IGCSE Class 5 > Class 5 Notes > Year 5 Mathematics (Cambridge) > Chapter Notes: Fractions and Percentages
Chapter Notes: Fractions and Percentages
Percentages
A percentage represents a proportion of a whole, expressed as parts per hundred (percent means "per hundred").Each part of a whole divided into 100 equal parts is 1% = 1/100.
Percentages can be written as fractions with a denominator of 100:
- Example: 20% = 20/100 = 1/5.
- Example: 25% = 25/100 = 1/4.
- Example: 60% = 60/100 = 3/5.
- Example: 15% = 15/100 = 3/20.
Equivalence between fractions and percentages:
- Example: 1/10 = 10/100 = 10%.
- Example: 7/100 = 7% = 0.07.
A 10x10 grid (100 squares) represents one whole:
- Each small square is 1/100 = 1%.
- Ten small squares are 10/100 = 10%.
Equivalent Fractions, Decimals and Percentages
Fractions, decimals, and percentages can represent the same proportion of a whole.
Example: In a picture with 100 symbols:
- Each symbol is 1/100 = 1% = 0.01.
- 10 symbols are 10/100 = 1/10 = 10% = 0.1.
Converting between forms:
- Fraction to decimal: 1/10 = 1 ÷ 10 = 0.1.
- Percentage to fraction: 10% = 10/100 = 1/10.
- Percentage to decimal: 10% = 10/100 = 0.1.
- Example: 50% = 50/100 = 1/2 = 0.5.
- Example: 30% = 30/100 = 3/10 = 0.3.
Comparing and Ordering Quantities
Quantities can be compared using symbols: < (less than), > (greater than), or = (equal to).Comparing fractions, decimals, and percentages requires converting to a common form:
Example: Compare 0.9 and 50%:
- 0.9 = 90/100 = 90%.
- 50% = 50/100 = 0.5.
- 0.9 > 50%.
- Example: 3/10 < 50% (since 3/10 = 30/100 = 30%, and 30% < 50%).
- Example: 0.4 > 1/10 (since 1/10 = 0.1, and 0.4 > 0.1).
- Example: 0.8 > 0.7.
- Example: 20% < 90% < 40% < 80% (ordering percentages directly).
Ordering quantities from smallest to largest:
- Example: Order 3/6, 4/6, 1/6, 5/6 becomes 1/6, 3/6, 4/6, 5/6.
- Example: Order 3/8, 7/8, 4/8, 2/8 becomes 2/8, 3/8, 4/8, 7/8.
- Example: Order 0.2, 0.5, 0.3, 0.7 becomes 0.2, 0.3, 0.5, 0.7.
- Example: Order 1.6, 0.9, 1.3, 0.6, 1.9 becomes 0.6, 0.9, 1.3, 1.6, 1.9.
Example: Order 0.2, 4/10, 0.8, 50%, 1/10, 90%, 1/10requires converting to decimals or percentages:
- 0.2 = 20%, 4/10 = 0.4 = 40%, 0.8 = 80%, 50% = 0.5, 1/10 = 0.1 = 10%, 90% = 0.9.
- Order: 1/10 (10%), 0.2 (20%), 4/10 (40%), 50% (50%), 0.8 (80%), 90% (90%).
Fractions as Operators
Fractions act as operators to find a portion of a quantity.Methods to calculate a fraction of a quantity:
Method 1: Divide by the denominator, then multiply by the numerator.
- Example: 7/10 of 400 = (400 ÷ 10) × 7 = 40 × 7 = 280.
Method 2: Multiply by the numerator, then divide by the denominator.
- Example: 7/10 of 400 = (400 × 7) ÷ 10 = 2800 ÷ 10 = 280.
Finding the whole from a fraction:
- Example: If 1/10 of a box = 60 cents, the whole box is 60 × 10 = 600 cents.
- Example: If 1/2 of a kg of onions = 75 cents, 1 kg costs 75 × 2 = 150 cents.
- Example: If 1/3 of a kg of carrots = 30 cents, 1 kg costs 30 × 3 = 90 cents.
- Example: If 1/8 of a box of oranges = 60 cents, the whole box costs 60 × 8 = 480 cents.
Using place value to find fractions of amounts:
Example: For 1/100 of 1 kg of spice costing $30:
- $30 ÷ 100 = $0.30 = 30 cents.
Ratio and Proportion
Proportion compares a part to the whole, expressed as a fraction or percentage.Example: In a group of 10 fruits with 6 oranges:
- Proportion of oranges: 6/10 = 3/5 = 60%.
- Ratio compares two quantities using the phrase "for every."
- Example: For every 1 box of limes, there are 3 boxes of lemons (ratio 1:3).
- Example: For every 2 boxes of grapes, there are 3 boxes of oranges (ratio 2:3).
Proportions in a market with 100 boxes of fruit:
- Apples: 20/100 = 20% = 1/5.
- Bananas: 50/100 = 50% = 1/2.
- Cherries: 10/100 = 10% = 1/10.
- Plums: 20/100 = 20% = 1/5.
The document Chapter Notes: Fractions and Percentages is a part of the Class 5 Course Year 5 Mathematics IGCSE (Cambridge).
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FAQs on Chapter Notes: Fractions and Percentages
| 1. What are the different ways to convert a fraction to a decimal? |  |
Ans. To convert a fraction to a decimal, you can divide the numerator (the top number) by the denominator (the bottom number). For example, to convert \( \frac{3}{4} \) to a decimal, divide 3 by 4, which equals 0.75. You can also use equivalent fractions to find a decimal by converting the fraction to a fraction with a denominator of 10, 100, or 1000 and then interpreting it as a decimal.
| 2. How do you calculate the percentage of a number using fractions? | |
Ans. To calculate the percentage of a number using fractions, first convert the fraction to a decimal and then multiply by 100. For instance, if you want to find 25% of 80, convert 25% to a fraction \( \frac{25}{100} \) or as a decimal, 0.25. Then, multiply: \( 0.25 \times 80 = 20 \). Thus, 25% of 80 is 20.
| 3. What is the relationship between fractions, decimals, and percentages? | |
Ans. Fractions, decimals, and percentages are different ways to express the same concept of a part of a whole. A fraction shows a part relative to a whole (e.g., \( \frac{1}{2} \)), a decimal represents the same part in a base-10 system (0.5), and a percentage is a fraction out of 100 (50%). They can be converted into one another through multiplication or division.
| 4. How can I compare and order fractions quickly? | |
Ans. To compare and order fractions, you can convert them to a common denominator, which allows you to easily see which fraction is larger or smaller. Alternatively, you can convert the fractions to decimals and then compare those. For example, \( \frac{1}{3} \approx 0.33 \) and \( \frac{1}{4} = 0.25 \), so \( \frac{1}{3} > \frac{1}{4} \).
| 5. What is the difference between ratio and proportion? | |
Ans. A ratio is a comparison of two quantities expressed as a fraction (e.g., the ratio of boys to girls in a class). Proportion, on the other hand, states that two ratios are equal (e.g., if there are 2 boys for every 3 girls in one class, and 4 boys for every 6 girls in another class, these two ratios are in proportion). Proportions can be solved using cross-multiplication.
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