Fractions Decimals And Percentages

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Notes

Converting Fractions, Decimals & Percentages

To convert a **percentage to a decimal**, divide by 100 (move digits two places right). $E.g., 6\% = 0.06$ .
To convert a **decimal to a percentage**, multiply by 100 (move digits two places left and add %). $E.g., 0.35 = 35\%$ .
To convert a **decimal to a fraction**: for 1 decimal place, write over 10; for 2 decimal places, over 100; for n decimal places, over 10ⁿ. Simplify if needed.
To convert a **percentage to a fraction**, write the percentage over 100 and simplify. $E.g., 37\% = \frac{37}{100}$ .
To convert a **fraction to a decimal**, rewrite with denominator 10, 100, 1000, etc., or use division. $E.g., \frac{3}{5} = \frac{6}{10} = 0.6$ .
To convert a **fraction to a percentage**, first convert to decimal then multiply by 100. $E.g., \frac{4}{5} = 0.8 = 80\%$ .
Common recurring decimals: 0.333... $= \frac{1}{3}, 0.666$ ... $= \frac{2}{3}$ .

Ordering Fractions, Decimals & Percentages

To order **only fractions**, write them with a common denominator (lowest common denominator) and compare numerators.
To order a **mixture** of fractions, decimals, and percentages, convert everything to decimals first.
Use symbols: $< (less than), > (greater than), \le (less$ than or $equal), \ge (greater$ than or $equal), = (equal), \ne (not equal)$ .
When ordering, write all numbers with the same number of decimal places to compare easily.
Example: $78\% < 0.8 < \frac{5}{6} < \frac{7}{8} (after$ converting all to decimals).

Common Conversions to Memorise

$\frac{1}{2} = 0.5 = 50\%$
$\frac{1}{4} = 0.25 = 25\%$
$\frac{3}{4} = 0.75 = 75\%$
$\frac{1}{5} = 0.2 = 20\%$
$\frac{2}{5} = 0.4 = 40\%$
$\frac{1}{10} = 0.1 = 10\%$
$\frac{1}{3}$ ≈ $0.333 = 33.3\%$

Tips for Exam Questions

A calculator can be used to check conversions between fractions and decimals (even if the question says 'without a calculator').
For ordering, convert all values to decimals (using calculator if allowed) then compare.
When shading a grid for a percentage, count total squares and shade the given percentage of them.
Always simplify fractions to lowest terms $(e.g., \frac{36}{124} = \frac{9}{31})$ .
To find a fraction of a quantity, multiply the quantity by the fraction $(e.g., \frac{5}{8}$ of $128 = 80)$ .

Worked Example: Ordering Fractions Only

Order $\frac{1}{3}, \frac{2}{5}, \frac{9}{25}, \frac{4}{15}$ from smallest to largest.
Find LCM of denominators: LCM of 3,5,25,15 is 75.
Rewrite: $\frac{1}{3} = \frac{25}{75}, \frac{2}{5} = \frac{30}{75}, \frac{9}{25} = \frac{27}{75}, \frac{4}{15} = \frac{20}{75}$ .
Order by numerators: $\frac{20}{75}, \frac{25}{75}, \frac{27}{75}, \frac{30}{75}$ .
Original order: $\frac{4}{15}, \frac{1}{3}, \frac{9}{25}, \frac{2}{5}$ .

Worked Example: Ordering Mixed FDP

Order $\frac{7}{8}, \frac{5}{6}, 0.8, 78\%$ from smallest to largest.
Convert all to decimals: $\frac{7}{8} = 0.875, \frac{5}{6}$ ≈ $0.8333, 0.8 = 0.800, 78\% = 0.78$ .
Compare: 0.78, 0.800, 0.8333, 0.875.
Original order: $78\% < 0.8 < \frac{5}{6} < \frac{7}{8}$ .

FDP Conversion Flowchart

Common FDP Equivalents

Ordering Fractions with Common Denominator

Ordering Mixed FDP

Practice questions

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1.Write $\frac{2}{5}$ as a decimal.
Easy
- A0.4
- B0.25
- C0.5
- D0.2
2.Write 15% as a decimal.
Easy
- A0.15
- B1.5
- C0.015
- D15
3.Write $\frac{3}{4}$ as a percentage.
Easy
- A75%
- B34%
- C0.75%
- D7.5%
4.Write 0.45 as a fraction in its simplest form.
Easy
- A $\frac{9}{20}$
- B $\frac{45}{100}$
- C $\frac{9}{10}$
- D $\frac{4}{5}$
5.Write $\frac{3}{10}$ as a percentage.
Easy
- A30%
- B3%
- C0.3%
- D13%
6.Write 26% as a decimal.
Easy
- A0.26
- B2.6
- C0.026
- D26
7.Write 0.48 as a fraction in its simplest form.
Easy
- A $\frac{12}{25}$
- B $\frac{48}{100}$
- C $\frac{12}{50}$
- D $\frac{24}{25}$
8.Write these numbers in order, starting with the smallest: $\frac{3}{4}, 72\%, 0.78, \frac{8}{10}$ .
Medium
- A $72\%, 0.78, \frac{3}{4}, \frac{8}{10}$
- B $72\%, \frac{3}{4}, 0.78, \frac{8}{10}$
- C $0.78, 72\%, \frac{3}{4}, \frac{8}{10}$
- D $\frac{3}{4}, 72\%, 0.78, \frac{8}{10}$

Summary

## Fractions, Decimals & Percentages ### Converting - **Percentage → Decimal:** ÷ 100 (move digits 2 places right). e.g., 6% = 0.06, 40% = 0.4, 350% = 3.5, 0.2% = 0.002 - **Decimal → Percentage:** × 100 (move digits 2 places left, add %). e.g., 0.35 = 35%, 1.32 = 132%, 0.004 = 0.4% - **Decimal → Fraction:** Write digits over 10^n (n = decimal places). e.g., 0.3 = 3/10, 0.07 = 7/100, 0.513 = 513/1000 - **Percentage → Fraction:** Write over 100. e.g., 37% = 37/100 - **Fraction → Decimal:** Convert to fraction over power of 10. e.g., 3/5 = 6/10 = 0.6, 7/20 = 35/100 = 0.35 - **Fraction → Percentage:** Convert to decimal then × 100. e.g., 4/5 = 0.8 = 80% ### Ordering - **All fractions:** Write with lowest common denominator, compare numerators. e.g., 3/5, 1/2, 13/20, 7/12 → LCD 60 → 36/60, 30/60, 39/60, 35/60 → order: 1/2, 7/12, 3/5, 13/20 - **Mixed FDP:** Convert everything to decimals, then compare. e.g., 7/8 = 0.875, 5/6 ≈ 0.8333, 0.8, 78% = 0.78 → order: 78% < 0.8 < 5/6 < 7/8 ### Key Symbols - < less than, > greater than, ≤ less than or equal, ≥ greater than or equal, ≠ not equal ### Tips - Use a calculator to check conversions (even if question says without). - For ordering, convert all to decimals (calculator can help).

Key terms

Fraction: A number expressed as a quotient a/b, where a is the numerator and b is the denominator.
Decimal: A number expressed in base 10 using a decimal point; e.g., 0.75.
Percentage: A fraction or ratio expressed as a number out of 100, denoted by the % sign.
Equivalent fraction: Fractions that represent the same value, e.g., 5/9 and 10/18.
Simplest form: A fraction where numerator and denominator have no common factors other than 1.
Lowest common denominator (LCD): The smallest multiple shared by the denominators of two or more fractions.

Slides

Converting Fractions, Decimals & Percentages

To convert a percentage to a decimal, divide by 100 (move digits two places right).
To convert a decimal to a percentage, multiply by 100 (move digits two places left) and add a % sign.
To convert a decimal to a fraction, write the digits over 10, 100, 1000 etc. depending on the number of decimal places.
To convert a fraction to a decimal, write it over a power of $10 (e.g., \frac{3}{5} = \frac{6}{10} = 0.6)$ .
To convert a fraction to a percentage, first convert to a decimal then multiply by 100.

Convert 15% to a decimal.

0.15
1.5
0.015
15.0

Converting Fractions to Decimals

If the denominator is a factor of 10, 100, 1000, write an equivalent fraction over that power of 10.
For example, $\frac{2}{5} = \frac{4}{10} =$ 0.4; $\frac{3}{4} = \frac{75}{100} =$ 0.75; $\frac{7}{20} = \frac{35}{100} = 0.35$ .
If not, use division $(numerator \div denominator)$ .

Write $\frac{2}{5}$ as a decimal: ____

Converting Decimals to Fractions

For one decimal place, write over 10. $E.g., 0.3 = \frac{3}{10}$ .
For two decimal places, write over 100. $E.g., 0.07 = \frac{7}{100}$ .
For three decimal places, write over 1000. $E.g., 0.513 = \frac{513}{1000}$ .
Always simplify the fraction if possible.

Write 0.45 as a fraction in its simplest form.

$\frac{45}{100}$
$\frac{9}{20}$
$\frac{45}{10}$
$\frac{9}{100}$

Converting Percentages to Fractions

Write the percentage over 100 and simplify if possible.
For example, $37\% =$ 37/100; $45\% = \frac{45}{100} =$ 9/20; $8\% = \frac{8}{100} = \frac{2}{25}$ .

Write 45% as a fraction in its simplest form: ____

Ordering Fractions, Decimals & Percentages

To compare a mix of fractions, decimals and percentages, convert everything to decimals.
Then compare the decimals (look at digits from left to right).
You can also convert all to fractions with a common denominator if only fractions are involved.

Which of these are true?

$0.8 < 78\%$
$78\% = 0.78$
$\frac{7}{8} = 0.875$
$\frac{5}{6} = 0.8333$ ...

Ordering Example

Example: Order $\frac{3}{4}, 72\%, 0.78, \frac{8}{10}$ from smallest.
Convert: $\frac{3}{4} = 0.75, 72\% = 0.72, 0.78 = 0.78, \frac{8}{10} = 0.8$ .
Order: 0.72, 0.75, 0.78, 0.8 → $72\%, \frac{3}{4}, 0.78, \frac{8}{10}$ .

Which is the smallest? $\frac{3}{4}, 72\%, 0.78, \frac{8}{10}$

$\frac{3}{4}$
72%
0.78
$\frac{8}{10}$

Using Symbols for Comparison

$Use < for 'less than', > for 'greater than', = for 'equal to'$ .
Example: $\frac{1}{4} < \frac{1}{3} < \frac{1}{2}$ .
When ordering, list from smallest to largest using < between them.

Order these from smallest: $0.51, \frac{1}{2}, 0.501, 48\%, \frac{2}{5}$ . Write $as: ____ < ____ < ____ < ____ < ____$

Common Recurring Decimals

0.333... $= \frac{1}{3}, 0.666$ ... $= \frac{2}{3}$ .
0.1666... $= \frac{1}{6}, 0.8333$ ... $= \frac{5}{6}$ .
These are useful to know for quick conversions.

What is 0.666... as a fraction?

$\frac{1}{3}$
$\frac{2}{3}$
$\frac{3}{4}$
$\frac{5}{6}$

Tips for Ordering

When ordering, write all numbers with the same number of decimal places (e.g., 3 d.p.) to compare easily.
A calculator can help convert fractions to decimals quickly.
Always double-check your order by converting back to original form.

Which of these are true?

$0.78 < 0.8$
$\frac{7}{8} > \frac{5}{6}$
$78\% > 0.8$
$\frac{5}{6} = 0.833$

Summary

You can convert between fractions, decimals and percentages.
To order a mix, convert all to decimals.
$Use <, >, = to$ compare and list in order.

Flashcards

How do you convert a percentage to a decimal?

Divide by 100 (move digits two places to the right). For example, $6\% = 0.06$ .

How do you convert a decimal to a percentage?

Multiply by 100 (move digits two places to the left and add a % sign). For example, $0.35 = 35\%$ .

Convert 0.3 to a fraction.

$0.3 = \frac{3}{10} (one$ decimal place, write over 10).

Convert 0.07 to a fraction.

$0.07 = \frac{7}{100} (two$ decimal places, write over 100).

Convert 0.513 to a fraction.

$0.513 = \frac{513}{1000} (three$ decimal places, write over 1000).

What is 0.33333... as a fraction?

0.33333... $= \frac{1}{3}$ .

What is 0.66666... as a fraction?

0.66666... $= \frac{2}{3}$ .

How do you convert a percentage to a fraction?

Write the percentage over 100. For example, $37\% = \frac{37}{100}$ .

Convert $\frac{3}{5}$ to a decimal.

$\frac{3}{5} = \frac{6}{10} = 0.6$ .

Convert $\frac{7}{20}$ to a decimal.

$\frac{7}{20} = \frac{35}{100} = 0.35$ .

Convert $\frac{4}{5}$ to a percentage.

$\frac{4}{5} = 0.8 = 0.8 \times 100 = 80\%$ .

Write $\frac{2}{5}$ as a decimal.

$\frac{2}{5} = 0.4$ .

Write 15% as a decimal.

$15\% = 0.15$ .

Write $\frac{3}{4}$ as a decimal.

$\frac{3}{4} = 0.75$ .

Write 0.45 as a fraction in its simplest form.

$0.45 = \frac{45}{100} = \frac{9}{20}$ .

Write $\frac{2}{5}$ as a percentage.

$\frac{2}{5} = 0.4 = 40\%$ .

Write $\frac{3}{10}$ as a percentage.

$\frac{3}{10} = 0.3 = 30\%$ .

Write 45% as a fraction in its simplest form.

$45\% = \frac{45}{100} = \frac{9}{20}$ .

How do you order a mixture of fractions, decimals, and percentages?

Convert everything to decimals, then compare. For example, $\frac{7}{8} = 0.875, \frac{5}{6}$ ≈ $0.833, 0.8, 78\% =$ 0.78; order: $78\% < 0.8 < \frac{5}{6} < \frac{7}{8}$ .

Order these numbers from smallest to largest: $\frac{3}{4}, 72\%, 0.78, \frac{8}{10}$ .

Convert: $\frac{3}{4} = 0.75, 72\% = 0.72, 0.78, \frac{8}{10} = 0.8$ . Order: $72\%, \frac{3}{4}, 0.78, \frac{8}{10}$ .

Order from smallest to largest: $0.51, \frac{1}{2}, 0.501, 48\%, \frac{2}{5}$ .

Convert: $\frac{1}{2} = 0.5, 48\% = 0.48, \frac{2}{5} = 0.4$ . Order: $\frac{2}{5}, 48\%, 0.501, 0.51, \frac{1}{2}$ .

Order from smallest to largest: $27\%, \frac{5}{17}, 0.268, \frac{7}{29}$ .

Convert: $27\% = 0.27, \frac{5}{17}$ ≈ $0.2941, 0.268, \frac{7}{29}$ ≈ 0.2414. Order: $\frac{7}{29}, 0.268, 27\%, \frac{5}{17}$ .

Order from smallest to largest: $\frac{11}{27}, 41\%, 0.4, \frac{16}{39}$ .

Convert: $\frac{11}{27}$ ≈ $0.4074, 41\% = 0.41, 0.4, \frac{16}{39}$ ≈ 0.4103. Order: $0.4, \frac{11}{27}, 41\%, \frac{16}{39}$ .

Which is largest: $0.3030, \frac{1}{3}, 0.0330, \frac{3}{10}, 33\%$ ?

Convert: $\frac{1}{3}$ ≈ $0.3333, \frac{3}{10} = 0.3, 33\% = 0.33$ . Largest is 1/3.

Order from smallest to largest: $\frac{9}{19}, \frac{3}{7}, 37\%, 0.43$ .

Convert: $\frac{9}{19}$ ≈ $0.4737, \frac{3}{7}$ ≈ $0.4286, 37\% = 0.37, 0.43$ . Order: $37\%, \frac{3}{7}, 0.43, \frac{9}{19}$ .

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