Basic 7 · Mathematics · Fractions
Multiplying Fractions and Finding a Fraction of a Quantity
In this topic you will explain how to multiply a fraction by a whole number and by another fraction, multiply percentages and decimal fractions, and find a fraction of a given quantity of money or objects.
By the end of these notes you should be able to:
- Multiply a fraction by a whole number
- Multiply a fraction by a fraction, including mixed numbers and decimals
- Find a fraction or percentage of a given quantity
What you already know
- Adding and subtracting fractions
- Benchmark 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
- B7.1.3.3.1 – Explain the process of multiplying a fraction (i.e. common percent and decimal fractions up to thousandths) by a whole number and by a fraction
Introduction
If each learner drinks ¾ of a litre of water a day, how much do 8 learners drink? This is multiplying a fraction by a whole number.
1. Repeated Addition
Multiplying a fraction by a whole number is repeated addition: 4 × ¾ = ¾ + ¾ + ¾ + ¾ = 12/4 = 3.
2. The Rule
- Write the whole number as a fraction over 1.
- Multiply the numerators.
- Multiply the denominators.
- Simplify and change to a mixed number if needed.
Multiply
Calculate 8 × ¾.
- 8/1 × ¾ = 24/4.
Answer: 6 litres
3. Cancelling First
You can simplify before multiplying by dividing a numerator and a denominator by a common factor. For 8 × ¾, divide 8 and 4 by 4: 2/1 × 3/1 = 6. Cancelling first keeps the numbers small.
4. Multiplying a Mixed Number by a Whole Number
Mixed number
Calculate 5 × 2⅓.
- Change to an improper fraction: 2⅓ = 7/3.
- 5/1 × 7/3 = 35/3.
Answer: 11⅔
5. Multiplying Decimals and Percentages by Whole Numbers
Decimal fractions and percentages follow the same idea. 6 × 0.25 = 1.5, because 0.25 is a quarter. 4 × 15% = 60%. Thinking of benchmarks helps: 3 × 0.5 is 3 halves, which is 1.5.
6. Real-Life Example
A school gives each of its 240 learners ⅜ of a loaf of bread at break. How many loaves are needed? 240 × ⅜ = (240 ÷ 8) × 3 = 30 × 3 = 90 loaves. Cancelling 240 with 8 first makes the calculation easy to do mentally.
New words
- repeated addition
- Adding the same number several times.
- cancel
- To divide a numerator and denominator by a common factor.
- improper fraction
- A fraction greater than one, such as 7/3.
- product
- The result of multiplication.
Summary
- Multiplying a fraction by a whole number is repeated addition.
- Write the whole number over 1, multiply numerators and denominators, then simplify.
- Cancel common factors first to keep numbers small.
Class activity: Water Survey
- Find how many cups of water each group member drinks a day, as a fraction of a litre.
- Multiply to find the total for the class for one week.
- Present your calculation.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 3 × ⅖ =
- A. 6/5
- B. 3/5
- C. 5/6
- D. 6/15
Show answer
Answer: 6/53 × 2 = 6.2. 12 × ⅚ =
- A. 10
- B. 12
- C. 6
- D. 72
Show answer
Answer: 10Cancel 12 and 6.3. 4 × ½ = 2.
- A. True
- B. False
Show answer
Answer: TrueFour halves.4. Dividing a numerator and denominator by a common factor is called ______.
Show answer
Answer: cancellingSimplifying first.5. Calculate 6 × 1¾.
Show answer
Answer: 10½6 × 7/4 = 42/4.6. A builder uses ⅗ of a 50 kg bag of cement each day. How much cement does he use in 5 days?
Show answer
Answer: 150 kg5 × ⅗ = 3 bags; 3 × 50 kg = 150 kg.
Lesson 2
- Strand
- Number
- Sub-strand
- Fractions
- Indicator
- B7.1.3.3.1 – Explain the process of multiplying a fraction (i.e. common percent and decimal fractions up to thousandths) by a whole number and by a fraction
Introduction
Half of a half is a quarter. In mathematics, "of" means multiply, so ½ of ½ = ½ × ½ = ¼.
1. The Rule
To multiply two fractions, multiply the numerators together and the denominators together, then simplify: ⅔ × ¾ = 6/12 = ½. Cancelling first: ⅔ × ¾, cancel the 3s and the 2 with the 4 to get 1/1 × 1/2 = ½.
2. Modelling with an Area Diagram
Shade ¾ of a rectangle with vertical lines. Then shade ⅔ of the whole rectangle with horizontal lines. The part shaded both ways is ⅔ × ¾. If the rectangle is divided into 12 equal parts, 6 are shaded twice, so ⅔ × ¾ = 6/12 = ½.
3. Multiplying Mixed Numbers
Mixed numbers
Calculate 1½ × 2⅔.
- Change to improper fractions: 3/2 × 8/3.
- Cancel: 3 with 3, and 2 with 8 (to 4).
- 1/1 × 4/1 = 4.
Answer: 4
4. Multiplying Decimals and Percentages
Decimal and percentage
Find 0.4 × 0.25 and 20% of 0.5.
- 0.4 × 0.25 = 0.1 (since 4 × 25 = 100, with 3 decimal places → 0.100).
- 20% of 0.5 = 0.2 × 0.5 = 0.1.
Answer: Both equal 0.1
5. Is the Answer Bigger or Smaller?
When you multiply a number by a fraction less than 1, the answer is smaller than the number. ⅔ × ¾ = ½, which is smaller than both. This surprises some learners who expect multiplication always to make numbers bigger.
New words
- of
- In fractions, "of" means multiply.
- area model
- A diagram using a shaded rectangle to show multiplication.
- proper fraction
- A fraction less than 1.
- cancel
- To simplify before multiplying.
Summary
- Multiply numerators together and denominators together, then simplify.
- Change mixed numbers to improper fractions before multiplying.
- Multiplying by a proper fraction makes a number smaller.
Class activity: Area Model Practice
- Use graph paper to model ½ × ¾ and ⅔ × ⅗.
- Check each result by calculation.
- Explain why the answer is smaller than both fractions.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. ½ × ⅓ =
- A. ⅙
- B. ⅔
- C. ⅖
- D. ⅚
Show answer
Answer: ⅙1 × 1 over 2 × 3.2. ⅘ × ⅝ =
- A. ½
- B. 20/40
- C. 32/20
- D. ⅞
Show answer
Answer: ½20/40 = ½.3. "Of" means multiply in fractions.
- A. True
- B. False
Show answer
Answer: True½ of 6 = ½ × 6.4. A fraction less than 1 is a ______ fraction.
Show answer
Answer: propere.g. ¾.5. Calculate 2¼ × 1⅓.
Show answer
Answer: 39/4 × 4/3 = 3.6. Explain why ⅔ × ¾ is smaller than ¾.
Show answer
Answer: Multiplying by ⅔ takes only two-thirds of ¾, which is less than the whole of ¾.Fraction of a fraction.
Lesson 3
- Strand
- Number
- Sub-strand
- Fractions
- Indicator
- B7.1.3.3.2 – Find a fraction of given quantity (i.e. money or given quantity of objects)
Introduction
"Three-quarters of the class passed." "40% of the money was saved." To find a fraction of a quantity, we multiply the quantity by the fraction.
1. Finding a Fraction of a Number
Divide the quantity by the denominator, then multiply by the numerator. To find ⅗ of 45: 45 ÷ 5 = 9, then 9 × 3 = 27.
2. Fractions of Money
Money
Find ¾ of GH₵240.
- 240 ÷ 4 = 60.
- 60 × 3 = 180.
Answer: GH₵180
3. Percentages of Quantities
Percentage
A school has 640 learners. 35% are in JHS 1. How many learners are in JHS 1?
- 35% = 35/100.
- 640 × 35/100 = 22,400/100.
Answer: 224 learners
4. Finding the Whole from a Fraction
Find the whole
⅖ of a sum of money is GH₵60. What is the whole sum?
- ⅖ of the whole is 60, so ⅕ is 60 ÷ 2 = 30.
- The whole is 5 × 30.
Answer: GH₵150
5. Problems in Context
Sharing harvest
A farmer harvests 1,200 tubers of yam. He sells ⅝ of them and gives ⅙ of the rest to his family. How many tubers does the family receive?
- Sold: ⅝ × 1,200 = 750.
- Rest: 1,200 − 750 = 450.
- Family: ⅙ × 450 = 75.
Answer: 75 tubers
6. Fractions and Percentages Together
Some problems mix forms. If 60% of 250 learners are girls and ⅓ of the girls are in the choir, then girls = 0.6 × 250 = 150, and choir girls = ⅓ × 150 = 50. Work one step at a time and write each result before moving on.
New words
- quantity
- An amount or number of something.
- percentage of
- A number of hundredths of a quantity.
- whole
- The complete amount.
- unitary method
- Finding one part first, then the whole or other parts.
Summary
- To find a fraction of a quantity, divide by the denominator and multiply by the numerator.
- To find a percentage of a quantity, multiply by the percentage over 100.
- To find the whole, find one part first using the unitary method.
Class activity: Fraction Shopping
- Use cedi play notes worth GH₵200.
- Find ¼, ⅖, 10% and 35% of the money.
- Model each amount with notes and check by calculation.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. ½ of 84 is…
- A. 42
- B. 168
- C. 21
- D. 40
Show answer
Answer: 4284 ÷ 2.2. 25% of GH₵360 is…
- A. GH₵90
- B. GH₵72
- C. GH₵120
- D. GH₵36
Show answer
Answer: GH₵90¼ of 360.3. To find ⅓ of a number, divide it by 3.
- A. True
- B. False
Show answer
Answer: TrueOne third.4. ⅘ of 50 is ______.
Show answer
Answer: 4050 ÷ 5 × 4.5. Find 15% of 800 kg.
Show answer
Answer: 120 kg800 × 15/100.6. ¾ of a number is 36. What is the number?
Show answer
Answer: 48¼ is 12, so whole is 48.
Homework
- Find ¼, ½ and ¾ of your family's weekly shopping budget (make up a sensible amount).
- Solve three percentage problems from adverts or newspapers.