GMAT Review: Problem Solving (Arithmetic, Algebra, Word Problems, Rate/Work, Mixtures)

GMAT – Problem Solving (Arithmetic, Algebra, Word Problems, Rate/Work, Mixtures)

GMAT Study Guide – Problem Solving (Arithmetic, Algebra, Word Problems, Rate/Work, Mixtures)

What This Is

Problem‑Solving (PS) questions are the “classic” quantitative items on the GMAT. They present a single‑answer, multiple‑choice problem that requires you to compute a numeric value (or choose the best estimate) using arithmetic, algebra, or reasoning about rates, work, and mixtures. For example, a typical PS stem might read:

“A factory can produce 120 widgets in 8 hours. If the production rate is increased by 25 %, how many widgets can be produced in 10 hours?”

Mastering PS is essential because it accounts for roughly 30 % of the Quantitative section and tests the logical‑numeric foundation needed for business‑school coursework.

Key Terms & Rules

• Linear Equation: An equation of the form ax + b = c. Solve by isolating x → x = (c‑b)/a.
• Quadratic Equation: ax² + bx + c = 0. Use the quadratic formula x = [‑b ± √(b²‑4ac)]/(2a) or factor when possible.
• Proportion: If a/b = c/d, then ad = bc. Frequently appears in rate and mixture problems.
• Rate/Work Formula: Work = Rate × Time. For combined workers, R₁ + R₂ = R_total.
• Mixture Equation: (Quantity₁ × %₁) + (Quantity₂ × %₂) = Total Quantity × Desired %.
• Percent Change: New Value = Original × (1 ± %/100). Use “+” for increase, “‑” for decrease.
• Integer Restriction: GMAT answers are always integers unless the question explicitly asks for a decimal or fraction.
• Estimation/Back‑Solving: Plug answer choices into the equation to see which satisfies the condition—often faster than full algebra.
• Data Sufficiency Mind‑Set (PS‑type): Even though PS is single‑answer, the same “no extra assumptions” rule applies; you cannot introduce information not given.
• Common‑Denominator Shortcut: When adding/subtracting fractions, look for a denominator that appears in answer choices to avoid full LCM calculations.
• Negative Numbers Rule: The GMAT never uses negative quantities for real‑world contexts (e.g., “negative people”); if a negative appears, the answer is wrong.
• Units Consistency: Always convert units (minutes ↔ hours, miles ↔ kilometers) before plugging numbers; mismatched units are a classic trap.

Step‑by‑Step / Process Flow

1. Read the Stem Carefully – Highlight numbers, units, and keywords (e.g., “increase by 25 %,” “combined rate”).
2. Translate Words into an Equation – Write a concise algebraic expression using the formulas above; keep variables minimal.
3. Simplify & Isolate the Unknown – Perform arithmetic (use mental math or a quick scratch pad) and solve for the target variable.
4. Check Answer‑Choice Plausibility – Verify that the computed value respects all constraints (positive, integer, realistic).
5. Back‑Solve if Stuck – Plug each answer choice into the simplified equation; the one that satisfies the condition is correct.

Common Mistakes

• Mistake: Forgetting to convert units (e.g., mixing minutes with hours).
  Correction: Convert all time (or distance) quantities to the same unit before using the rate/work formula.

• Mistake: Assuming a linear relationship when the problem involves a quadratic (e.g., “area of a square” vs. side length).
  Correction: Identify the underlying shape or relationship; use area = side² for squares, leading to a quadratic equation.

• Mistake: Ignoring the “integer only” rule and selecting a fractional answer when the question asks for a count of items.
  Correction: If the answer must be a whole number, round only after confirming the exact integer satisfies the equation.

• Mistake: Adding percentages directly (e.g., “increase by 20 % then another 30 % → total 50 %”).
  Correction: Apply successive changes multiplicatively: New = Original × 1.20 × 1.30.

• Mistake: Over‑complicating a mixture problem by using three variables when two suffice.
  Correction: Use the total quantity as a known value and solve for the unknown component directly.

Exam Insights

1. Most‑Tested Concept – Rate/Work: Expect variations on “combined work” and “time saved” problems; the GMAT loves swapping workers to test proportional reasoning.
2. Mixture Traps: Answer choices often include the correct total quantity but the wrong percentage; double‑check the weighted‑average equation.
3. Percent‑Change Double‑Shift: A “25 % increase then a 20 % decrease” does not return you to the original value; the net change is asymmetric.
4. Back‑Solving Preference: The GMAT designs many PS items so that only one answer choice satisfies a simple plug‑in test, saving you from lengthy algebra.

Quick Check Questions

1. (PS – Rate/Work) A printer can print 150 pages in 5 minutes. If two identical printers work together, how many pages can they print in 8 minutes?
   Answer: B) 480 – One printer’s rate = 30 pp/min; two printers = 60 pp/min → 60 × 8 = 480.

2. (PS – Mixture) A 20‑liter solution is 30 % acid. How many liters of pure acid must be added to obtain a 45 % acid solution?
   Answer: C) 6 – Let x be added acid: (0.30·20 + x) / (20 + x) = 0.45 → 6 L acid needed.

3. (Quantitative Comparison) Quantity A: The number of integers between 1 and 100 inclusive that are multiples of 4.
   
   Quantity B: The number of integers between 1 and 100 inclusive that are multiples of 6.
   Answer: A is greater – Multiples of 4: ⌊100/4⌋ = 25; multiples of 6: ⌊100/6⌋ = 16.

Last‑Minute Cram Sheet (10 One‑Liners)

1. Linear Equation Shortcut: x = (c‑b)/a – just move constants to the right, then divide.
2. Quadratic Quick‑Factor: Look for two numbers that multiply to ac and add to b; if none, use the formula.
3. Rate/Work Core: Work = Rate × Time – for combined workers, add rates. ⚠️ Don’t multiply times together.
4. Mixture Formula: (Qty₁·%₁) + (Qty₂·%₂) = Total·Desired %.
5. Percent Change Multiplicative: New = Original × (1 ± %/100) – apply each change sequentially. ⚠️ Never add percentages directly.
6. Back‑Solve First: Plug answer choices into a simplified equation; the one that works is the answer.
7. Integer Rule: If the question asks for “how many items,” the answer must be a whole number.
8. Common Denominator Trick: When answer choices share a denominator, use it to avoid full LCM.
9. Units Consistency: Convert all time, distance, or weight units before calculation. ⚠️ Mismatched units = wrong answer.
10. Negative‑Number Guard: GMAT never expects a negative count or length in real‑world contexts; a negative result signals a set‑up error.

Good luck—remember: the GMAT rewards clean, logical set‑ups more than brute‑force computation!