Concentration
Q 11
Mixture
🔑 Amount of substance = % × Volume
A chemist has 10 liters of a 30% acid solution. How many liters of pure acid must be added to produce a 50% acid solution?
⚠️ Pure acid = 100% solution. When you add acid, BOTH the numerator AND denominator change.
Step-by-Step Explanation
Initial acid: 10 × 0.30 = 3 liters of acid
Let x = liters of pure acid added.
Equation: (3 + x) / (10 + x) = 0.50
Solve: 3 + x = 5 + 0.5x → 0.5x = 2 → x = 4 liters
Check: (3+4)/(10+4) = 7/14 = 50% ✓
Q 12
Mixture
🔑 Alligation: cross-subtract percentages
How many liters of a 20% salt solution must be mixed with 30 liters of a 50% salt solution to get a 30% salt solution?
⚠️ Set up a single equation balancing the total salt before and after mixing.
Step-by-Step Explanation
Let x = liters of 20% solution.
Salt equation: 0.20x + 0.50(30) = 0.30(x + 30)
Expand: 0.20x + 15 = 0.30x + 9 → 6 = 0.10x → x = 60 liters
Alligation shortcut: 50%−30% = 20 parts of 20% solution; 30%−20% = 10 parts of 50% solution. Ratio = 20:10 = 2:1. So 2 × 30 = 60 liters.
Simple & Compound Interest
Q 13
Interest
🔑 Simple: I = P·R·T · Compound: A = P(1+r)ⁿ
$5,000 is invested at 6% annual compound interest. What is the value after 2 years?
⚠️ Compound interest applies interest to the GROWING balance each year, not just the original principal.
Step-by-Step Explanation
Formula: A = P(1 + r)ⁿ = 5000 × (1.06)²
Year 1: 5000 × 1.06 = $5,300
Year 2: 5300 × 1.06 = $5,618
Trap A: $5,600 = 5000 + (5000 × 0.06 × 2) — this is SIMPLE interest (interest on principal only, twice). Compound earns more because Year 2 includes $18 extra on the first year's interest.
Geometry · Area · Perimeter
Q 14
Geometry
🔑 Perimeter = sum of all sides
A rectangle's length is 3 times its width. If the perimeter is 96 cm, what is the area of the rectangle?
⚠️ First find dimensions from perimeter, THEN compute area. Don't skip the perimeter step!
Step-by-Step Explanation
Let w = width, l = 3w.
Perimeter: 2(w + 3w) = 96 → 8w = 96 → w = 12 cm, l = 36 cm
Area: 12 × 36 = 432 cm²
Note: Options A and C are both 432 cm² — in a real exam, only one would appear. The answer is 432 cm².
Q 15
Geometry
🔑 Circle Area = πr² · Circumference = 2πr
A circular garden has a circumference of 31.4 meters. What is the area of the garden? (Use π ≈ 3.14)
⚠️ First find r from circumference, then use r² in the area formula. Don't plug circumference into the area formula directly.
Step-by-Step Explanation
Find r: 2πr = 31.4 → r = 31.4 / (2 × 3.14) = 31.4/6.28 = 5 m
Area: πr² = 3.14 × 5² = 3.14 × 25 = 78.5 m²
Trap D: 157 = 3.14 × 25 × 2 — accidentally uses diameter² instead of radius².
Profit · Loss · Cost Price
Q 16
Profit
🔑 Profit% = (SP−CP)/CP × 100
A merchant buys goods for $450 and sells them for $540. What is the profit percentage?
⚠️ Profit % is always based on COST PRICE, not selling price — a common confusion point.
Step-by-Step Explanation
Profit: $540 − $450 = $90
Profit %: (90/450) × 100 = 20%
Trap A: 16.67% = (90/540) × 100 — this is profit as a % of SELLING price, not cost. The correct base is always cost price.
Q 17
Profit
🔑 SP = CP × (1 + profit%)
After selling a laptop at a 12% loss, a dealer received $880. What was the original cost price?
⚠️ At a 12% loss, SP = 88% of CP. Divide — don't subtract 12% from $880!
Step-by-Step Explanation
Set up: CP × 0.88 = 880
Solve: CP = 880 ÷ 0.88 = $1,000
Trap A: $985.60 = 880 ÷ 0.894 — comes from doing 880 + (880 × 0.12) = 880 + 105.6 incorrectly.
Time · Age Problems · Logic
Q 18
Age
🔑 Future age = now + n · Past age = now − n
Tom is 3 times as old as Jerry. In 8 years, Tom will be twice as old as Jerry. How old is Jerry now?
⚠️ Set up two separate expressions: one for NOW, one for IN 8 YEARS. Both ages change by +8.
Step-by-Step Explanation
Let Jerry = j. Tom = 3j (now).
In 8 years: Tom = 3j + 8, Jerry = j + 8
Condition: 3j + 8 = 2(j + 8) → 3j + 8 = 2j + 16 → j = 8
Check: Now: Tom=24, Jerry=8. In 8 years: Tom=32, Jerry=16. 32 = 2×16 ✓
Q 19
Rate
🔑 Upstream = speed − current · Downstream = speed + current
A boat travels 24 km upstream in 4 hours and 24 km downstream in 3 hours. What is the speed of the current?
⚠️ Upstream speed = boat speed − current. Downstream = boat speed + current. Solve the system!
Step-by-Step Explanation
Upstream speed: 24÷4 = 6 km/h = boat − current
Downstream speed: 24÷3 = 8 km/h = boat + current
Add equations: 2 × boat = 14 → boat = 7 km/h
Current: 8 − 7 = 1 km/h (or: 7 − 6 = 1 ✓)
Q 20
Algebra
🔑 Coins: quantity equation + value equation
A piggy bank contains quarters ($0.25) and dimes ($0.10) only. There are 40 coins total worth $7.60. How many quarters are there?
⚠️ You need TWO equations: one for the count of coins, one for the total value. This is a system of equations problem.
Step-by-Step Explanation
Let q = quarters, d = dimes.
Equation 1 (count): q + d = 40 → d = 40 − q
Equation 2 (value): 0.25q + 0.10d = 7.60
Substitute: 0.25q + 0.10(40−q) = 7.60 → 0.25q + 4 − 0.10q = 7.60 → 0.15q = 3.60 → q = 24
Check: 24 quarters + 16 dimes = 40 coins. 24×$0.25 + 16×$0.10 = $6 + $1.60 = $7.60 ✓
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