Basic 9 · Mathematics · Fractions

Order of Operations with Fractions, Decimals and Percentages

In this topic you will add, subtract, multiply and divide fractions, decimals and percentages using the order of operations.

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

Apply BODMAS to fractions
Apply BODMAS to decimals
Solve mixed problems with percentages

What you already know

Operations on fractions, decimals and percentages

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: Fractions
Indicator: B9.1.3.1.2 – Add and/or subtract multiply and/or divide given fractions using the principle of order of operations including the use of the BODMAS or, PEMDAS rule and apply the understanding of these to solve problems

Introduction

Mixed fraction expressions can look frightening. BODMAS gives you a clear path through them.

1. BODMAS and PEMDAS

Order of operations

Step	BODMAS	PEMDAS
1	Brackets	Parentheses
2	Orders (powers)	Exponents
3	Division and Multiplication	Multiplication and Division
4	Addition and Subtraction	Addition and Subtraction

Division and multiplication have equal rank, and are done from left to right. So are addition and subtraction. The word "of" is treated as multiplication, and it is often done early, since it acts as if it had brackets around it. For example, ½ of ⅔ + ¼ means (½ × ⅔) + ¼.

2. Worked Examples

Multiply first

Simplify ½ + ⅓ × ¾.

Multiply: ⅓ × ¾ = ¼.
Add: ½ + ¼.

Answer: ¾

Brackets first

Simplify (⅔ + ⅙) ÷ 5/12.

Brackets: 4/6 + 1/6 = 5/6.
Divide: 5/6 × 12/5.

Answer: 2

Mixed numbers

Simplify 1¾ − ½ × 2.

Multiply: ½ × 2 = 1.
Subtract: 1¾ − 1.

Answer: ¾

A student working a fraction problem

Figure 1: Working with fractions Write a BODMAS expression with fractions.

3. Common Errors

Common BODMAS errors

Expression	Wrong	Correct
½ + ½ × 4	(½ + ½) × 4 = 4	½ + 2 = 2½
6 ÷ 2 × 3	6 ÷ 6 = 1	3 × 3 = 9
10 − 4 + 2	10 − 6 = 4	6 + 2 = 8

Working strictly from left to right, without regard for order, or doing all multiplication before division, are the two commonest errors. Rewrite the expression at each step, and do one operation at a time. Line up your equals signs. A neat layout makes errors far less likely.

4. Complex Fractions

In a fraction with an expression on top and bottom, treat each as if it were in brackets. For (½ + ¼) ÷ (⅔ − ⅙), the top bracket is ¾, and the bottom one is ½. Then ¾ ÷ ½ = ¾ × 2 = 1½. The fraction bar works like a pair of brackets.

New words

BODMAS: The order of operations.
rank: The priority of an operation.
left to right: The order for operations of equal rank.
complex fraction: A fraction containing fractions.

Summary

Follow brackets, orders, then division and multiplication, then addition and subtraction.
Treat "of" as multiplication.
Work one step at a time.

Class activity: BODMAS Battle

Each writes three expressions.
Swap and solve.
Check for errors.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

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

1. Which comes first?
- A. brackets
- B. addition
- C. subtraction
- D. none
Show answer
Answer: bracketsOrder.
2. ½ + ¼ × 2 =
- A. 1
- B. 1½
- C. ¾
- D. ½
Show answer
Answer: 1½ + ½.
3. Multiplication and division have equal rank.
- A. True
- B. False
Show answer
Answer: TrueLeft to right.
4. 6 ÷ 2 × 3 = ______.
Show answer
Answer: 9Left to right.
5. Simplify ¾ − ⅓ × ¾.
Show answer
Answer: ½¾ − ¼.
6. Simplify (⅖ + ⅕) ÷ (1 − ¼).
Show answer
Answer: ⅘⅗ ÷ ¾.

Lesson 2

Strand: Number
Sub-strand: Decimals
Indicator: B9.1.3.1.2 – Add and/or subtract multiply and/or divide given fractions using the principle of order of operations including the use of the BODMAS or, PEMDAS rule and apply the understanding of these to solve problems

Introduction

With decimals, as with fractions, the order of operations prevents mistakes.

1. Worked Examples

Multiply first

Simplify 2.5 + 1.5 × 0.4.

Multiply: 1.5 × 0.4 = 0.6.
Add: 2.5 + 0.6.

Answer: 3.1

Brackets and powers

Simplify (3.2 − 1.2)² ÷ 0.5.

Brackets: 2.
Power: 2² = 4.
Divide: 4 ÷ 0.5.

Answer: 8

Mixed fractions and decimals

Simplify ¾ × 0.8 + 1.4.

Change ¾ to 0.75.
0.75 × 0.8 = 0.6.
0.6 + 1.4.

Answer: 2

2. Using a Calculator

Scientific calculators follow BODMAS, but simple ones do not. Test yours by entering 2 + 3 × 4. If the answer is 14, then it follows the order. If it is 20, it does not. Use the bracket keys for long calculations, and use the fraction key if you have one. Estimate first, in case you make a keying error. Do not depend entirely on a calculator, because some questions forbid its use.

A scientific calculator on a desk

Figure 2: Scientific calculator Test your calculator with 2 + 3 × 4.

3. Decimals in Formulas

Substitution

Find the value of A = πr² for r = 2.5, using π = 3.14. Then find the area of a ring between circles with radii 2.5 and 1.5.

A = 3.14 × 2.5² = 3.14 × 6.25 = 19.625.
Small circle: 3.14 × 1.5² = 3.14 × 2.25 = 7.065.
Ring: 19.625 − 7.065.

Answer: 19.625 square units, and a ring of 12.56 square units

Formulas follow the same order. Find the power first, then multiply, and subtract last. Writing out each step keeps you from errors, such as squaring 3.14 × 2.5 together.

4. Word Problems with Order

Shopping

A learner buys 3 pens at GH₵2.50 each and 2 books at GH₵12.75 each. She has a voucher for GH₵5. How much does she pay?

3 × 2.50 + 2 × 12.75 − 5.
7.50 + 25.50 − 5.

Answer: GH₵28

Writing a single expression for a problem helps you to see the structure. Multiplication is done first, because each item's cost must be found before adding. Many real-life bills work in this way.

New words

expression: A mathematical phrase.
substitution: Putting numbers in place of letters.
formula: A rule written with letters.
voucher: A ticket that reduces a price.

Summary

BODMAS applies to decimals too.
Scientific calculators follow the order, but simple ones may not.
Formulas are worked in the same order.

Class activity: Bill Builder

Write a bill as one expression.
Solve it using BODMAS.
Check with a calculator.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

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

1. 1.5 + 0.5 × 2 =
- A. 2.5
- B. 4
- C. 3
- D. 2
Show answer
Answer: 2.51.5 + 1.
2. (0.5 + 0.3) × 10 =
- A. 8
- B. 3.5
- C. 0.8
- D. 80
Show answer
Answer: 80.8 × 10.
3. Scientific calculators follow BODMAS.
- A. True
- B. False
Show answer
Answer: TrueOrder.
4. 0.2² = ______.
Show answer
Answer: 0.04Power.
5. Simplify 4.8 ÷ 1.2 + 0.5.
Show answer
Answer: 4.54 + 0.5.
6. Find the circle area when r = 3, using π = 3.14.
Show answer
Answer: 28.26 square units3.14 × 9.

Lesson 3

Strand: Number
Sub-strand: Percentages
Indicator: B9.1.3.1.2 – Multiply and/or divide given fractions using the principle of order of operations including the use of the BODMAS or PEMDAS rule and apply the understanding of these to solve problems

Introduction

In real life, percentages come mixed with fractions, decimals and several steps.

1. Order of Operations with Percentages

Treat a percentage as a fraction or a decimal. Then follow BODMAS. The word "of" means multiply.

Order with percentages

Calculate 10% of 200 + 25% of 80.

10% of 200 = 20.
25% of 80 = 20.
20 + 20.

Answer: 40

Brackets

Calculate 20% of (150 + 50).

Brackets: 200.
20% of 200.

Answer: 40

Fractions and percentages

Find ¾ of 40% of 200.

40% of 200 = 80.
¾ of 80.

Answer: 60

2. Successive Percentages

Two changes

A price of GH₵100 rises by 10%, then falls by 10%. Find the final price.

After the rise: 100 × 1.1 = 110.
After the fall: 110 × 0.9.

Answer: GH₵99

The price does not go back to GH₵100, since the 10% fall is of a bigger amount. Two percentage changes must be applied one after the other, and never added together.

Prices written on a board at a market

Figure 3: Market prices Solve a two-step percentage problem.

3. Household Budgets

Sample household budget of GH₵2,400

Item	Percentage	Amount
food	40%	GH₵960
rent	25%	GH₵600
transport	15%	GH₵360
school	10%	GH₵240
savings	10%	GH₵240

Budgets use percentages to show how income is shared. The percentages must total 100%, and the amounts must total the income. Check both.

4. Comparing Options

Better deal

Shop A offers 20% off a GH₵500 radio. Shop B sells it for GH₵420 less GH₵10. Which is cheaper?

Shop A: 80% of 500 = 400.
Shop B: 420 − 10 = 410.

Answer: Shop A is cheaper by GH₵10

Mathematics helps you to make wise decisions. Always convert offers to the same form, then compare them. A larger percentage does not always mean a better deal, and that is why you must look at the amounts as well.

New words

successive: Following one after another.
budget: A plan for money.
income: Money received.
inflation: A general rise in prices.

Summary

Treat percentages as fractions or decimals in BODMAS.
Apply successive percentages one at a time.
Compare offers using amounts.

Class activity: Budget Planner

Plan a monthly budget.
Use percentages that total 100%.
Calculate each amount.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

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

1. 50% of 10% of 200 =
- A. 10
- B. 20
- C. 5
- D. 100
Show answer
Answer: 1050% of 20.
2. GH₵200 rises by 50%, then falls by 50%. The result is…
- A. GH₵150
- B. GH₵200
- C. GH₵100
- D. GH₵250
Show answer
Answer: GH₵150300, then 150.
3. Budget percentages should total 100%.
- A. True
- B. False
Show answer
Answer: TrueWhole.
4. 15% of 100 + 5% of 100 = ______.
Show answer
Answer: 20Add.
5. Calculate 30% of (120 + 80).
Show answer
Answer: 6030% of 200.
6. A GH₵1,000 item is cut by 10%, then by 20%. Find the final price.
Show answer
Answer: GH₵720900 × 0.8.

Homework

Make a monthly budget for a family, using percentages.