Working With Fdp

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Notes

Converting Fractions, Decimals & Percentages

**Percentage to decimal**: Divide by 100 (move digits two places right). $E.g., 6\% = 0.06, 350\% = 3.5$ .
**Decimal to percentage**: Multiply by 100 (move digits two places left, add %). $E.g., 0.35 = 35\%, 0.004 = 0.4\%$ .
**Decimal to fraction**: Write digits over 10ⁿ $(n =$ decimal places). $E.g., 0.3 = \frac{3}{10}, 0.07 = \frac{7}{100}, 0.513 = \frac{513}{1000}$ .
**Percentage to fraction**: Write over 100. $E.g., 37\% = \frac{37}{100}$ .
**Fraction to decimal**: Convert to fraction over power of 10. $E.g., \frac{3}{5} = \frac{6}{10} =$ 0.6; $\frac{7}{20} = \frac{35}{100} = 0.35$ .
**Fraction to percentage**: Convert to decimal then multiply by 100. $E.g., \frac{4}{5} = 0.8 = 80\%$ .
**Learn simple recurring decimals**: 0.333… $= \frac{1}{3}$ , 0.666… $= \frac{2}{3}$ .

Recurring Decimals

**Recurring decimal**: A decimal with a repeating pattern; e.g., 0.323232… = 0.\dot{3}2\dot{3}.
**Notation**: Dot over single repeating digit; dots over first and last of repeating block (e.g., 0.\dot{1}\dot{2} for 0.121212…).
**Converting to fraction**: Set $x =$ decimal, multiply by 10ⁿ until two lines have same repeating part, subtract, solve for x.
**Worked example**: 0.\dot{3}0\dot{7} → $x =$ 0.307307…, $1000x =$ 307.307…, subtract: $999x = 307$ , so $x = \frac{307}{999}$ .
**Always simplify** fraction to lowest terms.

Ordering Fractions, Decimals & Percentages

**Ordering fractions only**: Write all with a **common denominator**, then compare numerators.
**Example**: $\frac{3}{5}, \frac{1}{2}, \frac{13}{20}, \frac{7}{12}$ → common denominator 60 → $\frac{36}{60}, \frac{30}{60}, \frac{39}{60}, \frac{35}{60}$ → order: $\frac{1}{2}, \frac{7}{12}, \frac{3}{5}, \frac{13}{20}$ .
**Ordering mixed FDP**: Convert everything to **decimals** (or percentages) for easy comparison.
**Example**: $\frac{7}{8}, \frac{5}{6}, 0.8, 78\%$ → decimals: 0.875, 0.833…, 0.8, 0.78 → order: $78\% < 0.8 < \frac{5}{6} < \frac{7}{8}$ .
**Use symbols**: $< (less than), > (greater than), \le , \ge , =, \ne$ .

Common Conversions & Tips

**Memorise**: $\frac{1}{2} = 0.5 = 50\%, \frac{1}{4} = 0.25 = 25\%, \frac{3}{4} = 0.75 = 75\%, \frac{1}{5} = 0.2 = 20\%, \frac{1}{10} = 0.1 = 10\%$ .
**Calculator tip**: Use calculator to check conversions even in non-calculator questions (for verification).
**Non-calculator tip**: For fractions, use short division or halving $(e.g., \frac{7}{8} =$ half of $\frac{7}{4}, etc.)$ .

FDP Conversion Flowchart

Recurring Decimal to Fraction Method

Ordering FDP: Convert to Decimals

Common FDP Equivalents

Practice questions

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1.Which of the following is the decimal equivalent of $\frac{3}{5}$ ?
Easy
- A0.35
- B0.6
- C1.67
2.Write 0.45 as a percentage.
Easy
- A4.5%
- B45%
- C0.45%
- D450%
3.Convert 72% to a fraction in its simplest form.
Easy
- A $\frac{72}{100}$
- B $\frac{18}{25}$
- C $\frac{36}{50}$
- D $\frac{7.2}{10}$
4.Which number has the largest value?
Easy
- A0.3030
- B $\frac{1}{3}$
- C0.0330
- D $\frac{3}{10}$
- E33%
5.Mr Mason asks 240 Year 11 students what they want to do next year. 15% want to go to college, $\frac{3}{4}$ want to stay at school, and the rest do not know. How many students do not know?
Medium
- A24
- B36
- C60
- D12
6.Karen got 32 out of 80 in a maths test and 38% in an English test. Did she get a higher percentage in maths or English?
Medium
- AMaths
- BEnglish
- CSame
- DCannot tell
7.Prove algebraically that the recurring decimal 0.2̇5 (0.25555...) equals which fraction?
Hard
- A $\frac{23}{90}$
- B $\frac{25}{99}$
- C $\frac{23}{99}$
- D $\frac{25}{90}$
8.Show that the recurring decimal 0.1̇7 (0.17777...) equals which fraction?
Hard
- A $\frac{8}{45}$
- B $\frac{17}{99}$
- C $\frac{16}{90}$
- D $\frac{8}{90}$

Summary

## Working with FDP – Revision Notes ### Converting Fractions, Decimals & Percentages - **Percentage to decimal**: Divide by 100 (move digits two places right). - 6% = 0.06, 40% = 0.4, 350% = 3.5, 0.2% = 0.002 - **Decimal to percentage**: Multiply by 100 (move digits two places left, add %). - 0.35 = 35%, 1.32 = 132%, 0.004 = 0.4% - **Decimal to fraction**: Write digits over 10^n (n = decimal places). - 0.3 = 3/10, 0.07 = 7/100, 0.513 = 513/1000 - **Percentage to fraction**: Write over 100. - 37% = 37/100 - **Fraction to decimal**: Convert to denominator 10, 100, 1000… - 3/5 = 6/10 = 0.6, 7/20 = 35/100 = 0.35 - **Fraction to percentage**: Convert to decimal then multiply by 100. - 4/5 = 0.8 → 80% ### Recurring Decimals - **Definition**: A decimal with a repeating pattern (e.g., 0.333… = 0.3̇). - **Converting to fraction**: 1. Let x = recurring decimal (e.g., x = 0.353535…) 2. Multiply by 10 repeatedly until two lines have same repeating part. 3. Subtract those lines: 100x – x = 35.3535… – 0.3535… → 99x = 35 4. Solve: x = 35/99 (simplify if needed). - **Common ones**: 0.3̇ = 1/3, 0.6̇ = 2/3. ### Ordering FDP - **All fractions**: Write over common denominator, compare numerators. - **Mixed FDP**: Convert everything to decimals, then order. - **Symbols**: < (less than), > (greater than), ≤, ≥, =, ≠. ### Key Tips - Use a calculator to check conversions (if allowed). - For ordering, write decimals to same number of places.

Key terms

Recurring decimal: A decimal with a repeating pattern of digits, e.g., 0.333… = 0.3̇. It is rational and can be written as a fraction.
Terminating decimal: A decimal that ends, e.g., 0.25.
Lowest common denominator: The smallest number that is a multiple of all denominators in a set of fractions, used to compare or add them.

Slides

Converting Between Fractions, Decimals & Percentages

To convert a percentage to a decimal, divide by $100 (e.g., 6\% = 0.06)$ .
To convert a decimal to a percentage, multiply by $100 (e.g., 0.35 = 35\%)$ .
To convert a decimal to a fraction, write digits over a power of $10 (e.g., 0.3 = \frac{3}{10})$ .
To convert a percentage to a fraction, write over $100 (e.g., 37\% = \frac{37}{100})$ .

What is 40% as a decimal?

0.04
0.4
4.0
40.0

Converting Fractions to Decimals and Percentages

To convert a fraction to a decimal, write it with denominator 10, 100, etc. $(e.g., \frac{3}{5} = \frac{6}{10} = 0.6)$ .
To convert a fraction to a percentage, first convert to decimal then multiply by $100 (e.g., \frac{4}{5} = 0.8 = 80\%)$ .

What is $\frac{7}{20}$ as a decimal?

0.35
0.7
0.07
3.5

Recurring Decimals

A recurring decimal has a repeating pattern (e.g., 0.333... = 0.3̇).
Recurring decimals are rational numbers and can be written as fractions.
To convert a recurring decimal to a fraction, set $x =$ decimal, multiply to align repeating parts, subtract, and solve.

To convert a recurring decimal to a fraction, set $x =$ decimal, multiply by a power of 10 to align $the ____$ parts, then subtract.

Example: Converting 0.3̇07̇ to a Fraction

Let $x = 0.307307307$ ...
Multiply by $1000: 1000x = 307.307307$ ...
Subtract: $1000x - x = 307$ → $999x = 307$ → $x = \frac{307}{999}$ .

If $x = 0.353535$ ..., what is 100x?

3.53535...
35.3535...
353.535...
0.353535...

Ordering Fractions, Decimals & Percentages

To order fractions only, write them with a common denominator.
To order a mix of FDP, convert everything to decimals.
Use symbols like $<, >, \le , \ge to$ compare.

Which of these are true?

$\frac{1}{4} < \frac{1}{3}$
$0.8 > 78\%$
$\frac{5}{6} = 0.8333$ ...

Worked Example: Ordering Fractions

Order $\frac{1}{3}, \frac{2}{5}, \frac{9}{25}, \frac{4}{15}$ from smallest to largest.
Common denominator $75: \frac{25}{75}, \frac{30}{75}, \frac{27}{75}, \frac{20}{75}$ .
Order: $\frac{20}{75}, \frac{25}{75}, \frac{27}{75}, \frac{30}{75}$ → $\frac{4}{15}, \frac{1}{3}, \frac{9}{25}, \frac{2}{5}$ .

To order $\frac{7}{8}, \frac{5}{6}, 0.8, 78\%$ , convert all $to ____$ .

Worked Example: Ordering Mixed FDP

Convert: $\frac{7}{8} = 0.875, \frac{5}{6}$ ≈ $0.8333, 0.8 = 0.8, 78\% = 0.78$ .
Order: 0.78, 0.8, 0.8333, 0.875.
Original: $78\% < 0.8 < \frac{5}{6} < \frac{7}{8}$ .

Which is the smallest? $\frac{3}{5}, 0.6, 62\%, \frac{5}{8}$

$\frac{3}{5}$
0.6
62%
$\frac{5}{8}$

Recurring Decimals: Common Patterns

0.333... $= \frac{1}{3}, 0.666$ ... $= \frac{2}{3}$ .
0.1̇7̇ $= \frac{17}{99}$ , 0.2̇5̇ $= \frac{25}{99}$ .
Always simplify fractions if possible.

What is 0.3̇8̇ as a fraction?

$\frac{38}{99}$
$\frac{38}{100}$
$\frac{7}{18}$
$\frac{19}{50}$

Using Algebra for Recurring Decimals

For 0.2̇6̇: let $x = 0.262626..., 100x = 26.2626$ ..., subtract: $99x = 26, x = \frac{26}{99} = \frac{4}{15}$ .
For 0.1̇7̇: $x = 0.171717..., 100x = 17.1717..., 99x = 17, x = \frac{17}{99}$ .

If $x =$ 0.3̇6̇, then $100x = 36.3636$ ..., so $99x = ____$ .

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