CIE IGCSE Mathematics (0580) Fractions, decimals and percentages Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Fractions, decimals and percentages Study Notes
LEARNING OBJECTIVE
- Fractions, decimals and percentages
Key Concepts:
proper fractions
improper fractions
mixed numbers
decimals
percentages
- Conversion
Fractions, Decimals, and Percentages
Fractions, Decimals, and Percentages
Proper Fractions
A proper fraction is a fraction where the numerator (top number) is less than the denominator (bottom number). The value of a proper fraction is always less than 1.
Examples: \( \frac{1}{2}, \quad \frac{3}{4}, \quad \frac{5}{8} \)
Improper Fractions
An improper fraction has a numerator greater than or equal to the denominator. The value is greater than or equal to 1.
Examples: \( \frac{5}{4}, \quad \frac{9}{7}, \quad \frac{6}{3} \)
Mixed Numbers
A mixed number consists of a whole number part and a proper fraction part. It is another way of writing an improper fraction.
Example: \( 1\frac{1}{2} = \frac{3}{2}, \quad 2\frac{3}{4} = \frac{11}{4} \)
To convert a mixed number to an improper fraction: Multiply the whole number by the denominator and add the numerator. Keep the same denominator.
Decimals
A decimal is another way to represent parts of a whole using base-10 place value. The decimal point separates whole numbers from fractional parts.
Examples: \( 0.25 = \frac{1}{4}, \quad 0.5 = \frac{1}{2}, \quad 1.75 = 1\frac{3}{4} \)
Decimals can be:
- Terminating (e.g. \( 0.5, 0.75 \))
- Recurring (e.g. \( 0.\overline{3}, 1.\overline{6} \))
Percentages
A percentage is a fraction out of 100, shown with the “%” symbol. It is useful for comparing proportions.
Examples: \( \frac{1}{2} = 0.5 = 50\%, \quad \frac{3}{4} = 0.75 = 75\% \)
Conversions Between Forms
You must be able to convert between:
- Fractions ↔ Decimals
- Fractions ↔ Percentages
- Decimals ↔ Percentages
Useful facts: \( \frac{1}{4} = 0.25 = 25\%, \quad \frac{1}{5} = 0.2 = 20\%, \quad \frac{1}{8} = 0.125 = 12.5\% \)
Example:
Convert \( \frac{3}{4} \) into a decimal and a percentage.
▶️ Answer/Explanation
Step 1: Convert to decimal:
\( \frac{3}{4} = 0.75 \)
Step 2: Convert to percentage:
\( 0.75 \times 100 = 75\% \)
Final Answer: \( \frac{3}{4} = 0.75 = 75\% \)
Example:
Convert \( 0.6 \) into a fraction and a percentage.
▶️ Answer/Explanation
Step 1: Convert to fraction:
\( 0.6 = \frac{6}{10} = \frac{3}{5} \)
Step 2: Convert to percentage:
\( 0.6 \times 100 = 60\% \)
Final Answer: \( 0.6 = \frac{3}{5} = 60\% \)
Example:
Convert \( 35\% \) into a fraction and a decimal.
▶️ Answer/Explanation
Step 1: Convert to decimal:
\( 35\% = \frac{35}{100} = 0.35 \)
Step 2: Convert to fraction (simplify):
\( \frac{35}{100} = \frac{7}{20} \)
Final Answer: \( 35\% = \frac{7}{20} = 0.35 \)
Example:
In a school of 500 students, 60% are boys. Of the boys, \( \frac{2}{5} \) play football. How many boys play football? Give your answer as a number and as a percentage of the whole school.
▶️ Answer/Explanation
Step 1: Find the number of boys.
\( \text{Boys} = 60\% \text{ of } 500 = \frac{60}{100} \times 500 = 300 \)
Step 2: Find how many boys play football.
\( \text{Football players} = \frac{2}{5} \times 300 = 120 \)
Step 3: Express 120 as a percentage of the whole school.
\( \frac{120}{500} \times 100 = 24\% \)
Final Answer: 120 boys play football, which is 24% of the school population.
Recognising Equivalence and Conversion Between Forms
Recognising Equivalence and Conversion Between Forms
Equivalence
Different number formats (fractions, decimals, percentages, mixed numbers) can represent the same value. Recognising equivalent values is essential in simplifying, comparing, and solving problems.
For example: \( \frac{1}{2} = 0.5 = 50\%, \quad \frac{3}{4} = 0.75 = 75\% \)
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator.
Example: \( \frac{3}{4} = 3 \div 4 = 0.75 \)
Converting Decimals to Fractions
To convert a decimal to a fraction, write it over the appropriate power of 10 and simplify.
Example: \( 0.6 = \frac{6}{10} = \frac{3}{5} \)
Converting Fractions to Percentages
Multiply the fraction by 100 and add the % symbol.
Example: \( \frac{3}{5} \times 100 = 60\% \)
Converting Percentages to Fractions
Write the percentage over 100 and simplify the fraction.
Example: \( 60\% = \frac{60}{100} = \frac{3}{5} \)
Converting Percentages to Decimals
Divide the percentage by 100.
Example: \( 25\% = \frac{25}{100} = 0.25 \)
Converting Decimals to Percentages
Multiply the decimal by 100 and add the % symbol.
Example: \( 0.4 \times 100 = 40\% \)
Converting Improper Fractions to Mixed Numbers
Divide the numerator by the denominator. The quotient becomes the whole number part, and the remainder becomes the new numerator.
Example: \( \frac{11}{4} = 2\frac{3}{4} \)
Converting Mixed Numbers to Improper Fractions
Multiply the whole number by the denominator and add the numerator.
Example: \( 2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4} \)
Example:
Match the equivalent forms:
- A. \( \frac{1}{5} \)
- B. \( 0.6 \)
- C. \( \frac{2}{3} \)
- D. \( 40\% \)
- 1. \( 0.666\ldots \)
- 2. \( \frac{2}{5} \)
- 3. \( 0.2 \)
- 4. \( \frac{3}{5} \)
▶️ Answer/Explanation
Step 1: Convert each option to decimals:
- A. \( \frac{1}{5} = 0.2 \) → matches 3
- B. \( 0.6 = \frac{3}{5} \) → matches 4
- C. \( \frac{2}{3} = 0.\overline{6} \) → matches 1
- D. \( 40\% = 0.4 = \frac{2}{5} \) → matches 2
Final Matching:
- A → 3
- B → 4
- C → 1
- D → 2
Example:
Complete the following table by converting between forms:
| Fraction | Decimal | Percentage |
|---|---|---|
| \( \frac{4}{5} \) | ? | ? |
| ? | 0.2 | ? |
| ? | ? | 75% |
▶️ Answer/Explanation
First row: \( \frac{4}{5} = 0.8 = 80\% \)
Second row: \( 0.2 = \frac{1}{5} = 20\% \)
Third row: \( 75\% = 0.75 = \frac{3}{4} \)
Final Completed Table:
| Fraction | Decimal | Percentage |
|---|---|---|
| \( \frac{4}{5} \) | 0.8 | 80% |
| \( \frac{1}{5} \) | 0.2 | 20% |
| \( \frac{3}{4} \) | 0.75 | 75% |