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

Question cluepercent of, what fraction, as a decimal, increased by, reduced by, or the new price
Core evidencethe value in its three forms, and whether the question wants a share or a changed total
Reasoning moveconvert to the easiest form, multiply for 'of', then add or subtract for a change
Trap checktreating a percent as dollars, or giving the share without applying the change
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 reasoner prove an answer, then lift it to scholarship level.

Read the problem and the question, then follow the worked thinking. The Selective answer proves the point efficiently; the Scholarship answer adds control while staying grounded in the facts.

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.

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.
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.
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.
Step 4 - The answer (Selective standard)
Accurate, concise and proven from the facts.
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).
- Direct answerThe first sentence answers the question instead of circling it.
- Proof from the factsThe model ties each step back to a stated fact or rule.
Step 5 - Aim higher: a Scholarship-level answer
The same answer with more control and nuance.
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.
- How the steps interactThe stronger answer connects the facts into a chain rather than listing them.
- Controlled reasoningIt goes further than the Selective answer but keeps pointing at the facts.

Step 6 - Why the Scholarship answer is stronger

Compare the two model answers like a marker would.

Selective Answer	Scholarship Answer	Why It Is Stronger
Takes 25% of 80, then subtracts.	Confirms with 3/4 of 80 = 60.	Stronger reasoning: a second method checks it.
Rejects 80 - 25.	Names both the dollars-trap and the stop-short trap.	Better exam control: both traps are understood.
Stays accurate and concise.	Adds why 'of' means multiply.	Controlled answer: richer but still proof-based.

Selective vs ScholarshipThe comparison shows the upgrade in thinking, not just a longer answer.
What to imitateStudents can copy the move: connect the facts and explain why the trap fails.

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.

Percent as dollars

80 - 25 = 55.

Percent of the amount

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

Reading the percent

Weaker: 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 change

Weaker: May stop at the 25% share.Stronger: Subtracts the share to get the new total.The question wants the new price, not the reduction.

Checking

Weaker: 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.