Problem Solving — Strategies for Math Word Problems
Overview
Problem solving is the heart of mathematics and a critical component of the TS TET Mathematics section. Unlike direct computation questions, word problems test whether a student (or future teacher) can translate real-world situations into mathematical language, select appropriate operations, and arrive at correct solutions. For aspiring teachers, mastering problem-solving strategies is doubly important: you must solve problems yourself and teach children systematic approaches.
The TS TET syllabus emphasises problem solving as a distinct skill area. Questions typically present everyday scenarios involving money, time, distance, age, or quantities — requiring candidates to identify what is given, what is asked, and which mathematical tools apply. Strong problem solvers don't just calculate; they reason, estimate, verify, and adapt when stuck.
This topic connects directly to arithmetic (percentage, ratio, profit-loss), algebra (forming equations), mensuration (word problems on area/volume), and even statistics. A clear problem-solving framework will improve performance across all Mathematics content areas.
Key Concepts
- Understanding the problem comes first: Read the problem twice. Identify what is given (data), what is unknown (to find), and any conditions or constraints. Restating the problem in your own words often reveals the path forward.
- Translation is the core skill: Convert words into mathematical expressions. "More than" suggests addition; "less than" suggests subtraction; "of" often means multiplication; "per" or "each" signals division or rate.
- Choose the right operation: Addition (combining), subtraction (difference or remaining), multiplication (repeated groups or scaling), division (sharing equally or finding rate). Many errors come from selecting the wrong operation.
- Work backwards when useful: If the problem gives the final result and asks for an initial value, reverse the operations. This is especially helpful in age problems and profit-loss questions.
- Use variables for unknowns: When direct arithmetic is unclear, let the unknown be x and form an equation. Solving the equation gives the answer.
- Estimation and reasonableness check: Before and after solving, estimate the answer's magnitude. If a problem asks how many sweets each child gets and your answer is 500, something is wrong.
- Draw diagrams or tables: Visual representation helps in geometry, distance-time, and mixture problems. Tables organise data in ratio, percentage, and comparison problems.
- Break complex problems into parts: Multi-step problems require solving intermediate values first. Identify sub-problems and solve sequentially.
Formulas / Key Facts
| Strategy | When to Use | Example Cue Words |
|---|---|---|
| Addition | Combining quantities | total, altogether, sum, in all |
| Subtraction | Finding difference or remainder | left, remaining, difference, how many more |
| Multiplication | Equal groups, scaling, area | times, product, each, per (with total) |
| Division | Sharing equally, finding rate | each, per, divide, ratio, average |
| Equation method | Unknown in relationship | is, equals, same as, results in |
| Unitary method | Finding value of one unit first | cost of 1, per item, rate per unit |
Polya's Four Steps (standard problem-solving model):
- Understand the problem
- Devise a plan (choose strategy)
- Carry out the plan (compute)
- Look back (verify and reflect)
Common formulas appearing in word problems:
- Distance = Speed × Time
- Simple Interest = (P × R × T) / 100
- Profit = SP − CP; Loss = CP − SP
- Percentage = (Part / Whole) × 100
Worked Examples
Example 1: Basic Two-Step Problem
Problem: A shopkeeper bought 50 kg of rice at Rs 40 per kg. He sold it at Rs 48 per kg. Find his total profit.
Solution:
- Step 1 (Understand): Given — 50 kg, CP per kg = 40, SP per kg = 48. Find — total profit.
- Step 2 (Plan): Find total CP, total SP, then profit = SP − CP.
- Step 3 (Execute):
- Total CP = 50 × 40 = Rs 2000
- Total SP = 50 × 48 = Rs 2400
- Profit = 2400 − 2000 = Rs 400
- Step 4 (Verify): Profit per kg = 8, so for 50 kg = 400. ✓
Answer: Rs 400
Example 2: Equation Method (Age Problem)
Problem: The sum of ages of a father and son is 56 years. The father is 4 times as old as the son. Find their ages.
Solution:
- Let son's age = x years.
- Father's age = 4x years.
- Given: x + 4x = 56
- 5x = 56 → x = 56/5 = 11.2
Wait — ages should be whole numbers. Re-read: if the problem intends whole numbers, check if given data is correct. Assuming it's valid:
- Son = 11.2 years, Father = 44.8 years.
If exam expects whole numbers, the typical version is sum = 60:
- 5x = 60 → x = 12 (son), 4x = 48 (father).
Key lesson: Always verify that your answer makes sense contextually.
Example 3: Distance-Time Problem
Problem: A car travels from city A to city B (180 km) at 60 km/h and returns at 45 km/h. Find the average speed for the whole journey.
Solution:
- Time for A to B = 180/60 = 3 hours
- Time for B to A = 180/45 = 4 hours
- Total distance = 180 + 180 = 360 km
- Total time = 3 + 4 = 7 hours
- Average speed = Total distance / Total time = 360/7 ≈ 51.43 km/h
Common trap: Students average the speeds (60 + 45)/2 = 52.5. This is wrong because time spent at each speed differs.
Answer: 360/7 km/h or approximately 51.43 km/h
Common Mistakes
- Mistake: Adding or averaging speeds directly in distance-time problems. Fix: Always calculate total distance and total time separately; average speed = total distance ÷ total time.
- Mistake: Misidentifying what the problem asks — solving for the wrong quantity. Fix: Underline or circle the question part before calculating. Re-read after solving to ensure you answered what was asked.
- Mistake: Ignoring units or mixing them (metres with kilometres, rupees with paise). Fix: Convert all quantities to the same unit at the start. State units in your final answer.
- Mistake: Performing operations in wrong order in multi-step problems. Fix: List intermediate steps explicitly. Solve sub-problems one at a time and label each result.
- Mistake: Not checking if the answer is reasonable (e.g., negative age, fractional people). Fix: After solving, ask: "Does this make real-world sense?" Estimate before detailed calculation.
Quick Reference
- Read twice, solve once — understanding prevents calculation waste.
- Translate keywords: total → add; left → subtract; each/per → divide; times → multiply.
- Use Polya's steps: Understand → Plan → Execute → Verify.
- Unitary method: Find value of one unit first, then scale.
- Average speed ≠ average of speeds — use total distance ÷ total time.
- When stuck, let unknown = x and form an equation from the given conditions.