Understanding Percentages In 11-plus Mathematics, percentages are a crucial concept that represents a part of a whole expressed in hundredths. Understanding per...
Understanding Percentages
In 11-plus Mathematics, percentages are a crucial concept that represents a part of a whole expressed in hundredths. Understanding percentages is essential for solving various mathematical problems, including those involving fractions and decimals.
What is a Percentage?
A percentage is defined as 'parts per hundred'. For example, 25% means 25 out of 100 or 25 parts of a whole divided into 100 equal parts.
Converting Between Percentages, Fractions, and Decimals
To convert between these forms:
Percentage to Fraction: Divide the percentage by 100. For example, 40% = 40/100 = 2/5.
Percentage to Decimal: Divide the percentage by 100. For example, 75% = 75/100 = 0.75.
Fraction to Percentage: Multiply the fraction by 100. For example, 1/4 = 0.25, so 1/4 = 25%.
Decimal to Percentage: Multiply the decimal by 100. For example, 0.6 = 60%.
Finding Percentages of Amounts
To find a percentage of an amount, you can use the formula:
Percentage of Amount = (Percentage / 100) × Amount
Worked Example
Problem: What is 20% of £50?
Solution:
Convert 20% to a decimal: 20/100 = 0.2.
Multiply by the amount: 0.2 × £50 = £10.
Percentage Increase and Decrease
Percentage increase and decrease are used to calculate how much a value has grown or shrunk:
Percentage Increase:New Value - Original Value divided by Original Value × 100.
Percentage Decrease:Original Value - New Value divided by Original Value × 100.
Worked Example
Problem: A jacket costs £40 and is now £30. What is the percentage decrease?
Solution:
Decrease = £40 - £30 = £10.
Percentage Decrease = (£10 / £40) × 100 = 25%.
Solving Problems Involving Percentages
Common problems include calculating discounts, profit, and loss:
Discounts: If an item is on sale, find the discount amount and subtract it from the original price.
Profit: Profit is calculated as the selling price minus the cost price, expressed as a percentage of the cost price.
Loss: Loss is calculated as the cost price minus the selling price, expressed as a percentage of the cost price.
Worked Example
Problem: A book originally costs £20 and is sold for £15. What is the percentage loss?
Solution:
Loss = £20 - £15 = £5.
Percentage Loss = (£5 / £20) × 100 = 25%.
Understanding percentages is vital for success in the 11-plus exams, as they frequently appear in various mathematical contexts. Practice with these concepts will enhance your problem-solving skills and prepare you for the challenges ahead.