Word Problems

โฑ 8 minโœ๏ธ Quiz at the end

Why Word Problems Matter

Word problems require you to translate a real-world situation into mathematical language, choose the right method, and solve it. They test not just calculation skills but also reading comprehension, reasoning, and checking.

Almost every topic in mathematics can appear in a word problem โ€” algebra, ratios, geometry, statistics, and more.

The RUCSAC Strategy

Use RUCSAC as a step-by-step framework for any word problem:

  1. Read the problem carefully โ€” all the way through before writing anything
  2. Understand what you need to find โ€” what is the question actually asking?
  3. Choose an operation or method โ€” which mathematical tool fits?
  4. Solve the problem โ€” show your working clearly
  5. Answer the question โ€” write a full sentence answer with units
  6. Check your answer makes sense โ€” does it fit the context?

Key Clue Words

Recognising clue words tells you which operation to use:

OperationClue words
Additiontotal, altogether, sum, more than, increased by, combined
Subtractiondifference, less, fewer, remain, left over, how many more, change, decreased by
Multiplicationtimes, product, per, each, of, area, at a rate of, times as many
Divisionshared equally, per person, each, quotient, split, how many groups

Note: "each" and "per" can signal either multiplication or division depending on context โ€” read carefully.

Setting Up Equations

The most powerful approach is to assign a variable to the unknown and write an equation.

Example 1 โ€” Ratio problem: Sam has 3 times as many marbles as Jo. Together they have 48.

Let Jo = x, then Sam = 3x. x + 3x = 48 โ†’ 4x = 48 โ†’ x = 12 Sam has 3 ร— 12 = 36 marbles

Check: 12 + 36 = 48 โœ“

Example 2 โ€” Two unknowns: Two numbers have a sum of 50 and a difference of 14. Find the larger number.

Let the numbers be x and y (x > y): x + y = 50 ... (1) x - y = 14 ... (2)

Add: 2x = 64 โ†’ x = 32, y = 50 - 32 = 18

Consecutive Integer Problems

Consecutive integers differ by 1: n, n+1, n+2, ...

Example: Five consecutive integers sum to 100. Find the smallest.

Let them be n, n+1, n+2, n+3, n+4: 5n + 10 = 100 โ†’ 5n = 90 โ†’ n = 18

The integers are 18, 19, 20, 21, 22. Check: 18+19+20+21+22 = 100 โœ“

Percentage Problems

Example: A shirt costs ยฃ24 after a 20% discount. What was the original price?

ยฃ24 represents 80% of the original (100% - 20% = 80%): 1% = 24 รท 80 = ยฃ0.30 100% = 0.30 ร— 100 = ยฃ30

Or: original = 24 รท 0.8 = ยฃ30

Reverse percentage tip: always identify what percentage the given value represents before dividing.

Speed, Distance, Time

The three quantities are linked by:

Speed = Distance รท Time

Distance = Speed ร— Time

Time = Distance รท Speed

Example: Train travels 360km in 3 hours. Speed = 360 รท 3 = 120km/h

Example: Car travels at 60km/h for 2.5 hours. Distance = 60 ร— 2.5 = 150km

Always check units are consistent (km and hours, or m and seconds).

Geometry Word Problems

Example: A rectangle has perimeter 46cm. Its length is 5cm more than its width. Find the width.

Let width = w, length = w + 5. Perimeter = 2(l + w) = 2(w + 5 + w) = 2(2w + 5) = 46 4w + 10 = 46 โ†’ 4w = 36 โ†’ w = 9cm

Length = 9 + 5 = 14cm. Check: 2(9 + 14) = 2 ร— 23 = 46 โœ“

Multi-Step Problems

Some problems require several operations. Work step by step and keep track of what each result represents.

Example: A baker makes 300 loaves. 127 are sold in the morning and 94 in the afternoon. The remaining loaves are packaged in boxes of 5. How many full boxes are made?

Step 1: total sold = 127 + 94 = 221 Step 2: remaining = 300 - 221 = 79 Step 3: boxes = 79 รท 5 = 15 remainder 4 โ†’ 15 full boxes

Checking Answers

Always verify your answer makes sense in context:

  • Is the number realistic? (e.g. a negative number of people makes no sense)
  • Does it satisfy the original conditions? Substitute back and check
  • Are the units correct?

A quick estimate before solving also helps catch large errors.

Common Mistakes

  • Not reading the whole problem before starting โ€” missing key information
  • Answering a different question to the one asked (e.g. finding Jo instead of Sam)
  • Forgetting units in the final answer
  • Not checking whether the answer satisfies all conditions in the problem
  • Using the wrong percentage base in reverse percentage problems

Tips and Tricks

  • Underline or highlight the key numbers and the question being asked
  • Write down what each variable represents before forming equations
  • For "times as many" problems, the smaller quantity is usually the variable (let the smaller = x)
  • For reverse percentage: identify what % the given value is, then scale to 100%
  • Always write a concluding sentence: "Sam has 36 marbles" โ€” not just "36"
  • If stuck, try a simpler version of the problem with easier numbers to understand the structure
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