Lesson 2 ~25 min Unit 1 · Financial Maths +85 XP

Converting Between FDP

Step-by-step methods for switching between fractions, decimals and percentages — including the tricky recurring decimals.

Today's hook: Your test score is 17/20. Your friend got 0.88. Who did better?

0/5QUESTS

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Think First

warm-up

Your test score is 17/20. Your friend got 0.88. Who did better? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

Every fraction can be turned into a decimal by long division, and every percentage can be turned into a fraction over 100. There is a method for each conversion, and a few that produce recurring decimals.

To go fraction $\to$ decimal, do the division: $\tfrac{a}{b} = a \div b$. To go decimal $\to$ percentage, multiply by 100. To go percentage $\to$ fraction, place it over 100 and simplify. Each route is a one-step rule.

$\tfrac{a}{b} \xrightarrow{\div} \text{decimal} \xrightarrow{\times 100} \text{percentage}$

Long division for any fraction

Even $\tfrac{7}{16}$ becomes a decimal: $7 \div 16 = 0.4375$.

Recurring patterns

$\tfrac{1}{3}, \tfrac{1}{6}, \tfrac{1}{7}, \tfrac{1}{9}$ all give recurring decimals.

Always simplify

$15\% = \tfrac{15}{100} = \tfrac{3}{20}$. Divide by HCF.

What You'll Master

objectives

Know

Fraction $\to$ decimal: numerator $\div$ denominator
Decimal $\to$ percentage: multiply by 100
Percentage $\to$ fraction: place over 100, then simplify
Recurring decimals use a dot or bar notation: $0.\overline{3}$

Understand

Why some decimals terminate and others recur
How the HCF lets you simplify a fraction in one go
Why percentages always sit over 100

Can Do

Convert any fraction to its decimal form by division
Simplify a percentage to a fraction in lowest terms
Recognise recurring decimals and write them correctly

Words You Need

vocabulary

Terminating decimalA decimal that has a finite number of digits, e.g. $0.375$.

Recurring decimalA decimal with a repeating block, e.g. $0.\overline{3}$.

Long divisionA step-by-step method to divide one number by another by hand.

SimplifyDivide top and bottom of a fraction by their HCF to get smallest terms.

HCFHighest Common Factor — the biggest number that divides into both.

Place valueThe value of a digit based on its position: tenths, hundredths…

Spot the Trap

heads-up

Wrong: "To convert $\tfrac{5}{8}$ to a decimal I divide $8 \div 5$" — NO. You divide the TOP by the BOTTOM.

Right: Fraction $\to$ decimal: numerator (top) $\div$ denominator (bottom). $\tfrac{5}{8} = 5 \div 8 = 0.625$.

Wrong: "$0.6 = \tfrac{6}{100}$" — NO. The $6$ is in the tenths place, so $0.6 = \tfrac{6}{10} = \tfrac{3}{5}$.

Right: Read the place value: $0.6$ is six TENTHS, so $\tfrac{6}{10} = \tfrac{3}{5}$.

Fraction to Decimal — Long Division

+5 XP

When the answer isn't obvious, use long division. Add a decimal point and zeros to the numerator and divide normally.

Take $\tfrac{3}{8}$. Write it as $3.000 \div 8$. Step through: $8$ into $3$ is $0$ remainder $3$. $8$ into $30$ is $3$ remainder $6$ (gives $0.3$). $8$ into $60$ is $7$ remainder $4$ (gives $0.37$). $8$ into $40$ is $5$ remainder $0$ (gives $0.375$). Done — a terminating decimal.

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

Add zeros as needed

Treat $3$ as $3.000$ — you can add zeros after the decimal point.

Stop when remainder is 0

That gives a terminating decimal.

Watch for repeating patterns

If a remainder repeats, the decimal recurs.

Recurring Decimals

+5 XP

Some fractions never give a clean terminating decimal. Their decimal form repeats forever — we mark this with a dot or bar.

Try $\tfrac{1}{3}$: $1 \div 3 = 0.333\ldots$. The 3 never stops. We write this as $0.\overline{3}$ (a bar over the repeating digit) or $0.\dot{3}$. For $\tfrac{1}{7}$, the pattern is longer: $0.\overline{142857}$ — six digits repeat. The bar marks where the cycle begins and ends.

$\tfrac{1}{3} = 0.\overline{3}, \quad \tfrac{1}{6} = 0.1\overline{6}, \quad \tfrac{1}{7} = 0.\overline{142857}$

Notation

$0.\overline{3}$ means $0.3333\ldots$ forever.

Round for practical use

For money, round $\tfrac{1}{3}$ to $0.33$.

Spot the trigger

Denominators with factors other than 2 and 5 give recurring decimals.

Watch Me Solve It · 3 examples

Watch Me Solve It · Compare a mark

+15 XP per step

PROBLEM

Your test score is $\tfrac{17}{20}$. Your friend got $0.88$. Who did better?

1

Convert $\tfrac{17}{20}$ to a decimal

$17 \div 20 = 0.85$

Long division: $17.00 \div 20$ gives $0.85$.
2

Compare

$0.85$ vs $0.88$

$0.88 > 0.85$.
3

State who did better

Friend: $0.88 = 88\%$. You: $0.85 = 85\%$.

Friend by $3$ percentage points.

AnswerYour friend ($88\%$ vs $85\%$).

Watch Me Solve It · Percentage to fraction

+15 XP per step

PROBLEM

Convert $36\%$ to a fraction in simplest form.

1

Write percentage as $\tfrac{?}{100}$

$36\% = \tfrac{36}{100}$

Every $\%$ is just “over 100”.
2

Find the HCF of 36 and 100

HCF$(36, 100) = 4$

Both divisible by 4.
3

Divide top and bottom by HCF

$\tfrac{36 \div 4}{100 \div 4} = \tfrac{9}{25}$

Cannot be simplified further.

Answer$36\% = \tfrac{9}{25}$

Watch Me Solve It · Recurring decimal

+15 XP per step

PROBLEM

Convert $\tfrac{1}{6}$ to a decimal. Write your answer using recurring notation.

1

Set up the division

$1.0000 \div 6$

Add zeros after the decimal point.
2

Long division

$10\div 6 = 1$ r $4$; $40 \div 6 = 6$ r $4$; $40 \div 6 = 6$ r $4$…

The remainder $4$ keeps repeating, so the $6$ keeps repeating.
3

Write with bar notation

$\tfrac{1}{6} = 0.1\overline{6}$

The bar shows the $6$ repeats forever.

Answer$\tfrac{1}{6} = 0.1\overline{6} = 0.1666\ldots$

Common Pitfalls

heads-up

Dividing the wrong way

For $\tfrac{5}{8}$, dividing $8 \div 5 = 1.6$ is wrong.

Fix: TOP $\div$ BOTTOM always. $\tfrac{5}{8} = 5 \div 8 = 0.625$.

Stopping too soon

For $\tfrac{1}{3}$, students may write $0.3$ and stop.

Fix: Keep going. If the remainder repeats, it's a recurring decimal: $0.\overline{3}$.

Not simplifying the final fraction

Writing $25\% = \tfrac{25}{100}$ as final answer.

Fix: Always simplify. $\tfrac{25}{100} = \tfrac{1}{4}$.

Copy Into Your Books

Fraction $\to$ Decimal

$\tfrac{a}{b} = a \div b$ (long division)
$\tfrac{3}{8} = 0.375$
$\tfrac{1}{3} = 0.\overline{3}$

Decimal $\to$ Percentage

Multiply by 100
$0.625 \to 62.5\%$
$0.05 \to 5\%$

Percentage $\to$ Fraction

Place over 100
Simplify by HCF
$36\% = \tfrac{36}{100} = \tfrac{9}{25}$

Recurring Notation

$0.\overline{3} = 0.333\ldots$
$0.1\overline{6}$ means the 6 repeats
$0.\overline{142857}$ — whole block repeats

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Converting Between FDP

4 problems

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

1 Convert $\tfrac{5}{8}$ to a decimal.

$5 \div 8 = 0.625$.$0.625$
2 Convert $0.45$ to a fraction in simplest form.

$0.45 = \tfrac{45}{100} = \tfrac{9}{20}$.$\tfrac{9}{20}$
3 Express $\tfrac{1}{9}$ as a recurring decimal.

$1 \div 9 = 0.\overline{1}$.$0.\overline{1}$
4 Which is larger: $\tfrac{7}{20}$ or $0.36$?

$\tfrac{7}{20} = 0.35$; $0.36 > 0.35$.$0.36$

Complete in your workbook.

Quick Check · 5 questions

$\tfrac{7}{8}$ as a decimal is:

+10 XP

$65\%$ as a fraction in simplest form is:

+10 XP

Which fraction gives a recurring decimal?

+10 XP

$0.12$ as a percentage and a fraction is:

+10 XP

$\tfrac{2}{3}$ as a percentage (to 1 dp) is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Convert each to a decimal (use bar notation if it recurs): (a) $\tfrac{9}{16}$ (b) $\tfrac{5}{6}$ (c) $\tfrac{11}{25}$

Answer in your workbook.

Understand Easy 2 MARKS

Q7. Mia ate $\tfrac{5}{12}$ of a pizza. Jay ate $42\%$. Who ate more, and by how many percentage points?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. (a) Convert $\tfrac{1}{7}$ to a recurring decimal. (b) Explain why the decimal expansion of $\tfrac{1}{7}$ has exactly 6 repeating digits, not more, not fewer. (Hint: think about the possible remainders when you divide by 7.)

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $7 \div 8 = 0.875$.

2. C — $\tfrac{65}{100} = \tfrac{13}{20}$.

3. D — $\tfrac{1}{6} = 0.1\overline{6}$.

4. C — $0.12 = 12\% = \tfrac{3}{25}$.

5. C — $\tfrac{2}{3} \approx 66.7\%$.

Show Your Working Model Answers

Q6 (3 marks): (a) $9 \div 16 = 0.5625$ [1]. (b) $5 \div 6 = 0.8\overline{3}$ [1]. (c) $11 \div 25 = 0.44$ [1].

Q7 (2 marks): $\tfrac{5}{12} = 5 \div 12 = 0.41\overline{6} \approx 41.7\%$ [1]. Jay ate more ($42\%$ vs $41.7\%$), by about $0.3$ percentage points [1].

Q8 (4 marks): (a) $1 \div 7 = 0.\overline{142857}$ [1]. (b) When dividing by 7, the possible non-zero remainders are 1, 2, 3, 4, 5, 6 — exactly 6 values [1]. Once a remainder repeats, the decimal cycle restarts [1]. Because all 6 remainders appear before any repeats, the cycle is exactly 6 digits long [1].

Stretch Challenge · +25 XP, +10 coins

The Conversion Chain

A fraction $\tfrac{a}{b}$ is in simplest form. When converted to a percentage and rounded to the nearest whole, it equals $58\%$. When converted to a decimal, it begins $0.58\overline{3}$. Find $a$ and $b$.

Reveal solution

$0.58\overline{3}$ is the recurring decimal for $\tfrac{7}{12}$ (check: $7 \div 12 = 0.583\overline{3}$). So $a = 7$, $b = 12$. As a percentage: $58.\overline{3}\% \approx 58\%$.

Quick Review

Top $\div$ bottom

Numerator divided by denominator gives the decimal

$\times 100$

Decimal to percentage

$\div 100$ + simplify

Percentage to fraction

Recurring

Bar notation: $0.\overline{3}, \;0.\overline{142857}$

HCF saves time

Simplify in one step using HCF

Always check

Reasonable size? $0.5 \to 50\%$, not $5\%$

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