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KS3 Maths · Year 9Chapter 3 · Recurring decimals, percentages and growth
Year 9 · Chapter 3

Recurring decimals, percentages and growth

By the end of this chapter you will be able to

  • Write a recurring decimal as a fraction, using the subtraction method
  • Write a fraction with a recurring decimal form, using long division and dot notation
  • Use a multiplier to work out a repeated percentage increase or decrease
  • Find the original amount before one or more percentage changes
  • Find the overall percentage change over several periods, and explain why it is not the sum of the percentages

You should already know

  • Writing a fraction as a decimal and back, and the place value of decimals (Year 8, Chapter 3)
  • Finding a percentage of an amount, and percentage increase and decrease using a multiplier (Year 8, Chapters 3 and 5)
  • Rates and simple interest (Year 8, Chapter 4)

Where this leads

  • Compound measures and rates (Year 9, Chapter 10)
  • Exponential graphs and growth models (Year 9, Chapter 6)
  • Sequences of repeated multiplication (Year 9, Chapter 7)

National Curriculum for England, Key Stage 3 mathematics, Year 9. Content: Convert a recurring decimal such as 0.3̇ or 0.1̇2̇ to a fraction, and convert simple fractions to recurring decimals, Calculate repeated percentage change, such as compound interest and depreciation, using multipliers, Find the original amount before a percentage change (reverse percentages), Find the overall percentage change over several periods, and solve simple growth and decay problems in context.

Exercise P  Do you remember?

Non-calculator
1
FWrite 0.25 as a fraction in its simplest form.
2
FWork out 3/8 as a decimal.
3
FA coat costs £80. It goes up by 10%. What is the new price?
4
FWhat is 40% of £250?
5
FWrite 0.5 as a percentage.
3.1

Recurring decimals to fractions

Year 9
Convert a recurring decimal to a fraction
Video lesson: Recurring decimals to fractionsComing soon
A baker divides 1 by 3 on a calculator and sees 0.3333... The 3s go on for ever. The baker needs an exact amount, so she writes the recurring decimal as a fraction, which is exact.

A recurring decimal has a digit or block of digits that repeats for ever. We mark it with a dot over the first and last digit of the repeating block. So 0.1̇2̇ means 0.121212…, and 0.3̇ means 0.333….

To change a recurring decimal to a fraction, let x be the number. Multiply x by a power of 10 so the repeating parts line up, then subtract the equations. The repeating part cancels, leaving an ordinary equation.

Key ideas

  • The dots show the repeating block. Dot the first and last digit only.
  • For a pure recurring decimal, multiply by 10 for each repeating digit, then subtract the original.
  • For a mixed recurring decimal, multiply first by 10 for the non-repeating digits, and then by 10 again for each repeating digit.
  • Simplify the fraction at the end.
Worked example 3.1.1
Write 0.3̇ as a fraction.

Method. Use the subtraction method: multiply so the repeating parts line up, then subtract.

  1. Let x be the number. Writing the number with a letter lets us use an equation.
    x = 0.3̇
  2. Multiply by 101. Multiplying by 101 moves one full repeat past the decimal point.
    101x = 3.333…
  3. Subtract the equation from the one before. The repeating part cancels, leaving a whole number on the right.
    (101 − 100)x = 3 − 0
  4. Divide and simplify. Divide both sides by the number in brackets, then cancel any common factor.
    x = 3/9 = 1/3
  5. Write the answer.
    1/3

Check. 1/3 = 0.3333..., which matches 0.3̇.

Watch out. Use one more multiplication than the number of repeating digits. The repeating part is the one with the dots.

Worked example 3.1.2
Write 0.1̇2̇ as a fraction.

Method. Use the subtraction method: multiply so the repeating parts line up, then subtract.

  1. Let x be the number. Writing the number with a letter lets us use an equation.
    x = 0.1̇2̇
  2. Multiply by 102. Multiplying by 102 moves one full repeat past the decimal point.
    102x = 12.121212…
  3. Subtract the equation from the one before. The repeating part cancels, leaving a whole number on the right.
    (102 − 100)x = 12 − 0
  4. Divide and simplify. Divide both sides by the number in brackets, then cancel any common factor.
    x = 12/99 = 4/33
  5. Write the answer.
    4/33

Check. 4/33 = 0.1212..., which matches 0.1̇2̇.

Watch out. Use one more multiplication than the number of repeating digits. The repeating part is the one with the dots.

Worked example 3.1.3
Write 0.23̇ as a fraction.

Method. Use the subtraction method: multiply so the repeating parts line up, then subtract.

  1. Let x be the number. Writing the number with a letter lets us use an equation.
    x = 0.23̇
  2. Multiply by 10 to move the non-repeating digits to the front. Multiplying by 101 moves the non-repeating part before the decimal point.
    101x = 2.333…
  3. Multiply by 102. Multiplying by 102 moves one full repeat past the decimal point.
    102x = 23.333…
  4. Subtract the equation from the one before. The repeating part cancels, leaving a whole number on the right.
    (102 − 101)x = 23 − 2
  5. Divide and simplify. Divide both sides by the number in brackets, then cancel any common factor.
    x = 21/90 = 7/30
  6. Write the answer.
    7/30

Check. 7/30 = 0.2333..., which matches 0.23̇.

Watch out. Use one more multiplication than the number of repeating digits. The repeating part is the one with the dots.

Watch out

Watch out: 0.1̇2̇ is 0.1212…, not 0.12. The repeating block 12 has two digits, so multiply by 100, not 10.

Exam tip

Exam tip. Count the digits in the repeating block. That tells you the power of 10 to use.

Exercise 3A  Recurring decimals to fractions

Non-calculator
1
FWrite 0.3̇ as a fraction in its simplest form.
2
FWrite 0.6̇ as a fraction in its simplest form.
3
FWrite 0.1̇2̇ as a fraction in its simplest form.
4
FWrite 0.4̇5̇ as a fraction in its simplest form.
5
FWrite 0.7̇2̇ as a fraction in its simplest form.
6
FWrite 0.23̇ as a fraction in its simplest form.
7
FWrite 0.16̇ as a fraction in its simplest form.
8
FWrite 0.83̇ as a fraction in its simplest form.
9
FWrite 0.45̇ as a fraction in its simplest form.
10
RWrite 0.9̇ as a fraction. Explain why this means 0.9̇ is the same as 1.
3.2

Fractions as recurring decimals

Year 9
Convert simple fractions to recurring decimals, and use dot notation
Video lesson: Fractions as recurring decimalsComing soon
A shop shares 2 cakes between 11 people. Dividing 2 by 11 with long division gives a decimal that never stops. The pattern of digits repeats, and the dot notation shows exactly which digits repeat.

Any fraction with a denominator that has factors other than 2 and 5 gives a recurring decimal. Use long division. Keep going until a remainder repeats, then the digits from that point form the repeating block.

Key ideas

  • Long division: keep dividing until a remainder repeats.
  • The repeating block is the digits from the first repeated remainder onwards.
  • Dot the first and last digit of the repeating block.
  • Denominators that only have factors 2 and 5 give terminating decimals.
Worked example 3.2.1
Write 2/11 as a recurring decimal.

Method. Long division gives the decimal, and a repeating block shows where to put the dots.

  1. Divide the numerator by the denominator. Use long division. Keep going until a remainder repeats.
    2 ÷ 11 = 0.1818...
  2. Find the repeating block. The digits 18 repeat over and over, so this block is the part to dot.
    repeating block = 18
  3. Write the answer with dots. Put a dot over the first and last digit of the repeating block.
    0.1̇8̇

Check. The block 18 repeats forever, so the dots go over the first and last digit of 18.

Watch out. A dot over only one digit means that digit repeats on its own: 0.3̇ = 0.333...

Worked example 3.2.2
Write 5/6 as a recurring decimal.

Method. Long division gives the decimal, and a repeating block shows where to put the dots.

  1. Divide the numerator by the denominator. Use long division. Keep going until a remainder repeats.
    5 ÷ 6 = 0.833...
  2. Find the repeating block. The digits 3 repeat over and over, so this block is the part to dot.
    repeating block = 3
  3. Write the answer with dots. Put a dot over the first and last digit of the repeating block.
    0.83̇

Check. The block 3 repeats forever, so the dots go over the first and last digit of 3.

Watch out. A dot over only one digit means that digit repeats on its own: 0.3̇ = 0.333...

Worked example 3.2.3
Write 5/12 as a recurring decimal.

Method. Long division gives the decimal, and a repeating block shows where to put the dots.

  1. Divide the numerator by the denominator. Use long division. Keep going until a remainder repeats.
    5 ÷ 12 = 0.4166...
  2. Find the repeating block. The digits 6 repeat over and over, so this block is the part to dot.
    repeating block = 6
  3. Write the answer with dots. Put a dot over the first and last digit of the repeating block.
    0.416̇

Check. The block 6 repeats forever, so the dots go over the first and last digit of 6.

Watch out. A dot over only one digit means that digit repeats on its own: 0.3̇ = 0.333...

Watch out

Watch out: 5/12 = 0.41666…, not 0.416. Only the 6 repeats, so the dot goes over the 6 alone.

Exam tip

Exam tip. Write enough digits to see the repeat clearly, so you do not stop too early.

Exercise 3B  Fractions as recurring decimals

Non-calculator
1
FWrite 2/11 as a recurring decimal.
2
FWrite 5/6 as a recurring decimal.
3
FWrite 1/7 as a recurring decimal.
4
FWrite 3/7 as a recurring decimal.
5
FWrite 4/9 as a recurring decimal.
6
FWrite 5/12 as a recurring decimal.
7
FWrite 7/22 as a recurring decimal.
8
FWrite 7/15 as a recurring decimal.
9
RExplain why 3/11 is a recurring decimal, not a terminating one. Write it with dots.
3.3

Repeated percentage change

Year 9
Calculate repeated percentage change using multipliers
Video lesson: Repeated percentage changeComing soon
Ruth buys a car for £12,000. It loses 15% of its value each year. To find its value after 3 years, multiply by the multiplier 0.85 three times. Each year the change builds on the value at the start of that year.

A multiplier is a number that does the whole percentage change in one step. An increase of 4% has multiplier 1.04. A decrease of 15% has multiplier 0.85. To repeat a change for n periods, multiply by the multiplier n times, or use a power: amount × multipliern.

Growth is an increase in a repeated period, such as compound interest. Decay is a decrease, such as depreciation.

Key ideas

  • Increase by r%: multiplier 1 + r/100. Decrease by r%: multiplier 1 − r/100.
  • Repeated change: amount × multipliern, where n is the number of periods.
  • Each period builds on the amount at the start of that period.
  • Do not add the percentage each period. The percentage is of the new amount.
Worked example 3.3.1
A car worth £12,000 loses 15% of its value each year. Find its value after 3 years.

Method. Multiply by the multiplier once for each period, which is 3 times.

  1. Write the multiplier. A 15% change multiplies the amount by 0.85.
    multiplier = 0.85
  2. Period 1. Multiply the starting value by 0.85.
    £12,000 × 0.85 = £10,200
  3. Period 2. Multiply the value from period 1 by 0.85 again.
    £10,200 × 0.85 = £8,670
  4. Period 3. Multiply the value from period 2 by 0.85 again.
    £8,670 × 0.85 = £7369.5
  5. Round the answer. Give the value to 2 decimal places.
    £7,369.50

Check. Check: 3 periods of multiplying by 0.85 gives £7369.5 before rounding.

Watch out. Do not add the percentage each time. Multiply by the multiplier each time, so the change builds on the new amount.

Worked example 3.3.2
£2500 is invested at 4% compound interest per year. Find the value after 2 years.

Method. Multiply by the multiplier once for each period, which is 2 times.

  1. Write the multiplier. A 4% change multiplies the amount by 1.04.
    multiplier = 1.04
  2. Period 1. Multiply the starting value by 1.04.
    £2,500 × 1.04 = £2,600
  3. Period 2. Multiply the value from period 1 by 1.04 again.
    £2,600 × 1.04 = £2,704
  4. Round the answer. Give the value to 2 decimal places.
    £2,704.00

Check. Check: 2 periods of multiplying by 1.04 gives £2,704 before rounding.

Watch out. Do not add the percentage each time. Multiply by the multiplier each time, so the change builds on the new amount.

Worked example 3.3.3
A population of 800 falls by 10% each year. Find the population after 2 years.

Method. Multiply by the multiplier once for each period, which is 2 times.

  1. Write the multiplier. A 10% change multiplies the amount by 0.9.
    multiplier = 0.9
  2. Period 1. Multiply the starting population by 0.9.
    800 × 0.9 = 720
  3. Period 2. Multiply the population from period 1 by 0.9 again.
    720 × 0.9 = 648
  4. Round the answer. Give the population to the nearest whole number.
    648

Check. Check: 2 periods of multiplying by 0.9 gives 648 before rounding.

Watch out. Do not add the percentage each time. Multiply by the multiplier each time, so the change builds on the new amount.

Watch out

Watch out: a 10% increase followed by a 10% decrease does not give the original amount. The second change is 10% of a bigger number.

Exam tip

Exam tip. Write the multiplier before you start, then work out each period on its own line.

Exercise 3C  Repeated percentage change (calculator)

Calculator allowed
1
FA car worth £12,000 loses 15% of its value each year. Find its value after 3 years.
2
F£2500 is invested at 4% compound interest per year. Find its value after 2 years.
3
FA town has 5000 people. The population grows by 5% each year. Find the population after 3 years, to the nearest whole number.
4
FA population of 800 falls by 10% each year. Find the population after 2 years.
5
FThe value of a shop is £20,000. It grows by 2% each year. Find its value after 4 years.
6
FA bike costs £350 and loses 20% of its value each year. Find its value after 3 years.
7
F£1500 is saved in an account that pays 12% interest each year. Find the amount after 2 years.
8
PSA laptop costs £1200 and loses 25% of its value each year. Find its value after 2 years.
9
PSA painting is worth £9000. It rises by 10% each year. Find its value after 3 years.
10
PSA colony of 300 bacteria doubles every hour. How many bacteria are there after 6 hours?
11
PSA coffee is at 80°C. It cools by 5% each minute. Find its temperature after 10 minutes, to 1 d.p.
12
PSA sum of £4000 is invested at 3% interest per year. Work out the value after 5 years, to the nearest penny.
13
PSA town has 40,000 people, and the population falls by 4% each year for 3 years. Find the population after 3 years, to the nearest whole number.
14
RAli says that a 20% increase followed by a 20% decrease gives the original price. Is Ali right? Show your working.
3.4

Reverse percentages

Year 9
Find the original amount before a percentage change
Video lesson: Reverse percentagesComing soon
A jacket is sold for £51 in a sale with a 15% discount. What was the price before the sale? Taking 15% off £51 would be the wrong method, because 15% was taken off the original price, not the sale price.

To find the original amount, divide by the multiplier. This is the reverse of multiplying. For a 15% decrease the multiplier is 0.85, so the original is the sale price divided by 0.85.

For repeated changes, undo one period at a time, starting from the final amount.

Key ideas

  • Reverse percentage: original = new amount ÷ multiplier.
  • After an increase of r%, divide by 1 + r/100. After a decrease of r%, divide by 1 − r/100.
  • For repeated periods, divide by the multiplier once for each period.
  • Do not find 20% of the new amount and add it. That is the wrong method.
Worked example 3.4.1
After a 20% increase, a price is £60. Find the original price.

Method. To find the original, divide by the multiplier, once for each period (the reverse of multiplying).

  1. Write the multiplier. The change multiplies the original by 1.2 each period.
    multiplier = 1.2
  2. Undo period 1. Divide by the multiplier to undo one period of change.
    £60.0000 ÷ 1.2 = £50.0000
  3. Round the answer. Give the original amount to 2 decimal places.
    £50.00

Check. Check: multiplying the original by 1.2 for 1 period(s) gives about £60.00.

Watch out. Do not take the percentage off the final amount. Divide by the multiplier, because the percentage is of the original.

Worked example 3.4.2
After a 15% discount, the price is £51. Find the original price.

Method. To find the original, divide by the multiplier, once for each period (the reverse of multiplying).

  1. Write the multiplier. The change multiplies the original by 0.85 each period.
    multiplier = 0.85
  2. Undo period 1. Divide by the multiplier to undo one period of change.
    £51.0000 ÷ 0.85 = £60.0000
  3. Round the answer. Give the original amount to 2 decimal places.
    £60.00

Check. Check: multiplying the original by 0.85 for 1 period(s) gives about £51.00.

Watch out. Do not take the percentage off the final amount. Divide by the multiplier, because the percentage is of the original.

Worked example 3.4.3
A value is £2704 after 2 years of 4% growth. Find the value at the start.

Method. To find the original, divide by the multiplier, once for each period (the reverse of multiplying).

  1. Write the multiplier. The change multiplies the original by 1.04 each period.
    multiplier = 1.04
  2. Undo period 1. Divide by the multiplier to undo one period of change.
    £2,704.0000 ÷ 1.04 = £2,600.0000
  3. Undo period 2. Divide by the multiplier to undo one period of change.
    £2,600.0000 ÷ 1.04 = £2,500.0000
  4. Round the answer. Give the original amount to 2 decimal places.
    £2,500.00

Check. Check: multiplying the original by 1.04 for 2 period(s) gives about £2,704.00.

Watch out. Do not take the percentage off the final amount. Divide by the multiplier, because the percentage is of the original.

Watch out

Watch out: 20% of £60 is £12, and £60 + £12 = £72 is not the original price. The original is found by dividing by 1.2.

Exam tip

Exam tip. Ask: "what is the original amount?" If the percentage is of the original, divide by the multiplier.

Exercise 3D  Reverse percentages (calculator)

Calculator allowed
1
FAfter a 20% increase, a price is £60. Find the original price.
2
FAfter a 15% discount, the price is £51. Find the original price.
3
FThe price of a car after 12% VAT is added is £134.40. Find the price before VAT.
4
FAfter a 6% increase, a score is 53. Find the original score.
5
FA bag of sugar has a mass of 36 kg after a 25% decrease. Find the original mass.
6
FA shop price is £345 after a 15% increase. Find the original price.
7
FA sale price is £42 after 30% off. Find the original price.
8
PSThe population of a town is 19 208 after it falls by 2% each year for 2 years. Find the population at the start.
9
PSThe value of a car after 2 years of 20% depreciation is £3200. Find its value when new.
10
PSA savings account has £2704 after 2 years of 4% compound interest. Find the amount that was first saved.
11
PSA painting is valued at £5788.13 after 3 years of 5% growth. Find the value when it was first bought, to the nearest penny.
12
PSThe price of a book is £7.20 after a 20% increase. Find the price before the increase.
13
PSA cup of tea has a temperature of 76.5°C after cooling by 10% in a minute. Find the temperature before cooling.
14
PSA mobile phone is sold for £540 after a 10% increase in price. Find the price before the increase.
15
RExplain why 20% off the sale price £60 does not give the original price of £75.
3.5

Overall change and growth or decay in context

Year 9
Find the overall percentage change over several periods
Video lesson: Overall change and growth or decay in contextComing soon
A share price goes up by 20% on Monday, then down by 20% on Tuesday. Many people think it is back to where it started. The overall change is not zero, because each change is applied to a different amount.

To find the overall change, multiply the multipliers for every change to get one overall multiplier. Then the overall percentage change is the overall multiplier minus 1, written as a percentage.

Overall change is not found by adding the percentages. A rise of 20% and a fall of 20% add to zero, but the overall change is −4%.

Key ideas

  • Overall multiplier = the product of all the multipliers.
  • Overall change = (overall multiplier − 1) as a percentage.
  • A negative answer means an overall decrease.
  • Adding the percentages is wrong: each change applies to a different amount.
Worked example 3.5.1
Find the overall percentage change after a 20% increase, then a 20% decrease.

Method. Multiply the changes together to get one overall multiplier, then find the overall percentage change.

  1. Change 1. Multiply the running multiplier by 1.2.
    1 × 1.2 = 1.2
  2. Change 2. Multiply the running multiplier by 0.8.
    1.2 × 0.8 = 0.96
  3. Find the overall change. The overall multiplier minus 1, then as a percentage, gives the overall change.
    0.96 − 1 = -0.04, so the change is −4%
  4. Write the answer.
    −4%

Check. A multiplier below 1 is a decrease, and above 1 is an increase.

Watch out. The overall change is not the sum of the percentages. Add the percentages and you get the wrong answer.

Worked example 3.5.2
Find the overall percentage change after a 15% increase, then a 10% increase.

Method. Multiply the changes together to get one overall multiplier, then find the overall percentage change.

  1. Change 1. Multiply the running multiplier by 1.15.
    1 × 1.15 = 1.15
  2. Change 2. Multiply the running multiplier by 1.1.
    1.15 × 1.1 = 1.265
  3. Find the overall change. The overall multiplier minus 1, then as a percentage, gives the overall change.
    1.265 − 1 = 0.265, so the change is 26.5%
  4. Write the answer.
    26.5%

Check. A multiplier below 1 is a decrease, and above 1 is an increase.

Watch out. The overall change is not the sum of the percentages. Add the percentages and you get the wrong answer.

Worked example 3.5.3
Find the overall percentage change after three decreases of 10%.

Method. Multiply the changes together to get one overall multiplier, then find the overall percentage change.

  1. Change 1. Multiply the running multiplier by 0.9.
    1 × 0.9 = 0.9
  2. Change 2. Multiply the running multiplier by 0.9.
    0.9 × 0.9 = 0.81
  3. Change 3. Multiply the running multiplier by 0.9.
    0.81 × 0.9 = 0.729
  4. Find the overall change. The overall multiplier minus 1, then as a percentage, gives the overall change.
    0.729 − 1 = -0.271, so the change is −27.1%
  5. Write the answer.
    −27.1%

Check. A multiplier below 1 is a decrease, and above 1 is an increase.

Watch out. The overall change is not the sum of the percentages. Add the percentages and you get the wrong answer.

Watch out

Watch out: 10% up then 10% down gives an overall change of −1%, not 0%. The second change is 10% of a smaller amount.

Exam tip

Exam tip. Multiply the multipliers, then take 1 away and write it as a percentage. Check the sign.

Exercise 3E  Overall change and growth or decay (calculator)

Calculator allowed
1
FA price rises by 10%, then falls by 10%. Find the overall percentage change.
2
FA share price goes up by 20%, then down by 20%. Find the overall percentage change.
3
FA value rises by 15%, then rises by 10%. Find the overall percentage change.
4
FA value falls by 10% three times in a row. Find the overall percentage change.
5
FA price increases by 15%, then decreases by 5%. Find the overall percentage change.
6
PSA population rises by 5% in one year and by 5% in the next. Find the overall percentage change.
7
PSThe price of a game rises by 12% and then falls by 12%. Find the overall percentage change to 2 d.p.
8
PSA car loses 15% of its value in its first year and then 20% in its second year. Find the overall percentage change in its value.
9
RAli says that a 10% increase followed by a 10% increase is the same as a 20% increase. Is Ali right? Explain your answer.

Chapter summary

SkillMethod
Recurring decimal to fractionLet x be the number, multiply to line up the repeats, subtract, then divide.
Fraction to recurring decimalLong division until a remainder repeats. Dot the first and last digit of the repeating block.
Repeated changeMultiply by the multiplier once per period: amount × multipliern.
Reverse percentageDivide by the multiplier (once per period) to find the original amount.
Overall changeMultiply the multipliers, then take 1 away and write as a percentage.

Can you…?

  • I can write a recurring decimal as a fraction using the subtraction method
  • I can write a fraction as a recurring decimal and use dots
  • I can use a multiplier to work out repeated percentage change
  • I can find an original amount by reversing a percentage change
  • I can find the overall percentage change over several periods

Key terms

recurring decimal
a decimal with a digit or block of digits that repeats for ever
repeating block
the digits that repeat in a recurring decimal
multiplier
a number that does a percentage change in one step
compound interest
interest added each period, so the interest earns interest too
depreciation
a decrease in value over time

Exercise 3Z  Check-up

Non-calculator
1
Write 0.7̇ as a fraction in its simplest form.
2
Write 0.1̇8̇ as a fraction in its simplest form.
3
Write 0.83̇ as a fraction in its simplest form.
4
Write 4/11 as a recurring decimal, with dots.
5
Write 5/6 as a recurring decimal, with dots.
6
A pair of shoes costs £60, after a 20% increase. Find the original price, using the reverse method.
7
The price of a book is £200, after a 20% discount. Find the price of the book after the discount, using a multiplier.
8
A price rises by 10%, then rises by 20%. Find the overall percentage change.
9
A value rises by 10%, then falls by 10%. Find the overall percentage change.
10
Explain why 0.3̇ + 0.6̇ = 1 is correct, using the fractions 1/3 and 2/3.

Total 25 marks. Allow about 25 minutes.

Exercise 3X  Practice test questions

Non-calculator
1
FWrite 0.2̇5̇ as a fraction in its simplest form.[2 marks]
2
FWrite 1/7 as a recurring decimal, with dots.[2 marks]
3
FWrite 3/11 as a recurring decimal.[2 marks]
4
FWrite 0.4̇5̇ as a fraction in its simplest form.[2 marks]
5
PSA company's profit falls by 25% in one year. Find the overall percentage change over two years in which it falls by 25% each year, then rises by 25%.[3 marks]
6
PSA price is £72 after a 20% increase. Find the original price.[2 marks]
7
RAli says that a 50% decrease followed by a 50% increase gives the original amount. Is Ali right? Show your working.[3 marks]
8
PSA sum of money is £2500 and grows by 4% each year. Find its value after 2 years, to the nearest penny.[3 marks]
9
FWrite 0.83̇ as a fraction. Show your working.[3 marks]
10
PSA price goes up by 10% in January and by another 10% in February. It is £363 at the end of February. Find the price at the start of January.[3 marks]
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