Word Problems

Strategy/Review

The Translation Method: English → Variables → Equation → Solve

This four-step pipeline turns any word problem into solvable math:

Step 1 — English: Read the problem. Identify what's being asked. Let that unknown be your variable (usually x).

Step 2 — Variables: Express all other quantities in terms of x using the relationships stated in the problem. "Three more than twice a number" → 2x + 3. "Five years ago" → x − 5.

Step 3 — Equation: Find the sentence that states an equality. "The sum is 42" → build your equation. "They arrive at the same time" → set times equal.

Step 4 — Solve: Use algebra. Check that your answer makes sense in the original context (no negative ages, no speeds exceeding the speed of light).

Archetype 1: Distance = Rate × Time

Formula: d = r × t → r = d/t → t = d/r

This is the most versatile QR formula. Every distance/rate/time problem uses it, often in multi-step scenarios.

Example — Opposite directions: Two cars start 300 miles apart and drive toward each other. Car A goes 60 mph, Car B goes 40 mph. When do they meet? → Combined rate = 60 + 40 = 100 mph → Time = distance / rate = 300 / 100 = 3 hours

Example — Catch-up: Car A leaves at 50 mph. Car B leaves 2 hours later at 70 mph. When does B catch A? → In 2 hours, A travels 100 miles (head start). → B gains 20 mph on A (relative speed: 70 − 50). → Time to close 100 miles at 20 mph = 5 hours after B departs.

Example — Round trip: A plane flies 600 miles at 300 mph with the wind and returns at 200 mph against the wind. What's the average speed? → NOT (300 + 200)/2 = 250 mph! → Time out = 600/300 = 2 hours. Time back = 600/200 = 3 hours. → Total distance = 1200 miles. Total time = 5 hours. → Average speed = 1200/5 = 240 mph.

Archetype 2: Work/Rate Problems

Formula: 1/t₁ + 1/t₂ + ... + 1/tₙ = 1/T Where t₁, t₂, ... are individual times to complete the job alone, and T is the time to complete it together.

Intuition: Each person contributes a RATE (what fraction of the job they complete per unit time). Rates add. 1/t is the rate (jobs per hour).

Example: Alice can paint a room in 4 hours. Bob can paint it in 6 hours. How long together? → Rate of Alice = 1/4 room/hr. Rate of Bob = 1/6 room/hr. → Combined rate = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 room/hr. → Time = 1 / (5/12) = 12/5 = 2.4 hours (or 2 hours 24 minutes).

Example — One helps then leaves: Alice works alone for 1 hour (completes 1/4 of the room). Bob joins. How much longer? → Remaining: 3/4 of the room. → Combined rate: 5/12 room/hr. → Time = (3/4) / (5/12) = (3/4) × (12/5) = 9/5 = 1.8 hours more.

Archetype 3: Mixture Problems

Formula: C₁V₁ + C₂V₂ = C_total × V_total Where C = concentration (or price, or percent) and V = volume (or quantity, or weight).

Example: How much pure alcohol (100%) must be added to 500 mL of a 20% solution to make a 50% solution? → Let x = mL pure alcohol added. → (1.00)(x) + (0.20)(500) = (0.50)(500 + x) → x + 100 = 250 + 0.5x → 0.5x = 150 → x = 300 mL

Example — Pricing: How many pounds of $4/lb coffee must be mixed with 10 pounds of $7/lb coffee to make a blend worth $5/lb? → 4x + 7(10) = 5(x + 10) → 4x + 70 = 5x + 50 → x = 20 pounds

Archetype 4: Age Problems

Strategy: Use a table or timeline. "Now" is your baseline. x = current age of the person the question asks about.

Example: John is twice as old as Mary. In 10 years, John will be 5 years older than Mary. Find John's current age. → Now: John = 2x, Mary = x → In 10 years: John = 2x + 10, Mary = x + 10 → Equation: 2x + 10 = (x + 10) + 5 → 2x + 10 = x + 15 → x = 5 → John is 10.

Trick: This problem's equation simplifies to x = 5 regardless of the number of years — the age difference is always constant! 2x − x = (2x+10) − (x+10) − 5 → x = 5.

Archetype 5: Simple Interest

Formula: I = P × r × t Where I = interest, P = principal, r = annual interest rate (as decimal), t = time in years.

Example: $5,000 invested at 4% simple interest for 3 years. Total interest? → I = 5000 × 0.04 × 3 = $600.

Total amount: A = P + I = P(1 + rt)

Compound interest (if tested): A = P(1 + r/n)^(nt) Where n = compounding periods per year. The DAT rarely requires compound interest calculations (they're calculator-intensive), but you may need to recognize the formula or compare simple vs. compound conceptually.

Archetype 6: Consecutive Integers

Consecutive integers: x, x+1, x+2, ... Consecutive even/odd: x, x+2, x+4, ... (start with an even x for evens, odd x for odds)

Example: The sum of three consecutive odd integers is 63. Find the smallest. → x + (x+2) + (x+4) = 63 → 3x + 6 = 63 → 3x = 57 → x = 19 → Integers: 19, 21, 23.

Archetype 7: Ratio and Proportion Word Problems

Example: The ratio of boys to girls in a class is 3:4. If there are 28 students, how many boys? → Let boys = 3k, girls = 4k. → 3k + 4k = 28 → 7k = 28 → k = 4 → Boys = 3(4) = 12.

Common Traps

Trap 1 — Average Speed ≠ Average of Speeds. Average speed = total distance / total time, NOT (speed1 + speed2)/2. The round-trip plane example above demonstrates this.

Trap 2 — Work Rates: Forgetting to Use Reciprocals. "Alice does the job in 4 hours" means her rate is 1/4. The most common error is writing "4" instead of "1/4" in the rate equation.

Trap 3 — Mixture: Forgetting Total Volume Changes. When you add pure alcohol, the total volume increases. The denominator in the final concentration is (original volume + added volume), not just the original.

Trap 4 — Age: Not Accounting for Everyone Aging. If a problem says "in 10 years," ALL people age 10 years. A student might add 10 to one person's age but forget the other.

Trap 5 — Interest: Confusing Simple and Compound. Simple interest = interest on principal only. Compound = interest on principal + accumulated interest. Compound grows faster.