Basic 8 · Mathematics · Mental Mathematics Strategies

Mental Mathematics with Number Properties and Word Problems

In this topic you will apply mental strategies and number properties to carry out calculations, work mentally with percentages and fractions, and solve word problems using mental mathematics.

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

Apply number properties to simplify calculations
Use mental strategies with fractions and percentages
Solve word problems using mental strategies

What you already know

Mental strategies and number properties (Basic 7)

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: Mental Mathematics Strategies
Indicator: B8.1.2.1.2 – Apply mental mathematics strategies and number properties to do calculation

Introduction

You do not need a calculator for 25 × 37 × 4. Rearranging the numbers makes it easy.

1. Properties in Action

Using properties

Property	Example
commutative	25 × 37 × 4 = 25 × 4 × 37
associative	(25 × 4) × 37 = 100 × 37 = 3,700
distributive	15 × 102 = 15 × 100 + 15 × 2 = 1,530
factorising out	37 × 8 + 37 × 2 = 37 × 10 = 370
identity	56 + 0 = 56; 89 × 1 = 89

2. Useful Tricks

To multiply by 5, multiply by 10 and halve. To multiply by 25, multiply by 100 and divide by 4. To multiply by 99, multiply by 100 and subtract the number once. To square a number ending in 5, multiply the tens digit by the next whole number and write 25 after it: 35² = (3 × 4) then 25 = 1,225.

Calculate mentally

Calculate 48 × 25 and 45².

48 × 25 = 48 × 100 ÷ 4 = 4,800 ÷ 4 = 1,200.
45²: 4 × 5 = 20, then write 25.

Answer: 1,200 and 2,025

A market trader calculating a price mentally

Figure 1: Quick calculations Write five calculations you can simplify using number properties.

3. Grouping to Make Tens

When adding many numbers, look for pairs that make 10 or 100: 47 + 36 + 53 + 64 = (47 + 53) + (36 + 64) = 100 + 100 = 200. This uses the commutative and associative properties, and saves time in the market as well as in examinations.

4. Differences of Squares

A useful pattern is a² − b² = (a + b)(a − b). So 53² − 47² = (53 + 47)(53 − 47) = 100 × 6 = 600. Patterns like this show how algebra makes arithmetic easier.

New words

commutative: Order does not matter.
associative: Grouping does not matter.
distributive: Multiplying over a sum.
factorise: To write as a product of factors.

Summary

Number properties let us rearrange calculations.
Tricks for 5, 25, 99 and squares ending in 5 speed up mental work.
Grouping to make tens and using patterns simplify calculations.

Class activity: Mental Maths Quiz

The teacher calls out calculations.
Learners solve them mentally and explain their strategies.
The class compares strategies to find the quickest.

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

Exercises

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

1. 25 × 8 × 4 =
- A. 800
- B. 400
- C. 80
- D. 1,600
Show answer
Answer: 80025 × 4 × 8.
2. 35² =
- A. 1,225
- B. 1,025
- C. 625
- D. 1,525
Show answer
Answer: 1,2253 × 4 then 25.
3. 37 × 8 + 37 × 2 = 370.
- A. True
- B. False
Show answer
Answer: TrueDistributive.
4. 16 × 25 = ______.
Show answer
Answer: 4001,600 ÷ 4.
5. Calculate 99 × 46 mentally.
Show answer
Answer: 4,554 (4,600 − 46)Compensation.
6. Use the difference of squares to calculate 62² − 38².
Show answer
Answer: (62 + 38)(62 − 38) = 100 × 24 = 2,400Pattern.

Lesson 2

Strand: Number
Sub-strand: Mental Mathematics Strategies
Indicator: B8.1.2.1.2 – Apply mental mathematics strategies and number properties to do calculation

Introduction

"20% off GH₵350!" Being able to work this out in your head helps you shop wisely.

1. Finding Percentages Mentally

Mental percentages of GH₵360

Percentage	Method	Answer
10%	divide by 10	GH₵36
5%	half of 10%	GH₵18
1%	divide by 100	GH₵3.60
20%	2 × 10%	GH₵72
25%	divide by 4	GH₵90
50%	divide by 2	GH₵180
15%	10% + 5%	GH₵54

2. Fractions of Amounts

To find a fraction of an amount mentally, divide by the denominator and then multiply by the numerator. ¾ of 80 is 80 ÷ 4 × 3 = 60. ⅖ of 150 is 150 ÷ 5 × 2 = 60. Use friendly numbers: 1/8 of 200 is half of half of half of 200, which is 25.

3. Discounts and Increases

Sale price

A pair of shoes costs GH₵240. There is a 15% discount. Find the sale price mentally.

10% of 240 = 24; 5% = 12.
15% = 36.
240 − 36 = 204.

Answer: GH₵204

Shoes on display with a discount label

Figure 2: Shoes on sale Calculate the sale price of three items mentally.

4. Reversing Percentages

You can often swap the numbers: 18% of 50 equals 50% of 18, which is 9. So x% of y = y% of x. Try 4% of 75: it is the same as 75% of 4, which is 3. Looking for such shortcuts makes mental work faster.

5. Why Percentages Help

Percentages let us compare amounts that have different totals. A discount of GH₵20 on a GH₵40 item is far better than GH₵20 off a GH₵400 item, because one is 50% and the other only 5%.

New words

percentage: A number out of 100.
discount: An amount taken off a price.
sale price: The price after a discount.
friendly number: A number that is easy to calculate with.

Summary

Build percentages from 10%, 5% and 1%.
Divide by the denominator and multiply by the numerator for fractions.
x% of y equals y% of x.

Class activity: Shop Sale

Make price tags for ten items.
Give each a discount and work out the sale price mentally.
Check using a calculator.

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

Exercises

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

1. 10% of GH₵250 is…
- A. GH₵25
- B. GH₵2.50
- C. GH₵250
- D. GH₵10
Show answer
Answer: GH₵25Divide by 10.
2. 15% of GH₵80 is…
- A. GH₵12
- B. GH₵8
- C. GH₵15
- D. GH₵20
Show answer
Answer: GH₵128 + 4.
3. 25% is the same as dividing by 4.
- A. True
- B. False
Show answer
Answer: TrueOne quarter.
4. An amount taken off a price is a ______.
Show answer
Answer: discountReduction.
5. Find ⅗ of 250 mentally.
Show answer
Answer: 150250 ÷ 5 × 3.
6. Use swapping to find 8% of 25.
Show answer
Answer: 2 (25% of 8)x% of y.

Lesson 3

Strand: Number
Sub-strand: Mental Mathematics Strategies
Indicator: B8.1.2.1.3 – Apply mental mathematics strategies to solve word problems

Introduction

Word problems in daily life rarely come with a calculator. Mental strategies help you to decide quickly.

1. Worked Problems

Airtime

Esi buys airtime worth GH₵20 and gets a 25% bonus. How much credit does she have?

25% of 20 = 5.
20 + 5 = 25.

Answer: GH₵25

Trotro fares

A family of 6 pays GH₵12.50 each for a trotro. Find the total fare.

6 × 12 = 72.
6 × 0.50 = 3.
72 + 3 = 75.

Answer: GH₵75

Mixing

A builder mixes 2 parts cement with 5 parts sand. How many head pans of sand are needed for 8 head pans of cement?

For every 2 of cement, 5 of sand.
8 ÷ 2 = 4 groups.
4 × 5 = 20.

Answer: 20 head pans

A mobile money agent kiosk in Ghana

Figure 3: Mobile money kiosk Write a word problem about airtime or mobile money.

2. Strategies for Multi-Step Problems

Break the problem into steps and keep track of each result. Write down short notes if needed. For example, if a shirt costs GH₵45 and trousers cost GH₵78, and you have GH₵150, then 45 + 78 = 123 and 150 − 123 = 27. Use rounding to add: 45 + 80 − 2 = 123.

3. Reasonableness

After solving, ask whether the answer is sensible. If six trotro fares come to GH₵750, you have made a place-value error. Estimating first, 6 × 10 = 60, shows that the answer should be around 60 to 80.

4. Mental Problem-Solving in Business

Traders work out profits in their heads. If a trader buys oranges at 3 for GH₵2 and sells them at GH₵1 each, she earns GH₵3 on goods that cost GH₵2, a profit of GH₵1 on every three oranges. Selling 300 oranges brings GH₵100 profit. Such quick thinking helps people to run businesses well.

New words

bonus: Something extra given.
ratio: A comparison of quantities.
fare: The price of a journey.
profit: Money gained after costs.

Summary

Break word problems into steps and use mental strategies for each.
Use ratios, percentages and rounding in real situations.
Check that the answer is reasonable.

Class activity: Market Problems

Make up five market problems.
Swap and solve them mentally.
Explain your strategies.

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

Exercises

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

1. 5 pens at GH₵4 each cost…
- A. GH₵20
- B. GH₵9
- C. GH₵15
- D. GH₵25
Show answer
Answer: GH₵205 × 4.
2. 3 oranges cost GH₵2. What do 12 oranges cost?
- A. GH₵8
- B. GH₵6
- C. GH₵12
- D. GH₵4
Show answer
Answer: GH₵84 groups.
3. Estimating can show place-value errors.
- A. True
- B. False
Show answer
Answer: TrueCheck.
4. The price of a journey is the ______.
Show answer
Answer: fareTransport.
5. A book marked GH₵40 is sold for GH₵36. What percentage discount was given?
Show answer
Answer: 10%, because GH₵4 is one tenth of GH₵40.Discount over marked price.
6. A trader buys 50 eggs at GH₵2 each and sells them at GH₵2.50 each. Find the profit.
Show answer
Answer: GH₵2550 × 0.50.

Homework

Solve ten mental mathematics problems from daily life.