Basic 9 · Mathematics · Number Operations
Word Problems with the Four Operations
In this topic you will solve operations involving addition, subtraction, multiplication and division using word problems, and round your answers appropriately.
By the end of these notes you should be able to:
- Translate word problems into calculations
- Solve multi-step word problems
- Round answers to suit the context
What you already know
- Operations on numbers
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.2.1 – Solve operations involving addition subtraction multiplication and division using word problems
Introduction
In real life, nobody hands you a calculation. You have to work out what the words are asking for.
1. Keywords
| Operation | Keywords |
|---|---|
| addition | total, altogether, sum, more than |
| subtraction | difference, left, less than, change |
| multiplication | each, times, product, per |
| division | share, per, average, split equally |
Keywords help, but they do not replace thinking. "How many more" asks for subtraction, even though it contains "more". Understand the situation, and draw a sketch if you need to.
2. Worked Examples
Addition and subtraction
A farmer harvests 1,250 kg of maize and 875 kg of cassava. She sells 1,400 kg in total. How much is left?
- Total harvest: 1,250 + 875 = 2,125 kg.
- Left: 2,125 − 1,400.
Answer: 725 kg
Multiplication and division
A printer prints 36 pages per minute. How long does it take to print 900 pages?
- Time = 900 ÷ 36.
Answer: 25 minutes
3. Writing Number Sentences
Translate each statement into symbols. "Three times a number, increased by 5, is 26" becomes 3x + 5 = 26. Subtract 5 to get 3x = 21, and so x = 7. "5 less than a number" means x − 5, not 5 − x. Reading word by word often leads to mistakes, so think about the meaning of the whole phrase.
4. Units
Check that all the quantities are in the same unit. If one length is in metres and another in centimetres, convert them first. Include the unit in your answer: GH₵, km, kg or litres. A good answer is a sentence that responds to the question. Do not write only "725". Write "725 kg is left". Check that your answer is reasonable. A bus cannot carry 2.4 people, and a farm cannot have a negative area.
New words
- keyword
- A word that gives a clue.
- number sentence
- A statement written in symbols.
- unit
- A standard of measurement.
- reasonable
- Making good sense.
Summary
- Keywords help, but understanding comes first.
- Write number sentences and keep units the same.
- State the answer in a sentence and check it.
Class activity: Story to Sum
- Write a word problem.
- Swap and write the number sentence.
- Solve and check.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. "Share equally" suggests…
- A. division
- B. addition
- C. subtraction
- D. indices
Show answer
Answer: divisionKeyword.2. "5 less than x" is…
- A. x − 5
- B. 5 − x
- C. 5x
- D. x + 5
Show answer
Answer: x − 5Order.3. Units should be the same before calculating.
- A. True
- B. False
Show answer
Answer: TrueConversion.4. 900 ÷ 36 = ______.
Show answer
Answer: 25Division.5. A tank holds 1,500 l. If 650 l are used, how much is left?
Show answer
Answer: 850 lSubtract.6. Twice a number plus 9 is 35. Find the number.
Show answer
Answer: 132x = 26.
Lesson 2
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.2.1 – Solve operations involving addition subtraction multiplication and division using word problems
Introduction
Many real problems need more than one step. The key is to take them one at a time.
1. Planning Multi-Step Problems
- Read the problem and note what is given.
- Identify the final question.
- Work out what you need to know first.
- Solve each step, and label it.
- Answer the question.
- Check.
2. Worked Examples
Shopping
Ama buys 3 kg of rice at GH₵18 per kg and 2 l of oil at GH₵35 per litre. She pays with GH₵200. Find her change.
- Rice: 3 × 18 = 54.
- Oil: 2 × 35 = 70.
- Total: 54 + 70 = 124.
- Change: 200 − 124.
Answer: GH₵76
Profit
A trader buys 50 shirts for GH₵1,500 and sells them at GH₵45 each. Find the profit per shirt.
- Cost per shirt: 1,500 ÷ 50 = 30.
- Profit per shirt: 45 − 30.
Answer: GH₵15
Travel
A bus travels 60 km/h for 2 hours and 80 km/h for 1.5 hours. Find the total distance.
- First part: 60 × 2 = 120 km.
- Second part: 80 × 1.5 = 120 km.
- Total: 120 + 120.
Answer: 240 km
3. Common Pitfalls
Learners often stop after the first step, or answer a different question. If the question asks for the change, do not stop at the total cost. Underline the final question, and go back to it at the end. Write down each step, and label it, for example "Cost of rice". This helps you to find mistakes, and it earns method marks.
4. Ratio and Rate Problems
Many multi-step problems involve rates and ratios. If 5 workers build a wall in 12 days, then 1 worker would take 60 days, so 10 workers would take 6 days. Here, more workers means fewer days. This is inverse proportion. In direct proportion, both quantities grow together: if 3 pens cost GH₵9, then 7 pens cost GH₵21. Decide which kind you have before calculating.
New words
- multi-step
- Needing more than one calculation.
- rate
- A quantity per unit.
- direct proportion
- Both quantities increase together.
- inverse proportion
- One rises as the other falls.
Summary
- Solve multi-step problems one labelled step at a time.
- Underline the final question.
- Distinguish direct from inverse proportion.
Class activity: Shop Problem
- Write a shopping list with prices.
- Create a multi-step problem.
- Swap and solve.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. 4 books at GH₵12 each cost…
- A. GH₵48
- B. GH₵16
- C. GH₵36
- D. GH₵24
Show answer
Answer: GH₵48Multiply.2. 6 workers take 10 days. How long do 3 workers take?
- A. 20 days
- B. 5 days
- C. 30 days
- D. 10 days
Show answer
Answer: 20 daysInverse.3. Labelling steps helps to find mistakes.
- A. True
- B. False
Show answer
Answer: TrueWorking.4. A quantity per unit is a ______.
Show answer
Answer: rateTerm.5. 5 pens cost GH₵15. Find the cost of 8 pens.
Show answer
Answer: GH₵24Direct proportion.6. A car uses 8 l per 100 km. Fuel is GH₵15 per litre. Find the cost of 250 km.
Show answer
Answer: GH₵30020 l × 15.
Lesson 3
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.2.2 – Solve word problems involving the four basic operations and round the answers to the nearest two decimal figures or to some significant figures
Introduction
A calculator may show 41.666667. Whether you write 41.67, 42 or 41 depends on the question.
1. Rounding to Two Decimal Places
Sharing money
GH₵1,000 is shared equally among 7 people. How much does each get, to the nearest pesewa?
- 1,000 ÷ 7 = 142.857...
- To 2 d.p.
Answer: GH₵142.86
Money is usually given to 2 d.p., because 1 pesewa is GH₵0.01. Round only at the end. In this case, 7 × 142.86 = 1,000.02, so a real sharing would need a small adjustment.
2. Rounding to Significant Figures
Population density
A district of 245,000 people covers 1,850 km². Find the number of people per km², to 3 s.f.
- 245,000 ÷ 1,850 = 132.43...
- To 3 s.f.
Answer: 132 people per km²
3. Rounding Up or Down
| Situation | Calculation | Answer |
|---|---|---|
| buses for 250 people, 45 seats | 250 ÷ 45 = 5.6 | 6 buses (round up) |
| shirts from 20 m, 1.5 m each | 20 ÷ 1.5 = 13.3 | 13 shirts (round down) |
| bags of cement, 7.2 needed | 7.2 | 8 bags (round up) |
| notes of GH₵20 from GH₵135 | 135 ÷ 20 = 6.75 | 6 notes (round down) |
Context decides. If everyone must travel, you need an extra bus, even if it is only part full. If you can only make whole shirts, you cannot count a part of one. Always ask what the answer means in the real world.
4. Accuracy of Results
An answer can never be more accurate than the data used. If a length is measured to the nearest metre, do not give an area to five decimal places. A good rule is to round your answer to the same number of significant figures as your least accurate measurement. State your accuracy, such as "to 3 s.f.", when you report results. In an examination, follow any instruction given in the question exactly, as marks may be lost if you do not.
New words
- round up
- To go to the next value.
- round down
- To go to the lower value.
- context
- The situation of a problem.
- density
- Amount per unit of area or volume.
Summary
- Money is given to 2 d.p.
- Context decides whether to round up or down.
- Answers should match the accuracy of the data.
Class activity: Round It Right
- Read five real situations.
- Decide how to round each answer.
- Explain your reasons.
You need: Place-value cards, Ghana cedi play notes, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 12.345 to 2 d.p. is…
- A. 12.35
- B. 12.34
- C. 12.3
- D. 12.4
Show answer
Answer: 12.35Next digit 5.2. 100 people need taxis of 4 seats. How many taxis?
- A. 25
- B. 24
- C. 26
- D. 20
Show answer
Answer: 25100 ÷ 4.3. Context affects rounding.
- A. True
- B. False
Show answer
Answer: TrueMeaning.4. 7.2 bags of cement means buying ______ bags.
Show answer
Answer: 8Round up.5. GH₵500 is shared by 3. Find each share to 2 d.p.
Show answer
Answer: GH₵166.67166.666...6. 130 learners travel in buses of 48 seats. How many buses are needed, and why?
Show answer
Answer: 3 buses, as 2 buses seat only 96.Round up.
Homework
- Solve five real-life problems, and explain how you rounded each answer.