Lesson 1 ~25 min Unit 1 · Financial Maths +85 XP

Fractions, Decimals, Percentages

Three ways of writing the same proportion — and how to switch between them at speed.

Today's hook: A shop advertises '1/3 off'. Another '33% off'. A third 'save 0.33 of the price'. Which deal is best?

0/5QUESTS

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Think First

warm-up

A shop advertises '1/3 off'. Another '33% off'. A third 'save 0.33 of the price'. Which deal is best? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

A fraction, a decimal and a percentage are three different ways of writing the same number — a part of a whole.

A fraction like $\tfrac{1}{4}$ shows parts of a whole. The same value can be written as the decimal $0.25$ or the percentage $25\%$. They are all the same amount. The percent sign $\%$ means “out of 100”, so $25\% = \tfrac{25}{100} = 0.25$.

$\tfrac{1}{4} = 0.25 = 25\%$

"Percent" = "per 100"

$\%$ literally means “divided by 100”, so $7\% = \tfrac{7}{100}$.

Same value, three faces

$\tfrac{1}{2} = 0.5 = 50\%$. Choose whichever form makes the problem easiest.

Money loves decimals

Calculators use decimals. Recipes use fractions. Shops use percentages. Switch as needed.

What You'll Master

objectives

Know

A fraction shows a part of a whole, written $\tfrac{a}{b}$
A decimal uses place value to show parts of a whole
A percentage is "out of 100" and uses the symbol $\%$
$\tfrac{1}{2}=0.5=50\%$, $\tfrac{1}{4}=0.25=25\%$, $\tfrac{3}{4}=0.75=75\%$

Understand

These three forms represent the same value
Why $\%$ is just a special fraction with denominator $100$
How common equivalents save calculation time

Can Do

Recall the most-used FDP equivalents from memory
Pick the easiest form to use for a given problem
Decide which of two values is bigger by switching forms

Words You Need

vocabulary

FractionA number written $\tfrac{a}{b}$ showing $a$ parts out of $b$ equal parts of a whole.

DecimalA number using place value to the right of the decimal point: tenths, hundredths…

PercentageA number "out of 100" — written with the $\%$ symbol. $30\% = \tfrac{30}{100}$.

EquivalentTwo different-looking values that have the same size. $\tfrac{1}{2}$ and $0.5$ are equivalent.

NumeratorThe TOP number of a fraction — how many parts you have.

DenominatorThe BOTTOM number of a fraction — how many equal parts make the whole.

Spot the Trap

heads-up

Wrong: "$0.5$ and $5\%$ are the same" — NO. $0.5 = 50\%$, not $5\%$. To convert decimal to percentage, move the decimal point TWO places right.

Right: A decimal $\to$ percentage: multiply by 100 (move point 2 places right). $0.7 = 70\%$.

Wrong: "$\tfrac{1}{3} = 33\%$ exactly" — Not quite. $\tfrac{1}{3} = 33.\overline{3}\%$, just slightly more than $33\%$.

Right: A percentage $\to$ decimal: divide by 100 (move point 2 places left). $45\% = 0.45$.

The Three Forms — Side by Side

+5 XP

Below are the equivalents you should know by heart. Memorising these saves time in every percentage problem you ever do.

The most-used equivalents are halves, quarters, fifths and tenths. Quarter equivalents are everywhere in sales: $25\%$ off means “take away a quarter”. Knowing $\tfrac{1}{5}=20\%$ unlocks GST-like calculations.

$\tfrac{1}{2}=0.5=50\%$ $\tfrac{1}{4}=0.25=25\%$ $\tfrac{1}{5}=0.2=20\%$ $\tfrac{1}{10}=0.1=10\%$

Halves and quarters

$\tfrac{1}{2}=50\%$, $\tfrac{1}{4}=25\%$, $\tfrac{3}{4}=75\%$.

Fifths and tenths

$\tfrac{1}{5}=20\%$, $\tfrac{2}{5}=40\%$, $\tfrac{1}{10}=10\%$.

Recurring thirds

$\tfrac{1}{3} \approx 33.3\%$, $\tfrac{2}{3} \approx 66.7\%$.

Comparing Mixed Forms

+5 XP

To compare fractions, decimals and percentages, first convert them ALL into the same form — usually decimals are easiest.

Suppose you want to order $\tfrac{3}{5}$, $0.55$, and $58\%$ from smallest to largest. Convert all to decimals: $\tfrac{3}{5} = 0.6$, $58\% = 0.58$, and $0.55$ is already a decimal. Now compare: $0.55 < 0.58 < 0.6$.

Convert all to decimals, then compare

One form rules

Choose decimals. They line up cleanly by place value.

Pad with zeros

Compare $0.5$ and $0.45$? Write as $0.50$ vs $0.45$. Now obvious.

Watch the percentage trap

$5\% = 0.05$, NOT $0.5$. Always two decimal places!

Watch Me Solve It · 3 examples

Watch Me Solve It · Compare three forms

+15 XP per step

PROBLEM

A shop offers a $\tfrac{1}{3}$ discount, another offers $33\%$ off, and a third advertises “save $0.33$ of the price”. Which is the biggest discount?

1

Convert each to a decimal

$\tfrac{1}{3} = 0.333\overline{3}\dots$

Divide $1 \div 3$ on a calculator (or by long division).
2

Compare the decimals

$0.333\overline{3} > 0.33 = 0.33$

$\tfrac{1}{3}$ is the biggest, $33\%$ and $0.33$ are equal.
3

State the winner

$\tfrac{1}{3} > 33\% = 0.33$

The $\tfrac{1}{3}$-off deal is the best — by a tiny but real margin.

Answer$\tfrac{1}{3}$ off is biggest (by about $0.3\%$). $33\%$ off and $0.33$ off are the same.

Watch Me Solve It · Convert a fraction

+15 XP per step

PROBLEM

Convert $\tfrac{3}{8}$ to a decimal and a percentage.

1

Divide top by bottom

$3 \div 8 = 0.375$

This gives the decimal form.
2

Multiply the decimal by 100

$0.375 \times 100 = 37.5$

This converts decimal to percentage.
3

Write the answer

$\tfrac{3}{8} = 0.375 = 37.5\%$

Always state all three when asked.

Answer$\tfrac{3}{8} = 0.375 = 37.5\%$

Watch Me Solve It · Order mixed values

+15 XP per step

PROBLEM

Order from smallest to largest: $\tfrac{2}{5}, \;0.38, \;42\%, \;\tfrac{1}{3}$.

1

Convert each to a decimal

$\tfrac{2}{5}=0.4, \;0.38=0.38, \;42\%=0.42, \;\tfrac{1}{3}=0.333\dots$

Use long division for $\tfrac{1}{3}$.
2

Order the decimals

$0.333 < 0.38 < 0.4 < 0.42$

Smallest to largest.
3

Rewrite in original forms

$\tfrac{1}{3} < 0.38 < \tfrac{2}{5} < 42\%$

Restore the original notation.

Answer$\tfrac{1}{3} < 0.38 < \tfrac{2}{5} < 42\%$

Common Pitfalls

heads-up

Confusing $5\%$ with $0.5$

$5\%$ means 5 per 100, so $5\% = 0.05$, not $0.5$. The decimal $0.5$ is actually $50\%$.

Fix: "Percent" means "divide by 100" — always move 2 places, not 1.

Forgetting the $\%$ sign

Writing $0.3 \times 100 = 30$ without the $\%$ — the answer is $30\%$, not 30.

Fix: When you multiply by 100 to convert to percentage, the $\%$ sign comes with it.

Treating $\tfrac{1}{3}$ as $0.33$ exactly

$\tfrac{1}{3}$ is a recurring decimal $0.\overline{3}$ — slightly bigger than $0.33$.

Fix: For thirds, keep fraction form for exactness, or use $0.\overline{3}$ notation.

Copy Into Your Books

Key Equivalents

$\tfrac{1}{2} = 0.5 = 50\%$
$\tfrac{1}{4} = 0.25 = 25\%$
$\tfrac{3}{4} = 0.75 = 75\%$
$\tfrac{1}{5} = 0.2 = 20\%$
$\tfrac{1}{10} = 0.1 = 10\%$

Conversions

Decimal $\to \%$: multiply by 100
$\% \to$ decimal: divide by 100
Fraction $\to$ decimal: numerator $\div$ denominator

Comparing Forms

Convert ALL to the same form first
Decimals are usually easiest
Pad with zeros so digits line up

Recurring Decimals

$\tfrac{1}{3} = 0.\overline{3} = 33.\overline{3}\%$
$\tfrac{2}{3} = 0.\overline{6} = 66.\overline{6}\%$
Approximate but never exact in decimal form

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Fractions, Decimals, Percentages

4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

1 Convert $\tfrac{3}{4}$ to a percentage.

$\tfrac{3}{4} = 3 \div 4 = 0.75 = 75\%$.$75\%$
2 Write $0.6$ as a fraction in simplest form.

$0.6 = \tfrac{6}{10} = \tfrac{3}{5}$.$\tfrac{3}{5}$
3 Express $12\%$ as a decimal.

$12\% \div 100 = 0.12$.$0.12$
4 Which is bigger: $\tfrac{2}{5}$ or $45\%$?

$\tfrac{2}{5}=0.4$; $45\%=0.45$. $0.45 > 0.4$.$45\%$ is bigger

Complete in your workbook.

Quick Check · 5 questions

Which of these is equivalent to $\tfrac{3}{4}$?

+10 XP

What is $40\%$ as a fraction in simplest form?

+10 XP

Convert $0.08$ to a percentage.

+10 XP

Which value is the largest?

+10 XP

What is $\tfrac{1}{3}$ as a percentage (to 1 decimal place)?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Convert each to ALL three forms (fraction in simplest form, decimal, percentage): (a) $0.85$ (b) $12\%$ (c) $\tfrac{7}{20}$

Answer in your workbook.

Understand Easy 2 MARKS

Q7. Tahlia scored $\tfrac{17}{20}$ on a test. Her friend Jay scored $82\%$. Who got the higher mark, and by how much (as a percentage)?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. Arrange these from smallest to largest, showing your conversions: $\tfrac{5}{8}$, $0.6$, $\tfrac{2}{3}$, $63\%$. Then explain why converting to one form makes the ordering reliable.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $\tfrac{3}{4} = 0.75$.

2. C — $40\% = \tfrac{40}{100} = \tfrac{2}{5}$.

3. C — $0.08 \times 100 = 8\%$.

4. B — $0.4$ is largest after converting all to decimals.

5. D — $\tfrac{1}{3} = 33.\overline{3}\% \approx 33.3\%$.

Show Your Working Model Answers

Q6 (3 marks): (a) $0.85 = \tfrac{85}{100} = \tfrac{17}{20} = 85\%$ [1]. (b) $12\% = 0.12 = \tfrac{12}{100} = \tfrac{3}{25}$ [1]. (c) $\tfrac{7}{20} = 7 \div 20 = 0.35 = 35\%$ [1].

Q7 (2 marks): $\tfrac{17}{20} = 0.85 = 85\%$ [1]. Tahlia's $85\%$ is higher than Jay's $82\%$ by $3\%$ [1].

Q8 (4 marks): $\tfrac{5}{8} = 0.625$; $0.6 = 0.600$; $\tfrac{2}{3} = 0.\overline{6} = 0.667$; $63\% = 0.63$ [2]. Order: $0.6 < \tfrac{5}{8} < 63\% < \tfrac{2}{3}$ [1]. Converting to decimals lets us compare digit-by-digit using place value, which is impossible when forms differ [1].

Stretch Challenge · +25 XP, +10 coins

The Petrol Price Problem

A litre of petrol cost $\$1.60$ last week. Today it has risen by $\tfrac{1}{8}$. Tomorrow it will rise a further $7.5\%$ from today's price. (a) What fraction has the petrol risen from last week to tomorrow (overall)? (b) Express your answer as a decimal AND a percentage.

Reveal solution

Today: $1.60 \times (1 + \tfrac{1}{8}) = 1.60 \times \tfrac{9}{8} = \$1.80$. Tomorrow: $1.80 \times 1.075 = \$1.935$. Overall rise: $1.935 - 1.60 = \$0.335$ from $\$1.60$, which is $\tfrac{0.335}{1.60} = \tfrac{67}{320} \approx 0.209 = 20.9\%$.

Quick Review

Same value

Fraction, decimal, percentage are three faces of one number

Convert it

Decimal $\to \%$: $\times 100$. $\% \to$ decimal: $\div 100$

Compare it

Convert all to decimals first

Remember $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{5}$

$50\%, 25\%, 20\%$ — memorise!

Watch zeros

$5\% = 0.05$, NOT $0.5$

Thirds recur

$\tfrac{1}{3} = 0.\overline{3} \approx 33.3\%$

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Unit overview · Maths subject page