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Basic 8 · Mathematics · Word Problems on Place Values

Word Problems on Place Value, Perfect Squares and Square Roots

In this topic you will create and solve word problems on place value, and use the concept of sets to identify perfect squares and determine square roots.

By the end of these notes you should be able to:

  • Solve real-life word problems involving place value
  • Create word problems on place value
  • Identify perfect squares and find square roots using sets

What you already know

  • Large numbers, rounding and powers

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
Word Problems on Place Values
Indicator
B8.1.1.1.6 – Create and solve word or real-life problems on place values

Introduction

When a company reports profits or the government announces a budget, place value tells us what the numbers really mean.

1. Steps for Word Problems

  1. Read the problem and underline the key numbers.
  2. Decide what is being asked.
  3. Choose the operation.
  4. Calculate carefully, keeping place values in line.
  5. Check that your answer makes sense.

2. Worked Problems

Budget problem

A district has GH₵4,500,000 for projects. It spends GH₵1,750,000 on a school and GH₵980,000 on boreholes. How much remains?

  1. Total spent: 1,750,000 + 980,000 = 2,730,000.
  2. Remaining: 4,500,000 − 2,730,000 = 1,770,000.

Answer: GH₵1,770,000

Digit problem

Using the digits 7, 3, 0, 9, 5 once each, write the largest and smallest five-digit numbers.

  1. Largest: put the biggest digits first: 97,530.
  2. Smallest: a number cannot start with 0, so start with 3: 30,579.

Answer: 97,530 and 30,579

A newly built school block in a Ghanaian community
Figure 1: School project Write a problem about the cost of a community project.

3. Rounding in Word Problems

Some problems ask you to round. If a report says 2,467,900 people voted, you could say "about 2.5 million". Think about what level of accuracy suits the situation. An accountant needs exact figures, but a newspaper headline does not.

4. Checking Answers

Use the inverse operation to check. In the budget problem, add the amount spent and the amount remaining: 2,730,000 + 1,770,000 = 4,500,000. If they match the total, your answer is correct.

5. Choosing the Operation

Key words can help but must be used with care. "Altogether" often suggests addition, "remains" suggests subtraction, "each" suggests multiplication or division and "shared" suggests division. Always think about the situation, not just the words, before you choose.

New words

word problem
A mathematics question set in a real situation.
key number
An important number in a problem.
inverse operation
An operation that undoes another.
profit
Money gained after costs are removed.

Summary

  • Read, choose the operation, calculate and check.
  • Place value decides the largest and smallest numbers from given digits.
  • Use the inverse operation to check, and round to a suitable accuracy.

Class activity: Problem Solvers

  1. Solve three budget or population problems using place-value cards.
  2. Check each with an inverse operation.
  3. Explain your method to another group.

You need: Place-value cards, Ghana cedi play notes, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. The largest number from 4, 1, 7 is…

    • A. 741
    • B. 147
    • C. 714
    • D. 417
    Show answer
    Answer: 741Biggest digit first.

  2. 2. A town had 3,250,000 people and gained 480,000. How many now?

    • A. 3,730,000
    • B. 3,700,000
    • C. 2,770,000
    • D. 3,750,000
    Show answer
    Answer: 3,730,000Add.

  3. 3. A five-digit number cannot start with 0.

    • A. True
    • B. False
    Show answer
    Answer: TrueIt would have four digits.

  4. 4. An operation that undoes another is an ______ operation.

    Show answer
    Answer: inverseUndo.

  5. 5. A bank has GH₵8,000,000 and lends GH₵2,450,000. How much remains?

    Show answer
    Answer: GH₵5,550,000Subtract.

  6. 6. Using 2, 0, 6, 8, 1, write the smallest five-digit even number.

    Show answer
    Answer: 10,268Start with 1 and end in an even digit.

Lesson 2

Strand
Number
Sub-strand
Word Problems on Place Values
Indicator
B8.1.1.1.6 – Create and solve word or real-life problems on place values

Introduction

Making your own problems shows that you truly understand a topic. It also helps you to see how mathematics connects to life.

1. Steps in Creating a Problem

  1. Choose a real situation: market, farm, school, football or government.
  2. Choose sensible numbers.
  3. Decide which operation you want the solver to use.
  4. Write a clear question.
  5. Solve your problem to make sure that it works.

2. Model Problems

Cocoa farmers

Create and solve a problem about cocoa. "A cooperative sold 12,500 bags of cocoa at GH₵2,400 per bag. How much did it receive?"

  1. 12,500 × 2,400.
  2. 125 × 24 = 3,000; add four zeros.

Answer: GH₵30,000,000

Stadium seats

"A stadium has 40,000 seats. On match day, 36,750 were filled. How many were empty?"

  1. 40,000 − 36,750.

Answer: 3,250 seats

Bags of cocoa beans stacked in a warehouse
Figure 2: Cocoa bags Create a word problem about bags of cocoa or rice.

3. Making Good Problems

A good problem has enough information, a clear question and realistic numbers. Avoid problems such as "A boy bought 5,000,000 mangoes for GH₵2", which make no sense. Check that the answer is possible: you cannot have 2.5 people.

4. Multi-Step Problems

Harder problems need more than one step. "A farmer sold 2,400 tubers of yam at GH₵8 each and spent GH₵5,600 on labour. How much profit did he make?" First find the income, 2,400 × 8 = 19,200, and then subtract: 19,200 − 5,600 = GH₵13,600. Try writing a multi-step problem for your partner.

New words

realistic
Possible in real life.
multi-step problem
A problem with more than one stage.
income
Money received.
cooperative
A group working together for shared benefit.

Summary

  • Good problems have a real situation, sensible numbers and a clear question.
  • Solve your own problem to check that it works.
  • Multi-step problems combine operations.

Class activity: Problem Exchange

  1. Write two word problems on place value.
  2. Swap with your partner and solve them.
  3. Check whether each problem is clear and realistic.

You need: Place-value cards, Ghana cedi play notes, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. A good word problem has…

    • A. a clear question
    • B. no numbers
    • C. no situation
    • D. impossible answers
    Show answer
    Answer: a clear questionClarity.

  2. 2. 12,500 × 2,400 =

    • A. 30,000,000
    • B. 3,000,000
    • C. 300,000,000
    • D. 2,500,000
    Show answer
    Answer: 30,000,000Multiply.

  3. 3. You should solve your own problem to check it.

    • A. True
    • B. False
    Show answer
    Answer: TrueCheck.

  4. 4. Money received is ______.

    Show answer
    Answer: incomeEarnings.

  5. 5. Write a word problem whose answer is found by subtracting.

    Show answer
    Answer: A tank holds 50,000 litres and 12,500 are used. How many remain? (any suitable)Subtraction.

  6. 6. A trader bought 1,500 bags for GH₵30,000 and sold them at GH₵25 each. Find the profit.

    Show answer
    Answer: GH₵7,500 (37,500 − 30,000)Multi-step.

Lesson 3

Strand
Number
Sub-strand
Sets
Indicator
B8.1.1.2.1 – Use the concept of sets to identify perfect squares and determine the square roots

Introduction

Four tiles can form a 2 by 2 square, and nine can form a 3 by 3 square. Numbers like 4 and 9 are perfect squares.

1. Perfect Squares

Perfect square: A number that is the product of a whole number multiplied by itself.

We can write the set of perfect squares as S = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...}. Using set notation means listing the members in curly brackets. Each member comes from multiplying a whole number by itself: 1 × 1, 2 × 2, 3 × 3 and so on.

2. Square Roots

Square root: A number that, when multiplied by itself, gives a stated number.

The symbol √ means square root. √49 = 7, because 7 × 7 = 49. Finding the square root is the inverse of squaring.

Squares and square roots
nn²√(n²)
5255
8648
1112111
1214412
1522515
2040020
Square floor tiles arranged in a larger square
Figure 3: Square tiles Use counters to show that 16 and 25 are perfect squares.

3. Finding Square Roots by Prime Factors

Find the square root

Find √324 using prime factors.

  1. 324 = 2² × 3⁴.
  2. Halve the indices: 2¹ × 3².
  3. 2 × 9 = 18.

Answer: √324 = 18

A number is a perfect square if all the indices in its prime factorisation are even. For example, 72 = 2³ × 3² is not a perfect square, because the index 3 is odd.

4. Using Square Roots

If a square garden has an area of 196 m², its side is √196 = 14 m. Square roots help in area problems and later in Pythagoras' theorem.

New words

perfect square
A whole number multiplied by itself.
square root
A number that multiplied by itself gives a stated number.
set
A collection of items listed in curly brackets.
member
An item in a set.

Summary

  • Perfect squares are products of whole numbers with themselves.
  • The square root is the inverse of squaring.
  • Using prime factors, halve the indices to find the square root.

Class activity: Square Builder

  1. Use bottle tops to build squares of sizes 1 to 10.
  2. List the set of perfect squares.
  3. Find the square roots of 64, 81 and 144.

You need: Bottle tops, Square-grid paper, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. √81 =

    • A. 9
    • B. 8
    • C. 18
    • D. 40.5
    Show answer
    Answer: 99 × 9.

  2. 2. Which is a perfect square?

    • A. 144
    • B. 150
    • C. 200
    • D. 72
    Show answer
    Answer: 14412 × 12.

  3. 3. 25 is a member of the set of perfect squares.

    • A. True
    • B. False
    Show answer
    Answer: True5 × 5.

  4. 4. The inverse of squaring is finding the square ______.

    Show answer
    Answer: rootInverse.

  5. 5. Find √225.

    Show answer
    Answer: 1515 × 15.

  6. 6. A square farm has area 2,500 m². Find its side and perimeter.

    Show answer
    Answer: Side 50 m; perimeter 200 m√2,500 = 50.

Homework

  1. Write three word problems on place value and solve them.
  2. List the perfect squares up to 400.
Word Problems on Place Value, Perfect Squares and Square Roots – Basic 8 Mathematics Notes | sirbus