Fractions, decimals and percentages as one quantity
See equivalent forms as different ways to describe the same share, then extend that reasoning to recurring decimals.
Core and Extended: equivalent forms and simplest fractions. The recurring-decimal notation and algebra sections are Extended only.
Before you begin
- Use place value in decimals.
- Multiply and divide numerator and denominator by the same number.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which change preserves the value of 2/3?
What you will learn
- Convert between fractions, decimals, percentages and mixed numbers.
- Simplify a fraction without changing its value.
- Convert recurring decimals to fractions using aligned subtraction.
Keep the whole fixed
A positive fraction describes a share of a whole divided into equal pieces. Its denominator cannot be zero. In 3/4, the denominator tells you the size of the pieces and the numerator tells you how many. To compare shares visually, use the same-sized whole. Three quarters of a large pizza can be more food than three quarters of a small one.
Splitting each quarter into two changes the label to eighths but not the shaded amount: 3/4 = 6/8. Multiplying or dividing both numerator and denominator by the same non-zero number preserves the ratio. Simplify 18/24 by dividing both by 6 to get 3/4. Dividing only one part changes the value.
A fraction can exceed one
A positive proper fraction has numerator smaller than denominator, such as 3/5. An improper fraction has numerator at least as large as denominator, such as 7/5. A mixed number combines a whole number and a proper fraction: 7/5 = 1 2/5 because five fifths make one whole and two fifths remain.
To convert 2 3/5, count fifths: two wholes contain 10 fifths, so 2 3/5 = 13/5. To reverse it, divide 13 by 5: quotient 2, remainder 3. The remainder becomes the new numerator and the denominator stays 5.
Decimals and percentages rename the share
A terminating decimal is a fraction with a power of ten as denominator: 0.375 = 375/1000 = 3/8. To turn a fraction into a decimal, divide the numerator by the denominator. For 3 ÷ 8, the digits give 0.375. You can check by multiplying 0.375 by 8 to get 3.
Percent means per hundred. Thus 0.375 = 37.5/100 = 37.5%. To change a decimal to a percentage, multiply by 100; to reverse it, divide by 100. Percentages can exceed 100%: 1.25 = 125% = 5/4. A percentage describes a ratio, not an amount by itself; 25% of 20 and 25% of 200 are different quantities.
Extended: show exactly which digits repeat
A recurring decimal repeats a block forever. We write 0.(27) here for 0.272727… and 0.1(6) for 0.16666…. In Cambridge notation, a dot above a single digit marks that repeating digit; dots above the first and last digits mark a repeating block. A bar above the whole block is another common convention. Dots above the 2 and 7 in 0.27 mean both digits repeat, not that only the 7 repeats.
Long division explains recurrence: dividing 2 by 11 gives a repeating cycle of remainders and the digits 0.181818… . A finite remainder cycle repeats the same digit block. Terminating decimals finish with remainder zero. Never replace a recurring number by a rounded decimal and call the result exact.
Extended: align the repeated tail, then subtract
Let x = 0.272727…. Multiplying by 100 shifts one whole two-digit block: 100x = 27.272727…. Subtract x and the identical infinite tails cancel, so 99x = 27 and x = 27/99 = 3/11. The number of repeating digits determines the power of ten.
For a non-repeating prefix, align two shifted copies. If x = 0.16666…, then 10x = 1.6666… and 100x = 16.6666…. Subtract these: 90x = 15, so x = 1/6. The denominators 9, 99 or 90 come from subtraction, not a rule to memorise without checking the repeating block.
Pause and explain
Which is exactly equal to 0.5%?
Worked example
A tank is 0.375 full. Express the filled share as a simplified fraction and a percentage, then find the share still empty.
Show the worked solution
- 0.375 = 375/1000. Divide numerator and denominator by 125: 3/8.
- Multiply the decimal by 100: 37.5% is filled.
- The empty share is 1 − 3/8 = 5/8, or 62.5%. Check: 37.5% + 62.5% = 100%.
Answer Filled: 3/8 = 37.5%. Empty: 5/8 = 62.5%.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Convert 7/8 to a decimal and percentage.
Give me a hint
Divide 7 by 8 or start from 1/8 = 0.125.
Compare my reasoning
- 7/8 = 7 × 0.125 = 0.875.
- 0.875 × 100 = 87.5, so the percentage is 87.5%.
- Check: the missing 1/8 is 12.5%, and the two percentages total 100%.
0.875; 87.5%.
Look for these in your work
- Kept the percentage sign attached to the percentage value.
- Checked against a full whole.
Convert 2.35 to a simplified improper fraction, a mixed number and a percentage.
Give me a hint
There are two decimal places, so begin with 235/100.
Compare my reasoning
- 2.35 = 235/100 = 47/20 after dividing both parts by 5.
- 47 ÷ 20 gives 2 with remainder 7, so the mixed number is 2 7/20.
- 2.35 × 100 = 235, so the percentage is 235%.
47/20; 2 7/20; 235%.
Look for these in your work
- Simplified the fraction without changing its value.
- Allowed the percentage to exceed 100%.
Extended: convert x = 0.2(7), meaning 0.27777…, into a simplified fraction. Show the two aligned copies.
Give me a hint
Use 10x and 100x so their repeating tails begin at the same decimal place.
Compare my reasoning
- 10x = 2.7777… and 100x = 27.7777….
- Subtract: 90x = 25.
- x = 25/90 = 5/18.
5/18
Look for these in your work
- Aligned the tails before subtracting.
- Simplified the final fraction.
Two identical tanks hold water. Tank A is 0.6 full and tank B is 5/8 full. A student says B holds '2.5% more water'. Find the percentage-point difference in fullness and the percentage increase in water relative to A. Explain why these differ.
Give me a hint
First express both fullness levels as percentages. For the relative increase, divide the difference by A's amount.
Compare my reasoning
- A is 60% full; B is 62.5% full. Difference = 2.5 percentage points.
- As fractions of capacity, the difference is 0.625 − 0.6 = 0.025.
- Relative to A, the increase is (0.025/0.6) × 100% = 25/6% = 4 1/6%. The denominators differ: capacity versus A's water.
2.5 percentage points; 4 1/6% more water relative to A.
Look for these in your work
- Kept the common tank capacity fixed.
- Named the reference whole for each percentage.
Common mistakes
- Changing only the numerator when simplifying.
- Treating 0.5% as 0.5: it is 0.005 as a decimal.
- Using 99 for every recurring decimal, even when there is a non-repeating prefix.
What can you explain now?
Convert 1 3/5 to an improper fraction, decimal and percentage.
Compare with the explanation
8/5; 1.6; 160%.
One whole is five fifths. Add three fifths, divide by five for the decimal, then multiply the decimal by 100 for the percentage.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, convert 3/8 three ways and explain why 0.1(6) is exactly 1/6, while 0.167 is only an approximation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
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