EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Fractions, decimals and percentages
Fractions, ratio and percentages · Lesson 5 of 5
Last Lesson and Before
Answer each one, then check.
1. Write ¼ as a decimal.
0.25
2. Write 30% as a decimal.
0.3
3. Last lesson: what multiplier decreases by 20%?
0.8
4. Which is bigger: 0.45 or 0.405?
0.45
Learning Objectives
1. Convert between fractions, decimals and percentages.
2. Order a mixture of fractions, decimals and percentages.
3. Know which fractions give terminating decimals and which recur.
4. Convert a recurring decimal to a fraction. (Higher)
Equivalents Worth Knowing
|
Fraction |
Decimal |
Percentage |
|---|---|---|
|
½ |
0.5 |
50% |
|
¼ |
0.25 |
25% |
|
¾ |
0.75 |
75% |
|
⅕ |
0.2 |
20% |
|
⅛ |
0.125 |
12.5% |
|
1/10 |
0.1 |
10% |
|
⅓ |
0.dot3 |
33⅓% |
|
⅔ |
0.dot6 |
66⅔% |
Converting Between the Three
Each conversion is one step.
|
1 Fraction to decimal Divide the top by the bottom: ⅜ = 3 ÷ 8 = 0.375. |
2 Decimal to percentage Multiply by 100: 0.375 = 37.5%. |
3 Percentage to fraction Write over 100 and simplify: 35% = 35/100 = 7/20. |
4 Decimal to fraction Use place value: 0.36 = 36/100 = 9/25. |
Terminating and Recurring Decimals
Some fractions end; others repeat for ever.
▸ Terminating. The decimal stops: ⅜ = 0.375.
▸ Recurring. A digit or group of digits repeats for ever: ⅓ = 0.333…= 0.dot3 and 1/7 = 0.dot14285dot7.
▸ Dot notation. A dot over one digit repeats it; dots over the first and last digit repeat the whole group.
▸ The rule. A fraction in its simplest form terminates only if the prime factors of its denominator are just 2s and 5s: 7/40 terminates because 40 = 2³ × 5.
Ordering a Mixture
|
Write in order, smallest first: ⅜, 0.38, 37%, 0.dot3 |
1. Change everything to decimals
⅜ = 0.375, 0.38, 37% = 0.37, 0.dot3 = 0.333…
2. Compare the decimals
0.333…< 0.37 < 0.375 < 0.38
3. Write the original forms in order
0.dot3, 37%, ⅜, 0.38
Answer: 0.dot3, 37%, ⅜, 0.38
PART TWO · HIGHER
Recurring Decimals to Fractions
An exact fraction for a decimal that never ends.
A Recurring Decimal to a Fraction
|
Show that 0.dot4dot5 = 5/11 |
1. Let x be the decimal
x = 0.454545…
2. Two digits repeat, so multiply by 100
100x = 45.454545…
3. Subtract to remove the repeating part
100x − x = 45, so 99x = 45
4. Divide and simplify
x = 45/99 = 5/11
Answer: 0.dot4dot5 = 45/99 = 5/11
When the Repeat Starts Later
|
Write 0.2dot3 as a fraction in its simplest form. |
1. Let x be the decimal
x = 0.2333…
2. Multiply by 10 and by 100, so both have the same repeating tail
10x = 2.333… and 100x = 23.333…
3. Subtract
100x − 10x = 21, so 90x = 21
4. Divide and simplify
x = 21/90 = 7/30
Answer: 7/30
Key Terms
|
Terminating decimal A decimal that stops, e.g. 0.375. |
Recurring decimal A decimal in which a digit or group of digits repeats for ever. |
|
Dot notation Dots above digits to show which digits recur. |
Equivalent Equal in value, though written differently. |
Your Task: Terminate or Recur?
10 minutes
|
Without dividing, predict whether each fraction terminates or recurs, then check with a calculator: 3/20, 5/12, 7/16, 2/15, 9/25, 4/9. Higher: write 0.dot7 and 0.1dot6 as fractions. 1. Write each denominator in prime factors. 2. Predict, then check. 3. Higher: use the algebra method. |
A good answer shows: Terminate: 3/20 (0.15), 7/16 (0.4375), 9/25 (0.36). Recur: 5/12 (0.41dot6), 2/15 (0.1dot3), 4/9 (0.dot4). Higher: 0.dot7 = 7/9, 0.1dot6 = 15/90 = 1/6.
Can I...?
☐ Convert fractions to decimals and percentages.
☐ Convert percentages to fractions.
☐ Order fractions, decimals and percentages.
☐ Use dot notation for recurring decimals.
☐ Say whether a fraction terminates or recurs.
☐ Convert a recurring decimal to a fraction. (Higher)
Summary
✓ Fraction to decimal: divide. Decimal to percentage: multiply by 100.
✓ To order a mixture, change everything to decimals.
✓ A fraction terminates only if its denominator's prime factors are 2 and 5.
✓ (Higher) Recurring to fraction: multiply to line up the repeats, subtract, divide.
|
EXAM FOCUS Show that 0.dot4dot5 = 5/11 (3 marks) For "show that" with recurring decimals, write out x = … and 100x = … with at least two repeats of the digits, and show the subtraction. The algebra is the method mark. |