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Daily Maths

Daily Maths

Percentages, fractions and decimals

Percentages- Formula Core + Extended 11 min read

On this page
  1. In this note
  2. Formula bank
  3. What a percentage is
  4. Converting between the three forms
  5. Putting fractions, decimals and percentages in order
  6. Finding a percentage of an amount
  7. Writing one amount as a percentage of another
  8. Ways they test it
  9. Mistakes to avoid
  10. Summary
  11. Practice
  12. Can I do this?
  13. Quiz

In this note

Syllabus 1.5 and 1.13
  • 1.5Convert between percentages, fractions and decimals, both ways.
  • 1.5Put a mixed list of fractions, decimals and percentages in order.
  • 1.13Find a percentage of an amount, with and without a calculator.
  • 1.13Write one amount as a percentage of another.

The first page on percentages: increase and decrease, reverse percentages and profit and loss all build on it.

Formula bank

Per centLearn it\(x\%=\dfrac{x}{100}\)per cent means "out of 100". The whole is always 100%.Use it when turning any percentage into a fraction or decimal.
Percentage → decimalLearn it\(\div100\)move the point 2 places left: 7% = 0.07.Use it when you need a multiplier for the calculator.
Decimal or fraction → percentageLearn it\(\times100\)\(0.3=30\%\); \(\tfrac38\times100=37.5\%\).Use it when "write as a percentage".
Percentage of an amountLearn it\(\dfrac{x}{100}\times\text{amount}\)"of" means multiply.Use it when "find 35% of 240".
One amount as a percentage of anotherLearn it\(\dfrac{\text{part}}{\text{whole}}\times100\)the whole is the amount after the word "of"; same units first.Use it when "write 42 as a percentage of 60".

What a percentage is

A 10 by 10 square with 37 of its 100 small squares shaded
37 of the 100 squares are shaded: 37%

Per cent means out of 100. Split something into 100 equal parts: each part is 1%, and 37 parts are 37%.

\[37\%=\frac{37}{100}=0.37\]

A percentage is always a percentage OF something, and that something is the whole: 100%.

PercentageFractionDecimal
100%\(1\)1
50%\(\tfrac12\)0.5
25%\(\tfrac14\)0.25
10%\(\tfrac1{10}\)0.1
5%\(\tfrac1{20}\)0.05
1%\(\tfrac1{100}\)0.01
12.5%\(\tfrac18\)0.125
\(33\tfrac13\%\)\(\tfrac13\)\(0.\dot3\)
150%\(\tfrac32\)1.5

Converting between the three forms

Conversion trianglePercentageDecimalFraction÷ 100× 100put over 100,simplify× 100place value, simplifytop ÷ bottom
Every conversion is ×100, ÷100, or turning a fraction into a division.

General methodHow to convert any number between the three forms

  1. Percentage → decimal: ÷ 100 (point 2 places LEFT). 7% = 0.07.
  2. Decimal → percentage: × 100 (point 2 places RIGHT), add %. 0.065 = 6.5%.
  3. Percentage → fraction: put it over 100, clear any decimal, simplify. 35% = \(\tfrac{35}{100}=\tfrac7{20}\).
  4. Fraction → percentage: × 100 (or make the bottom 100). \(\tfrac7{25}=\tfrac{28}{100}=28\%\).
  5. Fraction → decimal: top ÷ bottom. \(\tfrac38=0.375\).

Example Write \(\tfrac{7}{40}\) as a percentage.

Multiply the fraction by 100.

  1. \(\tfrac{7}{40}\times100=\tfrac{700}{40}\)
  2. \(=17.5\)

Answer 17.5%

Try it yourself

1
Write 4% as a decimal.
Show answer

Answer 0.04

2
Write \(\tfrac{3}{20}\) as a percentage.
Hint
Make the bottom 100: × 5.
Show answer

Answer 15%

3
Write 65% as a fraction in its simplest form.
Hint
Over 100, then divide top and bottom by 5.
Show answer

Answer \(\tfrac{13}{20}\)

Putting fractions, decimals and percentages in order

General methodHow to order any mixed list

  1. Change every value to a decimal.
  2. Write them all to the same number of decimal places (3 is usually enough).
  3. Order the decimals.
  4. Write the answer with the ORIGINAL values, in the order the question asks (smallest or largest first).

Example Order, smallest first: \(0.62,\ \tfrac58,\ 61\%,\ 0.\dot6,\ \tfrac35\)

Change all to decimals with 3 places, order, then write the originals.

  1. \(0.620,\ 0.625,\ 0.610,\ 0.666\ldots,\ 0.600\)
  2. order: \(0.600<0.610<0.620<0.625<0.666\)

Answer \(\tfrac35,\ 61\%,\ 0.62,\ \tfrac58,\ 0.\dot6\)

Try it yourself

1
Which is biggest: \(\tfrac25\), 41.5%, 0.42 or \(\tfrac37\)?
Hint
\(3\div7=0.4285\ldots\)
Show answer

Answer \(\tfrac37\) (= 0.4285…)

Finding a percentage of an amount

General methodHow to find any percentage of an amount

  1. No calculator: find 10% (÷ 10), 5% (half of 10%) and 1% (÷ 100), then add the pieces you need. 35% = 10% + 10% + 10% + 5%.
  2. Calculator: change the percentage to a decimal and multiply. 35% of 240 = 0.35 × 240.
  3. Money: give the answer to 2 decimal places ($0.36).

Trick: \(x\%\) of \(y\) = \(y\%\) of \(x\). 8% of 25 = 25% of 8 = 2.

Example Find 35% of $240 without a calculator.

Build 35% from 10% and 5%.

  1. \(10\%=24\)
  2. \(5\%=12\)
  3. \(35\%=24+24+24+12=84\)

Answer $84

Try it yourself

1
Find 15% of 360.
Hint
10% = 36, 5% = 18.
Show answer

Answer 54

2
Find 17.5% of 80.
Hint
10% + 5% + 2.5%.
Show answer

Answer 14

3
Find 150% of 46.
Hint
100% + 50%.
Show answer

Answer 69

Writing one amount as a percentage of another

General methodHow to write any amount as a percentage of another

  1. Find the whole: the total, or the amount after the word "of".
  2. Make the units the same (minutes with minutes, cents with cents).
  3. Divide: \(\dfrac{\text{part}}{\text{whole}}\).
  4. Multiply by 100 and add %.

Example Write 45 minutes as a percentage of 2 hours.

Same units first, then part ÷ whole × 100.

  1. 2 hours = 120 minutes
  2. \(\tfrac{45}{120}\times100=37.5\)

Answer 37.5%

Try it yourself

1
Sana scores 42 out of 60. What percentage is this?
Show answer

Answer 70%

2
Write 40 cents as a percentage of $2.
Hint
$2 = 200 cents.
Show answer

Answer 20%

3
A plant grows from 20 cm to 50 cm. Its new height is what percentage of its old height?
Hint
The whole is the OLD height.
Show answer

Answer 250%

Ways they test it

Type 1 Convert

"Write 0.085 as a percentage." / "Write 0.35% as a fraction in its simplest form." (1-2 marks)

Spot it "write ... as a ...".

  1. Use the conversion rule.
  2. Fractions: simplest form for the last mark.

For example Write 0.35% as a fraction in its simplest form.

  1. \(\tfrac{0.35}{100}=\tfrac{35}{10000}\)
  2. divide by 5: \(\tfrac{7}{2000}\)

Answer \(\tfrac7{2000}\)

Type 2 Order a mixed list

"Write these in order of size, starting with the smallest." (2 marks)

Spot it "in order of size".

  1. All to decimals.
  2. Order.
  3. Answer in the original forms.

Type 3 Percentage of an amount

"Work out 12% of 650." (1-2 marks)

Spot it "% of".

  1. Build from 10%, 5%, 1% (no calculator) or × decimal.

Type 4 One amount as a % of another

"Write 650 g as a percentage of 2.5 kg." (2 marks)

Spot it "as a percentage of".

  1. Same units.
  2. part ÷ whole × 100.

Type 5 Compare using percentages

"Sara scores 54/75 in Physics and 47/64 in Chemistry. Which test went better?" (3 marks)

Spot it different totals to compare.

  1. Change each to a percentage.
  2. Compare and say which, with both percentages written.

For example 54/75 and 47/64

  1. \(\tfrac{54}{75}\times100=72\%\)
  2. \(\tfrac{47}{64}\times100=73.4\%\)

Answer Chemistry

Type 6 Budgets: "the rest"

"He spends 30% on rent, ¼ on food and $6800 on transport. What percentage does he save?"

Spot it several parts and "the rest".

  1. Change every part to a percentage.
  2. Take the total away from 100%.

Mistakes to avoid

  • ✗ 7% = 0.7✓ 7% = 0.07÷ 100 moves the point TWO places.
  • ✗ \(\tfrac38=3.8\%\)✓ \(\tfrac38=37.5\%\)A fraction is a division.
  • ✗ 42 as % of 60 = \(\tfrac{60}{42}\times100\)✓ \(\tfrac{42}{60}\times100\)The whole (after "of") goes on the bottom.
  • ✗ 45 min of 2 h = \(\tfrac{45}{2}\times100\)✓ \(\tfrac{45}{120}\times100\)Units must match first.
  • ✗ Ordered list written as decimals✓ write the original valuesThe answer is the given list re-ordered.

Summary

  • \(x\%=\tfrac{x}{100}\); the whole is 100%.
  • % → decimal: ÷ 100. Decimal or fraction → %: × 100. % → fraction: over 100, simplify.
  • Order: all to decimals, answer in the original forms.
  • % of an amount: 10%, 5%, 1% building blocks, or × decimal on a calculator.
  • A as a % of B: A ÷ B × 100, same units first.

If you see …

  • "write as a percentage"× 100 and add %.
  • "% of"multiply.
  • "as a percentage of"part ÷ whole × 100.
  • two different unitsconvert first.

Practice

Practice: easy, medium, hard

Easy

1
Write \(\tfrac18\) as a percentage.
Hint
\(\tfrac18\times100\).
Show working
  1. \(\tfrac{100}{8}=12.5\)

Answer 12.5%

2
Find 20% of 45.
Hint
10% = 4.5.
Show working
  1. \(10\%=4.5\)
  2. \(20\%=9\)

Answer 9

3
Write 18 out of 24 as a percentage.
Show working
  1. \(\tfrac{18}{24}=\tfrac34=75\%\)

Answer 75%

Medium

1
A shop sold 1500 phones. 330 were black. What percentage were black?
Guide me

What is it asking? a percentage

What do we know? 330 of 1500

Plan part ÷ whole × 100

Show working
  1. \(\tfrac{330}{1500}\times100=22\)

Answer 22%

2
Write 650 g as a percentage of 2.5 kg.
Hint
2.5 kg = 2500 g.
Guide me

What is it asking? 650 g as a % of 2.5 kg

What do we know? different units

Plan convert, then part ÷ whole × 100

Show working
  1. \(\tfrac{650}{2500}\times100=26\)

Answer 26%

3
Find 12% of 650 without a calculator.
Hint
10% = 65, 1% = 6.5.
Guide me

What is it asking? 12% of 650

What do we know? no calculator

Plan 10% + 1% + 1%

Show working
  1. \(65+6.5+6.5=78\)

Answer 78

Hard

1
Bilal scores 62 out of 80, then 70 out of 80 in a retest. Find both scores as percentages.
Guide me

What is it asking? two percentages

What do we know? both out of 80

Plan part ÷ whole × 100 each time

Show working
  1. \(\tfrac{62}{80}\times100=77.5\)
  2. \(\tfrac{70}{80}\times100=87.5\)

Answer 77.5% and 87.5%

PartAnswer / workingMarksGuidance
\(\tfrac{62}{80}\times100\)M1
77.5A1
87.5B1
Total: 3 marks
2
A school has 1200 students. 40% come by bus, 240 walk, 300 come by car and the rest cycle. What percentage cycle?
Hint
Turn each group into a percentage first.
Guide me

What is it asking? the percentage who cycle

What do we know? bus 40%, walk 240, car 300 out of 1200

Plan all as percentages, then 100% minus the total

Show working
  1. walk: \(\tfrac{240}{1200}=20\%\); car: \(\tfrac{300}{1200}=25\%\)
  2. \(100-40-20-25=15\)

Answer 15%

PartAnswer / workingMarksGuidance
walk 20% or car 25% seenM1
\(100-40-20-25\) or \(\tfrac{180}{1200}\)M1
15A1
Total: 3 marks
3
Order, smallest first: \(0.42,\ \tfrac25,\ 41.5\%,\ \tfrac37\).
Guide me

What is it asking? the list in order

What do we know? four values close to 0.4

Plan decimals to 4 places, then the original forms

Show working
  1. \(0.4000,\ 0.4150,\ 0.4200,\ 0.4285\)

Answer \(\tfrac25,\ 41.5\%,\ 0.42,\ \tfrac37\)

PartAnswer / workingMarksGuidance
at least three correct decimalsM1
\(\tfrac25,\ 41.5\%,\ 0.42,\ \tfrac37\)A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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