Basic 9 · Mathematics · Fractions
Reviewing Fractions, Decimals and Percentages
In this topic you will review fractions and solve problems involving the basic operations on fractions, decimals and percentages.
By the end of these notes you should be able to:
- Solve problems with operations on fractions
- Solve problems with operations on decimals
- Solve problems involving percentages
What you already know
- Fractions, decimals and percentages (Basic 8)
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
- B9.1.3.1.1 – Review fractions and solve problems involving basic operations on fractions
Introduction
Recipes, time and land sharing all use fractions. Being confident with them is vital.
1. Review of the Rules
| Operation | Rule | Example |
|---|---|---|
| adding | use a common denominator | ⅔ + ¾ = 8/12 + 9/12 = 17/12 |
| subtracting | use a common denominator | ⅚ − ¼ = 10/12 − 3/12 = 7/12 |
| multiplying | multiply tops and bottoms | ⅗ × 10/9 = ⅔ |
| dividing | multiply by the reciprocal | ⅘ ÷ ⅔ = ⅘ × 3/2 = 6/5 |
2. Mixed Numbers
Mixed numbers
Calculate 2½ + 1¾.
- Add whole numbers: 2 + 1 = 3.
- Add fractions: ½ + ¾ = 5/4 = 1¼.
- 3 + 1¼.
Answer: 4¼
Multiplying mixed numbers
Calculate 1½ × 2⅔.
- Change to improper fractions: 3/2 × 8/3.
- Cancel: 24/6 = 4.
Answer: 4
Always change mixed numbers to improper fractions before you multiply or divide. Multiplying the whole numbers and the fractions separately gives a wrong answer.
3. Word Problems
Sharing land
A farmer plants ⅖ of his land with maize and ¼ with cassava. What fraction is left?
- Used: ⅖ + ¼ = 8/20 + 5/20 = 13/20.
- Left: 1 − 13/20.
Answer: 7/20
Fraction of an amount
A learner spends ⅓ of GH₵60 on food, and ¼ of what remains on books. How much is left?
- Food: 20, so 40 remains.
- Books: ¼ of 40 = 10.
- Left: 40 − 10.
Answer: GH₵30
4. Common Mistakes
Do not add the denominators: ½ + ⅓ is not 2/5. Find a common denominator: 3/6 + 2/6 = 5/6. Always simplify your final answer. Note, too, that multiplying by a fraction less than 1 makes a number smaller, while dividing by it makes the number larger. The word "of" usually means multiply. Be careful with "of what remains", which refers to a new whole.
New words
- common denominator
- A shared bottom number.
- reciprocal
- A fraction turned upside down.
- mixed number
- A whole number and a fraction.
- improper fraction
- A fraction with a top larger than its bottom.
Summary
- Add and subtract with common denominators.
- Multiply across and divide by the reciprocal.
- Change mixed numbers before multiplying.
Class activity: Fraction Relay
- Solve a chain of fraction problems.
- Pass your answer on.
- Check the final result.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. ½ + ¼ =
- A. ¾
- B. 2/6
- C. ⅕
- D. 1
Show answer
Answer: ¾Common denominator.2. ⅔ ÷ ⅘ =
- A. ⅚
- B. 8/15
- C. 6/5
- D. 2/5
Show answer
Answer: ⅚Reciprocal.3. The word "of" usually means multiply.
- A. True
- B. False
Show answer
Answer: TrueRule.4. The reciprocal of ¾ is ______.
Show answer
Answer: 4/3Flip.5. Calculate 3¼ − 1½.
Show answer
Answer: 1¾13/4 − 6/4.6. Ama spends ¼ of GH₵80, then ½ of the rest. How much is left?
Show answer
Answer: GH₵3060, then 30.
Lesson 2
- Strand
- Number
- Sub-strand
- Decimals
- Indicator
- B9.1.3.1.1 – Review fractions and solve problems involving basic operations on fractions
Introduction
Prices, fuel, measures and body temperatures are all recorded as decimals.
1. Adding and Subtracting
Line up the decimal points, fill empty places with zeros and calculate as for whole numbers. For example, 12.5 + 3.47 = 15.97, and 20 − 7.65 = 12.35. Keep the point in the same position in your answer.
2. Multiplying
Multiplying
Calculate 0.4 × 0.06.
- Ignore the points: 4 × 6 = 24.
- Count decimal places: 1 + 2 = 3.
- Place the point: 0.024.
Answer: 0.024
Multiplying by a decimal less than 1 makes a number smaller. Check with an estimate: 2.3 × 4.8 ≈ 2 × 5 = 10, and the exact answer is 11.04.
3. Dividing
Dividing
Calculate 1.44 ÷ 1.2.
- Multiply both numbers by 10 to make the divisor whole: 14.4 ÷ 12.
- Divide.
Answer: 1.2
Long division
Calculate 7.2 ÷ 0.08.
- Multiply both by 100: 720 ÷ 8.
Answer: 90
4. Word Problems
Cloth
A tailor buys 4.5 m of cloth at GH₵28.50 per metre. Find the total cost.
- 28.50 × 4.5.
- 28.50 × 4 = 114.
- 28.50 × 0.5 = 14.25.
- 114 + 14.25.
Answer: GH₵128.25
Unit price
2.5 kg of beef costs GH₵112.50. Find the price per kg.
- 112.50 ÷ 2.5 = 1,125 ÷ 25.
Answer: GH₵45
5. Decimals and Fractions Together
Sometimes a calculation has both fractions and decimals. Choose one form. ½ + 0.75 can be written 0.5 + 0.75 = 1.25. Where a fraction gives a recurring decimal, such as ⅓, it is better to stay in fractions: ⅓ × 0.6 = ⅓ × 3/5 = 1/5. Using exact forms prevents rounding errors in your work.
New words
- decimal point
- The dot separating whole numbers from parts.
- divisor
- The number you divide by.
- unit price
- The price of one unit.
- estimate
- A rough calculation.
Summary
- Line up points when adding and subtracting.
- Count decimal places when multiplying.
- Make the divisor whole when dividing.
Class activity: Shopping Decimals
- Write a list with decimal prices.
- Calculate the total and the change.
- Check by estimating.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 0.3 × 0.2 =
- A. 0.06
- B. 0.6
- C. 6
- D. 0.5
Show answer
Answer: 0.06Two places.2. 3.6 ÷ 0.4 =
- A. 9
- B. 0.9
- C. 90
- D. 1.44
Show answer
Answer: 936 ÷ 4.3. Decimal points are lined up when adding.
- A. True
- B. False
Show answer
Answer: TrueRule.4. 5 − 1.75 = ______.
Show answer
Answer: 3.25Subtract.5. Find the cost of 3.5 m of cloth at GH₵20 per metre.
Show answer
Answer: GH₵703.5 × 20.6. A 7.5 m rope is cut into 0.25 m pieces. How many pieces?
Show answer
Answer: 307.5 ÷ 0.25.
Lesson 3
- Strand
- Number
- Sub-strand
- Percentages
- Indicator
- B9.1.3.1.1 – Review fractions and solve problems involving basic operations on fractions
Introduction
Discounts, interest, taxes and exam marks all use percentages. They are among the most useful ideas in mathematics.
1. Review
| Task | Method | Example |
|---|---|---|
| percentage of an amount | multiply | 15% of 240 = 36 |
| one amount as a percentage of another | divide and × 100 | 30 out of 120 = 25% |
| increase by a percentage | multiply by 1 + rate | 200 + 15% = 230 |
| decrease by a percentage | multiply by 1 − rate | 200 − 15% = 170 |
| reverse percentage | divide by the multiplier | 230 ÷ 1.15 = 200 |
2. Profit and Loss
Profit percentage
A trader buys a bag for GH₵80 and sells it for GH₵100. Find the profit percentage.
- Profit: 100 − 80 = 20.
- Percentage: 20/80 × 100.
Answer: 25%
Always use the cost price as the base, unless told otherwise. Using 100 as the base here would give 20%, which is wrong.
3. Simple Interest
Simple interest
Find the simple interest on GH₵2,000 at 15% per year for 3 years.
- I = PRT/100.
- I = 2,000 × 15 × 3 / 100.
Answer: GH₵900
4. Taxes and Levies
VAT
A phone costs GH₵1,000 before a 15% tax. Find the price paid.
- Tax: 15% of 1,000 = 150.
- Price: 1,000 + 150.
Answer: GH₵1,150
Tax rates change over time, so always use the rate given in the question.
5. Reverse Percentages
Original price
After a 20% discount, a shirt costs GH₵80. Find the original price.
- The sale price is 80% of the original.
- 0.8 × original = 80.
- Original = 80 ÷ 0.8.
Answer: GH₵100
A common mistake is to add 20% of 80 to the sale price, giving GH₵96. This is wrong, since the 20% was taken from the original price, and not from the sale price. Always ask which amount the percentage refers to.
New words
- percentage
- A number out of 100.
- profit
- Selling price less cost price.
- simple interest
- Interest on the original sum only.
- discount
- A reduction in price.
Summary
- Base percentages on the correct amount.
- Use I = PRT/100 for simple interest.
- Reverse percentages divide by the multiplier.
Class activity: Market Percentages
- Make price cards.
- Apply discounts and profits.
- Find the original prices.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 25% of 80 is…
- A. 20
- B. 25
- C. 40
- D. 60
Show answer
Answer: 20¼ of 80.2. 40 out of 160 as a percentage is…
- A. 25%
- B. 40%
- C. 20%
- D. 16%
Show answer
Answer: 25%40/160.3. Profit percentage is usually based on the cost price.
- A. True
- B. False
Show answer
Answer: TrueBase.4. Reducing GH₵50 by 10% gives GH₵______.
Show answer
Answer: 4550 − 5.5. Find the simple interest on GH₵500 at 10% for 2 years.
Show answer
Answer: GH₵100PRT/100.6. After a 25% rise, a price is GH₵250. Find the old price.
Show answer
Answer: GH₵200250 ÷ 1.25.
Homework
- Solve five problems on discount, profit and interest.