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Basic 7 · Mathematics · Fractions

Benchmark Fractions, Decimals and Percentages

In this topic you will recall the decimal and percentage equivalents of benchmark fractions, find equivalent fractions and use them to compare quantities, and compare and order fractions, decimals and percentages.

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

  • Recall the decimal and percentage forms of halves, thirds, fourths, fifths and tenths
  • Find equivalent fractions and use them to compare quantities
  • Compare and order common fractions, decimals and percentages

What you already know

  • Fractions and decimals (Upper Primary)

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
B7.1.3.1.1 – Determine and recall the percentages and decimals of given benchmark fractions (i.e. tenths fifths fourths thirds and halves) and use these to compare quantities

Introduction

"Half price!" "25% off!" "0.5 litres." Shops and adverts use fractions, decimals and percentages all the time. Knowing the common ones by heart makes everyday calculations quick.

1. What Are Benchmark Fractions?

Benchmark fractions: Common fractions used as reference points, such as ½, ⅓, ¼, ⅕ and 1/10.

2. Converting Benchmark Fractions

Benchmark fractions, decimals and percentages
FractionDecimalPercentage
½0.550%
⅓0.333... (0.3 recurring)33⅓%
⅔0.666...66⅔%
¼0.2525%
¾0.7575%
⅕0.220%
⅖0.440%
1/100.110%
3/100.330%

3. How to Convert

To change a fraction to a decimal, divide the numerator by the denominator: ¾ = 3 ÷ 4 = 0.75. To change a decimal to a percentage, multiply by 100: 0.75 × 100 = 75%. To change a fraction to a percentage directly, multiply by 100%: ⅖ × 100% = 40%.

A shop sign showing a percentage discount
Figure 1: Percentage sale Write the fraction and decimal for three discounts on shop signs.

4. Using Benchmarks to Compare Quantities

Compare

Which is more: ¾ of GH₵80 or 70% of GH₵80?

  1. ¾ = 75%.
  2. 75% > 70%.

Answer: ¾ of GH₵80 (GH₵60) is more than 70% of GH₵80 (GH₵56).

5. Percentages in Everyday Life

Percentages appear everywhere in Ghana. Banks quote interest rates such as 15% per year. Examination results are given as percentages. The government charges VAT as a percentage of prices. Mobile networks advertise bonuses such as "20% extra airtime". Knowing that 20% is ⅕ helps you see quickly that on GH₵50 of airtime, the bonus is GH₵10.

New words

benchmark fraction
A common reference fraction such as ½ or ¼.
percentage
A number out of 100, written with %.
equivalent
Having the same value.
recurring decimal
A decimal with a repeating digit, such as 0.333...

Summary

  • Benchmark fractions such as ½, ¼ and ⅕ have well-known decimal and percentage equivalents.
  • Divide numerator by denominator to get a decimal; multiply by 100 to get a percentage.
  • Converting to the same form makes comparing quantities easy.

Class activity: Fraction Strip Match

  1. Use fraction strips to show ½, ¼, ⅕ and 1/10.
  2. Match each strip to its decimal and percentage card.
  3. Use the strips to compare ¾ and ⅗.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

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

  1. 1. ¼ as a percentage is…

    • A. 25%
    • B. 40%
    • C. 4%
    • D. 75%
    Show answer
    Answer: 25%0.25 × 100.

  2. 2. 0.4 as a fraction is…

    • A. ⅖
    • B. ¼
    • C. ⅓
    • D. ⅘
    Show answer
    Answer: ⅖4/10 = 2/5.

  3. 3. ½ = 0.5 = 50%.

    • A. True
    • B. False
    Show answer
    Answer: TrueSame value.

  4. 4. ⅕ written as a decimal is ______.

    Show answer
    Answer: 0.21 ÷ 5.

  5. 5. Convert ¾ to a decimal and a percentage.

    Show answer
    Answer: 0.75 and 75%Divide, then × 100.

  6. 6. Which is greater: ⅓ or 30%? Explain.

    Show answer
    Answer: ⅓ = 33⅓%, which is greater than 30%.Same form.

Lesson 2

Strand
Number
Sub-strand
Fractions
Indicator
B7.1.3.1.1 – Determine and recall the percentages and decimals of given benchmark fractions (i.e. tenths fifths fourths thirds and halves) and use these to compare quantities

Introduction

Half a pizza is the same as two quarters or four eighths. Fractions that look different but have the same value are equivalent fractions.

1. Finding Equivalent Fractions

Multiply or divide the numerator and denominator by the same non-zero number. Multiplying: ⅔ = (2 × 3)/(3 × 3) = 6/9. Dividing (simplifying): 12/16 = (12 ÷ 4)/(16 ÷ 4) = ¾.

Equivalent fractions of ¾
× 2× 3× 4× 5
6/89/1212/1615/20

2. Simplest Form

Simplest form: A fraction whose numerator and denominator have no common factor except 1.

Simplify

Write 36/48 in its simplest form.

  1. HCF of 36 and 48 is 12.
  2. 36 ÷ 12 = 3; 48 ÷ 12 = 4.

Answer: ¾

Coloured fraction strips showing halves, thirds and quarters
Figure 2: Fraction strips Use a fraction wall to find three fractions equal to ½.

3. Checking Equivalence by Cross-Multiplication

Two fractions a/b and c/d are equivalent if a × d = b × c. For 3/5 and 12/20: 3 × 20 = 60 and 5 × 12 = 60, so they are equivalent. For 2/3 and 5/8: 2 × 8 = 16 and 3 × 5 = 15, so they are not equivalent.

4. Using Equivalent Fractions to Compare Quantities

Compare test scores

Ama scored 18/25 in Science and 34/50 in Maths. In which subject did she do better?

  1. Change 18/25 to fiftieths: 36/50.
  2. Compare 36/50 and 34/50.

Answer: She did better in Science (36/50 > 34/50).

5. Equivalent Fractions in Measurement

Equivalent fractions are used in measuring. Half a metre is the same as 50/100 of a metre, or 50 cm. Three-quarters of an hour is 45/60 of an hour, or 45 minutes. A recipe needing ½ cup of sugar can be measured as 2/4 cup with a quarter-cup measure. Recognising these equivalences makes practical tasks easier.

New words

equivalent fractions
Fractions with the same value.
simplest form
A fraction that cannot be simplified further.
numerator
The top number of a fraction.
denominator
The bottom number of a fraction.

Summary

  • Multiply or divide the numerator and denominator by the same number to get equivalent fractions.
  • Simplify using the HCF to reach the simplest form.
  • Use common denominators or cross-multiplication to compare fractions.

Class activity: Equivalence Chains

  1. Start with ⅖ and write five equivalent fractions.
  2. Simplify 24/36, 45/60 and 28/42.
  3. Check two pairs for equivalence by cross-multiplying.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

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

  1. 1. Which is equivalent to ½?

    • A. 4/8
    • B. 3/8
    • C. 2/5
    • D. 5/8
    Show answer
    Answer: 4/8Multiply by 4.

  2. 2. 20/45 in simplest form is…

    • A. 4/9
    • B. 2/9
    • C. 4/5
    • D. 5/9
    Show answer
    Answer: 4/9Divide by 5.

  3. 3. ⅔ and 8/12 are equivalent.

    • A. True
    • B. False
    Show answer
    Answer: TrueMultiply by 4.

  4. 4. The bottom number of a fraction is the ______.

    Show answer
    Answer: denominatorNumber of parts.

  5. 5. Write 42/56 in simplest form.

    Show answer
    Answer: ¾HCF 14.

  6. 6. Kofi ate 3/8 of a cake and Yaw ate 5/12. Who ate more?

    Show answer
    Answer: Yaw; 3/8 = 9/24 and 5/12 = 10/24.Common denominator 24.

Lesson 3

Strand
Number
Sub-strand
Fractions
Indicator
B7.1.3.1.2 – Compare and order fractions (i.e. common percent and decimal fractions up to thousandths) limit to the benchmark fractions

Introduction

Which is the best offer: ⅓ off, 0.3 off or 35% off? To decide, we must write all the values in the same form.

1. Strategy: Convert to One Form

To compare a mix of fractions, decimals and percentages, convert them all to decimals (or all to percentages). Then compare as you would compare decimals.

Order from least to greatest

Order ⅗, 0.55, 62% and ½.

  1. ⅗ = 0.6.
  2. 0.55 stays 0.55.
  3. 62% = 0.62.
  4. ½ = 0.5.

Answer: ½, 0.55, ⅗, 62%

2. Comparing Decimals up to Thousandths

Line up the decimal points and compare digit by digit from the left. For 0.375 and 0.38: tenths are both 3; hundredths are 7 and 8; so 0.375 < 0.38. Adding zeros helps: 0.380 versus 0.375.

Measuring jugs with different amounts of liquid
Figure 3: Measuring jugs Compare the amounts in three jugs written as fractions, decimals and percentages.

3. Using a Number Line

Placing values on a number line from 0 to 1 shows their order clearly. ¼ (0.25) lies a quarter of the way along, ½ (0.5) in the middle and ¾ (0.75) three-quarters of the way. Values close to these benchmarks are easy to place.

4. Real-Life Decision

Best offer

Shop A gives ⅓ off, Shop B gives 0.3 off and Shop C gives 35% off the same GH₵300 phone. Which is the best offer?

  1. ⅓ ≈ 33.3%; 0.3 = 30%; 35% = 35%.
  2. The largest discount is 35%.

Answer: Shop C (35% of GH₵300 = GH₵105 off)

New words

order
To arrange from least to greatest or greatest to least.
convert
To change from one form to another.
discount
An amount taken off the price.
number line
A line showing numbers in order.

Summary

  • Convert fractions, decimals and percentages to the same form before comparing.
  • Compare decimals digit by digit after lining up the decimal points.
  • Number lines and benchmarks help to order values.

Class activity: Line Up the Values

  1. Each learner holds a card with a fraction, decimal or percentage.
  2. Convert your value to a decimal.
  3. Line up from least to greatest and check on a number line.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

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

  1. 1. Which is greatest?

    • A. 0.7
    • B. ½
    • C. 60%
    • D. ⅗
    Show answer
    Answer: 0.70.7 > 0.6 > 0.5.

  2. 2. Order from least: 0.45, ⅖, 42%.

    • A. ⅖, 42%, 0.45
    • B. 0.45, 42%, ⅖
    • C. 42%, ⅖, 0.45
    • D. ⅖, 0.45, 42%
    Show answer
    Answer: ⅖, 42%, 0.450.40, 0.42, 0.45.

  3. 3. 0.375 is less than 0.38.

    • A. True
    • B. False
    Show answer
    Answer: TrueHundredths 7 < 8.

  4. 4. An amount taken off the price is a ______.

    Show answer
    Answer: discountReduction.

  5. 5. Arrange in descending order: ¾, 0.8, 70%.

    Show answer
    Answer: 0.8, ¾, 70%0.8, 0.75, 0.70.

  6. 6. Which is a better score: 17/20 or 84%? Explain.

    Show answer
    Answer: 17/20 = 85%, which is better than 84%.Same form.

Homework

  1. Collect three discount offers from adverts and decide which is the best.
  2. Write ten equivalent fractions for ⅔ and ⅗.
Benchmark Fractions, Decimals and Percentages – Basic 7 Mathematics Notes | sirbus