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Maths Foundations

Fractions, Decimals and Percentages

This lesson teaches you to move between fractions, decimals and percentages of the same value, to find a percentage of an amount by converting to the easiest form, and to apply a percentage increase or decrease to get the new total.

  • Convert between fractions, decimals and percentages of the same value.
  • Find a percentage of an amount by converting to the easiest form.
  • Apply a percentage increase or decrease, not just the percentage itself.
Free sample lesson — reading only

Lesson overview

What this free sample teaches

Find the percentage of the amount and apply the change to get the new total.

Focus

  • Equivalent forms: fraction, decimal, percentage.
  • Percentage of: a share found by multiplying.
  • Percentage change: find the change, then apply it.

What gets tested

  • Converting between the three forms.
  • Finding a percentage of an amount.
  • Applying a percentage increase or decrease.

Quick guide

  • Convert a percent to a friendly fraction (25% = 1/4).
  • 'Of' means multiply the amount.
  • For a change, find the change then add or subtract.

Success criteria

  • You can convert between the three forms.
  • You can find a percentage of an amount.
  • You can explain why a percent is not a fixed number of dollars.

teach

Teach: three views of one value, then change it correctly

A fraction, a decimal and a percentage can be the same number — convert to the easiest form, then take the percentage OF the amount.

Fraction, decimal and percentage questions reward two moves: converting between the three forms, and handling 'percentage of' and 'percentage change' correctly. The trap is treating a percentage as a number of dollars rather than a share of an amount.

A fraction, a decimal and a percentage are three ways of writing the same value. 1/4, 0.25 and 25% are identical. Knowing the common pairs by heart — 1/2 = 0.5 = 50%, 1/4 = 25%, 1/10 = 10%, 3/4 = 75% — lets you switch to whichever form makes a calculation easiest, which is the heart of these questions.

'A percentage of an amount' means multiply. 25% of 80 is 0.25 x 80, or the easier 1/4 of 80 = 20. Convert the percentage to a friendly fraction or decimal first, then multiply the amount. Reading 'percent' as a fixed number of dollars — taking 25 off an 80-dollar price — is the classic mistake.

A percentage change is a two-part move: find the change, then apply it. A 25% reduction on $80 means find 25% of 80 (which is 20), then subtract it (80 - 20 = 60). An increase adds instead. The percentage alone (20) is not the answer — the question asks for the new amount, so you must add or subtract.

The most tempting wrong answers stop one step short or confuse the forms: giving the percentage of the amount but forgetting to subtract it, or subtracting the percent as dollars. Always ask what the question wants — the share itself, or the new total after the change — and convert to the easiest form before calculating.

Equivalent forms

a fraction, decimal and percentage of the same value.Switch to whichever form is easiest to calculate with.

Percentage of

a share of an amount, found by multiplying.'Of' means multiply, not subtract a number of dollars.

Percentage change

find the change, then add it on or take it off.The new total, not the percentage itself, is the answer.

Form-confusion trap

treating a percent as dollars, or stopping at the share.Check what the question actually asks for.
Anatomy of a percentage problem
  1. Question cluepercent of, what fraction, as a decimal, increased by, reduced by, or the new price
  2. Core evidencethe value in its three forms, and whether the question wants a share or a changed total
  3. Reasoning moveconvert to the easiest form, multiply for 'of', then add or subtract for a change
  4. Trap checktreating a percent as dollars, or giving the share without applying the change
  5. Answer shape... % of ... is ... , so the new amount is ...

The moveMove from the percentage to an easy fraction/decimal, take it of the amount, then add or subtract for a change.

  • You can convert between fractions, decimals and percentages.
  • You can find a percentage of an amount.
  • You can apply a percentage increase or decrease.

show

Show: a worked example

Watch a strong mathematician prove the answer, extract the method, and prepare to reuse it.

Read the problem and the question, then follow the worked solution. First prove the correct answer. Then name the reusable method, recognise its common traps, and turn the solution into actions for the next question.

Full problem: take the percentage, then change

Question: A jacket costs $80. In a sale it is reduced by 25%. What is the new price?

The Sale Price

A jacket normally costs $80.

In a sale, the price is reduced by 25 percent.

We want the new sale price — which means finding 25 percent of $80 and then taking it off.

  1. Step 1 - Decode the question and name what to look for

    Name the exact thinking the question wants before you start.

    This is a percentage-change problem: the price drops by 25%, and I want the new price. So I must find 25% of 80 first, then subtract it — not subtract 25, and not stop at the 25% amount.

    • What the question is really askingThe student names what to look for first, so the search has a clear target.
    • What not to confuseNaming the question type heads off the nearest wrong move before it starts.

  2. Step 2 - Reasoning chain: clue -> rule -> step -> answer

    Follow the strong reasoner's path from the facts to the answer.

    Convert, take the percentage, then apply the change.

    • Convert: 25% = 1/4 = 0.25, the easiest form here is 1/4.
    • Percentage of the amount: 1/4 of 80 = 20. So the reduction is $20.
    • Apply the change (a reduction): 80 - 20 = 60.
    • Check: $60 is less than $80, as a reduction should be.
    • Answer: the new price is $60.
    • Clue to ruleThe proof step names the exact fact or rule, not a general impression.
    • Step and effectThe student shows what each step forces and how it decides the answer.

  3. Step 3 - Common wrong answer: spot the trap

    See why a tempting answer is wrong before choosing.

    Common wrong answer: $55.

    Why students choose it: they read '25%' as 25 dollars and subtract it: 80 - 25 = 55.

    Why it is wrong: a percentage is a share of the amount, not a fixed number of dollars. 25% of 80 is 20, not 25. (Another near-miss is answering $20 — the reduction — and forgetting to subtract it from 80.)

    Corrected reasoning: 25% of 80 = 20, then 80 - 20 = 60.

    • Why it tempts studentsA trap usually uses a real fact, which is why students choose it too quickly.
    • Corrected reasoningThe fix shows which fact or rule changes the answer, not just that it is wrong.

  4. Step 4 — The answer, and why it is correct

    State the answer and give the shortest complete proof.

    The new price is $60. First convert 25% to the easy fraction 1/4 and take it of the amount: 1/4 of 80 = 20, so the reduction is $20. Because the price is reduced, subtract: 80 - 20 = 60. The answer is the new total ($60), not the reduction ($20), and not 80 - 25 (which wrongly treats the percent as dollars).

    • The answerThe result is stated directly, with its unit when the question requires one.
    • Why it is correctEvery important step is tied to the relevant property, operation, relationship or check.

  5. Step 5 — Going deeper: the reusable method and the traps

    Name the method, explain why it works, and turn mistakes into warnings.

    The new price is $60, reached in two deliberate moves: take the percentage of the amount, then apply the change. Converting 25% to 1/4 makes the first move trivial — 1/4 of 80 = 20 — and since the price falls, the second move subtracts: 80 - 20 = 60. Two classic errors hide here: treating '25%' as $25 (giving 55) confuses a share with a fixed amount, and stopping at $20 answers 'how much off?' rather than 'what is the new price?'. A sense check seals it: a quarter off $80 should leave three quarters, and 3/4 of 80 is indeed 60.

    • Reusable methodThe explanation names the mathematical structure that makes the method work again.
    • Trap patternsTempting mistakes become recognisable patterns that can be caught next time.

  6. Step 6 — How to use this method next time

    Turn this solution into concrete actions for a new question.

    In this questionOn the next questionWhy it helps
    Found 25% of $80, then subtracted the $20 discount.Underline whether the question asks for the change amount or the amount left after the change.This prevents stopping at a correctly calculated but unrequested intermediate value.
    Replaced 25% with the friendly fraction 1/4.Convert a percentage to the easiest equivalent fraction or decimal before calculating.A convenient representation reduces arithmetic and makes the method clearer.
    Checked that a quarter off leaves three quarters: 3/4 of 80 = 60.Use the complementary percentage as an independent check after an increase or decrease.The second route catches percent-as-dollars and wrong-direction errors.

compare

Compare: percent-as-dollars vs percent-of-amount

The difference between subtracting 25 and finding a quarter.

Both students looked at the same problem. One stops at a first impression; the other proves the answer from the facts and rules. Markers reward the second.

The whole response

Percent as dollars

80 - 25 = 55.

Percent of the amount

1/4 of 80 = 20, then 80 - 20 = 60.

Point-by-point breakdown3
  • Reading the percentWeaker: Subtracts 25 as if it were dollars.Stronger: Finds 25% of the actual amount (20).A percentage is a share of the amount, not a fixed number.
  • Finishing the changeWeaker: May stop at the 25% share.Stronger: Subtracts the share to get the new total.The question wants the new price, not the reduction.
  • CheckingWeaker: Accepts 55 without a check.Stronger: Notes 3/4 of 80 = 60 as a check.A sense check catches the dollars confusion.

guide

Guide: student checkpoint

Do one small reasoning move before independent practice.

Reread the worked problem and question. Now pause like a strong reasoner: find the rule, name the trap, or upgrade a basic answer before you work on your own.

A $50 book is increased by 10%. In one sentence, give the new price and how you found it.

Want feedback on your own answer? Get started to practise with instant marking.

  • Convert the percentage to an easy fraction or decimal.
  • Find the percentage of the amount.
  • Add or subtract to get the new total.