Word problems account for a large portion of Quant questions. The key skill is translating English sentences into equations systematically, then solving. Speed comes from recognizing the problem type instantly and applying the right template.

A) Distance, Rate, Time

Distance = Rate × Time (D = RT). Units must be consistent (mph with hours, km/h with hours, etc.).

• Same direction: relative rate = |r₁ − r₂|. Time to close a gap = gap ÷ relative rate.
• Opposite directions (approaching): relative rate = r₁ + r₂. Time to meet = distance ÷ (r₁ + r₂).
• Average speed for the whole trip = total distance ÷ total time. NOT the average of the two speeds.
• For equal distances at two speeds r₁ and r₂: average speed = 2r₁r₂ / (r₁ + r₂) (harmonic mean). Example: 60 mph there, 40 mph back → avg = 2(60)(40)/(60+40) = 4800/100 = 48 mph.
• Round trips: total distance = 2d; time = d/r₁ + d/r₂.

B) Work Problems

Rate of work = 1 job / time to complete job. Combined rates of independent workers add directly.

• Machine A: rate 1/4 job/hr. Machine B: rate 1/6 job/hr. Together: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 job/hr. Time = 12/5 = 2.4 hrs.
• General formula: combined time = t₁t₂ / (t₁ + t₂) (for two workers).
• Partial work: in x hours, a worker with time t completes x/t of the job.
• Workers can sometimes work against each other (one fills a tank, one drains it): net rate = fill rate − drain rate.

C) Mixture Problems

Two solution types: mixing concentrations (chemistry), and weighted cost problems (pricing).

• Equation approach: (vol₁ × conc₁) + (vol₂ × conc₂) = total volume × final concentration. Set up and solve for the unknown.
  Example: Mix 20 L of 10% salt solution with 30 L of 40% salt solution. Total salt = (20×0.10) + (30×0.40) = 2 + 12 = 14 L. Final concentration = 14/50 = 28%.
• Alligation shortcut: draw a box with the two concentrations and the target. The ratio of volumes = (target − conc₂) : (conc₁ − target).
  Example: Need 30% from 10% and 40%. Ratio = (30−40):(10−30) = 10:20 = 1:2. So use 1 part of 40% and 2 parts of 10%.

D) Interest Problems

• Simple interest: I = Prt. Total amount: A = P(1 + rt).
• Compound interest (annual): A = P(1 + r)^t.

E) Profit, Loss & Discount

• Profit = Selling Price (SP) − Cost Price (CP)
• Profit % = (Profit / CP) × 100
• Loss % = (Loss / CP) × 100
• Discount % = (Discount / Marked Price) × 100
• SP = CP × (1 + profit%/100) = Marked Price × (1 − discount%/100)

Example: A shirt costs $40 (CP). Sold for $52 (SP). Profit = $12. Profit% = (12/40)×100 = 30%. If it was marked at $65 and sold at $52, Discount% = (13/65)×100 = 20%.

F) Ratio and Proportion Problems

• Ratios with a total: a:b:c with total T → each part = (ratio value / sum of ratio) × T.

G) Min/Max Word Problems

Min/max questions ask for the highest or lowest possible value of a quantity given a set of constraints. The setup is a word problem; the method is systematic.

• To maximize one value: set every other quantity to its minimum allowed value. The remaining value absorbs the rest.
• To minimize one value: set every other quantity to its maximum allowed value.
• Always verify your extreme assignment satisfies all constraints simultaneously (sum, count, range, distinct, positive, etc.).

Example — maximizing one element: 5 positive integers sum to 100. To maximize the largest value, minimize the other four. Since they must be positive integers, set them to 1 each: max = 100 − 4 = 96.

Example — weighted average: 30 students average 70 on a test. At most how many scored 90 or above? Maximize the count above 90 by having those students score exactly 90 and the rest score as low as possible (0). Let k students score 90: 90k = 30×70 = 2100 → k = 23.3 → at most 23 students.

Example — range constraint: 7 distinct positive integers are less than 20 and their sum is at most 50. To maximize the largest value, minimize the other six. Smallest six distinct positive integers: 1+2+3+4+5+6 = 21. Remaining budget: 50 − 21 = 29. But it must be distinct from 6 and less than 20, so max = 19. Check: is 19 distinct from 1–6? Yes. Sum = 21+19 = 40 ≤ 50. ✓

H) Strategy: 3-Step Approach

1. Identify the problem type (rate, work, mixture, percent, ratio).
2. Define variables with explicit units ("Let x = speed in mph" not just "let x = speed").
3. Translate each sentence into one equation. Solve the system.

Use smart numbers when exact computation takes too long. For ratios, plug in numbers that make fractions clean (use multiples of all denominators).