Basic 7 · Mathematics · Number Operations

Story Problems with Decimals and the Meaning of Powers

In this topic you will create and solve story problems involving the four operations on decimals, and explore the meaning of repeated factors and powers using objects.

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

Add and subtract decimals in story problems
Multiply and divide decimals in story problems
Explain repeated factors and write them as powers

What you already know

Operations on whole numbers
Decimal place value

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: B7.1.2.2.3 – Create and solve story problems involving decimals on the four basic operations

Introduction

Money, measurements and sports scores often involve decimals. To solve these problems, we must line up the decimal points carefully.

1. Adding and Subtracting Decimals

Line up the decimal points one below the other. Fill empty places with zeros so that every number has the same number of decimal places. Then add or subtract as with whole numbers, and place the decimal point in the answer directly below the others.

Add decimals

Ama bought rice for GH₵45.50, tomatoes for GH₵12.75 and fish for GH₵38.20. Find the total.

Line up: 45.50 + 12.75 + 38.20.
Hundredths: 0 + 5 + 0 = 5. Tenths: 5 + 7 + 2 = 14; write 4, carry 1.
Ones: 5 + 2 + 8 + 1 = 16; write 6, carry 1. Tens: 4 + 1 + 3 + 1 = 9.

Answer: GH₵96.45

Subtract decimals

Ama paid with a GH₵100 note. How much change did she receive?

100.00 − 96.45.

Answer: GH₵3.55

Food items with price tags in cedis

Figure 1: Prices with decimals Write a shopping list with decimal prices and find the total.

2. Measurement Problems

Lengths

A tailor has 5.6 m of cloth. She uses 2.75 m for a dress and 1.8 m for a shirt. How much cloth is left?

Used: 2.75 + 1.80 = 4.55 m.
Left: 5.60 − 4.55 = 1.05 m.

Answer: 1.05 m

3. Creating Story Problems

To create a story problem, choose a real situation (shopping, measuring, travelling), choose sensible decimal numbers, and ask a clear question. For example: "Kwame ran 2.4 km on Monday and 3.15 km on Tuesday. How far did he run altogether?" Then solve your own problem to check that it works.

New words

decimal point: The dot separating whole numbers from parts.
line up: Arrange digits so that places match.
story problem: A word problem based on a real situation.
total: The result of adding.

Summary

Line up decimal points and fill empty places with zeros before adding or subtracting.
Place the decimal point in the answer below the others.
Create story problems from real situations and solve them to check.

Class activity: Shopping Story Problems

Use a price list with decimal prices.
Write two addition and two subtraction story problems.
Swap with another pair and solve.

You need: Bottle tops, Ghana cedi play notes, Number cards

Exercises

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

1. 3.4 + 2.15 =
- A. 5.55
- B. 5.19
- C. 5.45
- D. 6.55
Show answer
Answer: 5.55Line up points.
2. 10 − 3.75 =
- A. 6.25
- B. 7.25
- C. 6.35
- D. 7.75
Show answer
Answer: 6.2510.00 − 3.75.
3. Decimal points should be lined up when adding.
- A. True
- B. False
Show answer
Answer: TrueMatching places.
4. The dot separating whole numbers from parts is the decimal ______.
Show answer
Answer: pointDecimal point.
5. Kojo has 4.5 litres of water. He drinks 1.25 litres. How much is left?
Show answer
Answer: 3.25 litresSubtraction.
6. Write and solve a story problem that involves adding three decimal amounts of money.
Show answer
Answer: Example: 12.50 + 8.75 + 4.20 = GH₵25.45.Any correct problem.

Lesson 2

Strand: Number
Sub-strand: Number Operations
Indicator: B7.1.2.2.3 – Create and solve story problems involving decimals on the four basic operations

Introduction

If one bottle of oil costs GH₵18.50, how much do 6 bottles cost? If 7.5 m of rope is cut into 3 equal pieces, how long is each piece? These problems use multiplication and division of decimals.

1. Multiplying a Decimal by a Whole Number

Multiply

Find the cost of 6 bottles of oil at GH₵18.50 each.

Multiply as whole numbers: 1,850 × 6 = 11,100.
18.50 has 2 decimal places, so the answer has 2 decimal places.

Answer: GH₵111.00

2. Multiplying Two Decimals

Multiply as whole numbers, then count the total number of decimal places in both numbers and put that many in the answer. For example, 2.4 × 1.5: 24 × 15 = 360, and there are 2 decimal places in total, so the answer is 3.60 = 3.6.

3. Dividing a Decimal by a Whole Number

Divide

7.5 m of rope is cut into 3 equal pieces. How long is each piece?

7.5 ÷ 3: 7 ÷ 3 = 2 remainder 1; 15 ÷ 3 = 5.
Keep the decimal point in line.

Answer: 2.5 m

A coil of rope being measured

Figure 2: Rope Write a division story problem about cutting rope or cloth.

4. Dividing by a Decimal

To divide by a decimal, change the divisor into a whole number by multiplying both numbers by the same power of ten. For 4.8 ÷ 0.6, multiply both by 10: 48 ÷ 6 = 8. For 3.75 ÷ 0.25, multiply both by 100: 375 ÷ 25 = 15.

5. Story Problem

Fuel

A taxi uses 0.08 litres of fuel per kilometre. How much fuel does it use for a 125 km journey? Fuel costs GH₵14.50 per litre. What is the cost?

Fuel: 125 × 0.08 = 10 litres.
Cost: 10 × 14.50 = GH₵145.00.

Answer: 10 litres costing GH₵145.00

New words

divisor: The number you divide by.
dividend: The number being divided.
decimal places in a product: The total decimal places of the numbers multiplied.
per: For each.

Summary

When multiplying, count the total decimal places and place them in the answer.
When dividing by a whole number, keep the decimal point in line.
To divide by a decimal, multiply both numbers by a power of ten to make the divisor whole.

Class activity: Decimal Market

Use cedi play notes and a price list with decimals.
Solve problems such as "the cost of 4 items" and "the price of one item when 5 cost GH₵62.50".
Present one problem and its solution.

You need: Bottle tops, Ghana cedi play notes, Number cards

Exercises

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

1. 0.4 × 3 =
- A. 1.2
- B. 0.12
- C. 12
- D. 1.02
Show answer
Answer: 1.2One decimal place.
2. 6.3 ÷ 0.7 =
- A. 9
- B. 0.9
- C. 90
- D. 0.09
Show answer
Answer: 963 ÷ 7.
3. 2.5 × 1.2 = 3.
- A. True
- B. False
Show answer
Answer: True25 × 12 = 300 → 3.00.
4. The number you divide by is the ______.
Show answer
Answer: divisore.g. 0.7.
5. Five pens cost GH₵12.50. What is the cost of one pen?
Show answer
Answer: GH₵2.5012.50 ÷ 5.
6. A bag of sugar weighs 2.25 kg. What is the mass of 8 bags?
Show answer
Answer: 18 kg2.25 × 8.

Lesson 3

Strand: Number
Sub-strand: Number Operations
Indicator: B7.1.2.3.1 – Illustrate with examples the meaning of repeated factors using counting objects such as bottle tops or bundle sticks

Introduction

When the same number is multiplied many times, writing it out is long: 2 × 2 × 2 × 2 × 2 × 2. Mathematicians use a short form called a power.

1. Repeated Factors

In 3 × 3 × 3 × 3, the number 3 is a repeated factor. We can model it with bottle tops: 3 groups of 3 make 9; 3 groups of those 9 make 27; 3 groups of 27 make 81.

2. Powers

Power (index form): A short way of writing repeated multiplication, such as 3⁴, where 3 is the base and 4 is the index (exponent).

Writing repeated factors as powers

Repeated factors	Power	Read as	Value
2 × 2 × 2	2³	2 to the power 3 (2 cubed)	8
5 × 5	5²	5 squared	25
3 × 3 × 3 × 3	3⁴	3 to the power 4	81
10 × 10 × 10 × 10 × 10	10⁵	10 to the power 5	100,000

Square grid paper with shaded squares

Figure 3: Squares on grid paper Shade squares on grid paper to show 2², 3² and 4².

3. Base and Index

In 7⁵, the base is 7 (the number being multiplied) and the index is 5 (how many times it is used as a factor). Do not confuse 7⁵ with 7 × 5: 7⁵ = 7 × 7 × 7 × 7 × 7 = 16,807, while 7 × 5 = 35.

4. Squares and Cubes

A number to the power 2 is called a square, because it gives the area of a square: a square with sides of 4 cm has area 4² = 16 cm². A number to the power 3 is called a cube, because it gives the volume of a cube: a cube with edges of 3 cm has volume 3³ = 27 cm³.

New words

power: Repeated multiplication written in short form.
base: The number that is multiplied in a power.
index (exponent): The small number showing how many times the base is used.
square number: A number multiplied by itself, such as 25.

Summary

Repeated factors can be written as powers, such as 2 × 2 × 2 = 2³.
In a power, the base is multiplied by itself as many times as the index shows.
Squares (power 2) and cubes (power 3) relate to area and volume.

Class activity: Bottle-Top Powers

Use bottle tops to build 2, then 2 × 2, then 2 × 2 × 2 and 2 × 2 × 2 × 2.
Record each as a power and find its value.
Repeat with the base 3.

You need: Bottle tops, Bundles of sticks, Printed factor-and-power cards

Exercises

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

1. 4 × 4 × 4 written as a power is…
- A. 4³
- B. 3⁴
- C. 4 × 3
- D. 12
Show answer
Answer: 4³Base 4, index 3.
2. The value of 2⁵ is…
- A. 32
- B. 10
- C. 25
- D. 16
Show answer
Answer: 322 × 2 × 2 × 2 × 2.
3. In 6², the base is 6.
- A. True
- B. False
Show answer
Answer: True6 is multiplied.
4. In 9⁴, the index is ______.
Show answer
Answer: 4Number of factors.
5. Write 5 × 5 × 5 × 5 as a power and find its value.
Show answer
Answer: 5⁴ = 625Power form.
6. Explain the difference between 3⁴ and 3 × 4.
Show answer
Answer: 3⁴ = 3 × 3 × 3 × 3 = 81, while 3 × 4 = 12.Repeated multiplication vs multiplication.

Homework

Write three story problems with decimals and solve them.
Write the first ten square numbers and the first five cube numbers.