Arithmetic Reasoning
Arithmetic Reasoning tests your ability to solve real-world word problems. The math itself is not advanced, but you must correctly identify what is being asked, set up the problem, and calculate accurately.
Ratios and Proportions
A ratio expresses the relationship between two quantities. The ratio of 3 to 4 can be written as 3:4 or 3/4.
A proportion states that two ratios are equal: a/b = c/d. To solve for an unknown, cross-multiply: a × d = b × c.
Rate is a ratio comparing two different units (miles per hour, dollars per pound). A unit rate reduces the denominator to 1 (e.g., $2.50 per pound).
Example: If 5 lb of apples cost $8.75, how much do 8 lb cost? Set up: 5/8.75 = 8/x → 5x = 70 → x = $14.00
Percentages
Core formula: Part = Percent × Whole. Rearrange to find any of the three values.
- Finding the part: 30% of 200 = 0.30 × 200 = 60
- Finding the percent: 45 out of 180 = 45/180 = 25%
- Finding the whole: 60 is 30% of what? 60 ÷ 0.30 = 200
Percent change: (New − Old) / Old × 100. Price rises $80 → $100: (100−80)/80 × 100 = 25% increase.
Tax: Final = original × (1 + tax rate). $50 item at 8% tax = 50 × 1.08 = $54.00
Discount: Final = original × (1 − discount rate). 20% off $50 = 50 × 0.80 = $40.00
Critical rule — consecutive percentage changes are NOT additive. A 15% increase then a 15% decrease = 1.15 × 0.85 = 0.9775 → a 2.25% net decrease, not zero.
Simple interest: I = P × r × t (Principal × annual rate × years)
Compound interest: A = P(1 + r/n)^(nt), where n = compounding periods per year. Quarterly → n = 4. Monthly → n = 12.
Fractions and Decimals
Adding/subtracting fractions: Find a common denominator, then add/subtract numerators.
Multiplying fractions: Multiply numerators together, multiply denominators together, then simplify.
Dividing fractions: Multiply by the reciprocal of the divisor — "flip and multiply."
Mixed number → improper fraction: 3½ = (3 × 2 + 1)/2 = 7/2
Comparing fractions: Cross-multiply. For 3/4 vs 5/7: compare 3×7=21 to 4×5=20. Since 21 > 20, 3/4 > 5/7.
Decimal ↔ fraction: 0.75 = 75/100 = 3/4. To convert a fraction to a decimal, divide numerator by denominator.
Distance, Rate, and Time
Formula: d = r × t (distance = rate × time). Rearrange: r = d/t and t = d/r.
Average speed for a whole trip = total distance ÷ total time. Do not average the speeds directly — that only works if equal time is spent at each speed.
Two-object problems: - Objects moving toward each other: add their speeds to find closing rate. - Objects moving same direction: subtract speeds to find closing rate.
Upstream/downstream: Boat speed downstream = boat speed + current. Upstream = boat speed − current.
Averages (Mean, Median, Mode, Range)
Mean (average): Sum all values ÷ count. Total = mean × count.
Finding a missing value: If 5 tests average 88, their total = 5 × 88 = 440. Subtract the known scores to find the missing one.
Median: Order all values; the middle value is the median. With an even count, average the two middle values.
Mode: The most frequently occurring value. A set can have multiple modes or none.
Range: Highest − lowest. Measures spread, not center.
Work Rate Problems
Express each worker's rate as "fraction of job per hour," then add rates for combined work.
- A finishes in 4 hours → rate = 1/4 job/hr
- B finishes in 6 hours → rate = 1/6 job/hr
- Together: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 job/hr
- Time together = 1 ÷ (5/12) = 12/5 = 2.4 hours
If one entity fills (adds) and another drains (removes), subtract their rates.
Mixture Problems
The amount of substance is conserved when solutions are combined:
(Concentration₁ × Volume₁) + (Concentration₂ × Volume₂) = Concentration_final × Volume_final
Example: 10 L of 40% alcohol — how much pure alcohol (100%) must be added to reach 60%?
(0.40 × 10) + (1.0 × x) = 0.60 × (10 + x) 4 + x = 6 + 0.6x → 0.4x = 2 → x = 5 liters
Geometry in Word Problems
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × w | 2l + 2w |
| Square | s² | 4s |
| Triangle | ½ × b × h | sum of sides |
| Circle | πr² | 2πr |
| Trapezoid | ½(b₁+b₂)h | sum of sides |
Volume formulas: Rectangular prism = l×w×h · Cylinder = πr²h · Cone = ⅓πr²h · Sphere = (4/3)πr³
Common unit conversions: - 1 yard = 3 feet = 36 inches - 1 mile = 5,280 feet - 1 gallon = 4 quarts - 1 ton = 2,000 pounds - 1 hour = 60 min = 3,600 sec
Probability
Basic probability = favorable outcomes ÷ total possible outcomes.
- Coin flip: P(heads) = 1/2
- Rolling a 4 on a standard die: P(4) = 1/6
- Drawing an ace from a standard deck: P(ace) = 4/52 = 1/13
Probability always falls between 0 (impossible) and 1 (certain).