Welcome to Numbers and the Number System: The Chapter on Percentages!
Hello future mathematicians! Don’t worry if the word "percentage" sounds complicated—it’s actually one of the most useful things you’ll learn. Percentages are everywhere: in sales, banking, population statistics, and even your test scores!
In this chapter, we will demystify percentages, turning them from confusing numbers into simple tools you can use every day. We will cover everything from basic conversions to complex financial problems like compound interest.
Ready to start saving money and boosting your score? Let’s dive in!
Section 1: The Foundation – What Exactly Is a Percentage?
1.1 Definition and Concept
The word Percentage comes from the Latin phrase per centum, which literally means "per one hundred" or "out of every hundred."
- A percentage is simply a fraction where the denominator is always 100.
- We use the symbol % to denote a percentage.
Analogy: Imagine taking a 100-question test. If you get 85 questions correct, you scored 85 out of 100. In percentage terms, that is 85%.
Did you know? Even if a quantity is greater than the original amount (like a 200% increase), the concept still means "200 parts out of 100 parts" of the original whole!
Key Takeaway
Percentages are just a way of standardising a number so it relates to a whole of 100.
Section 2: The Core Skill – Converting Between Forms
To use percentages effectively in calculations, you often need to convert them into decimals or fractions. This is a crucial skill for both non-calculator and calculator papers.
2.1 Percentage to Decimal (P \(\to\) D)
To convert a percentage to a decimal, you must divide by 100.
Trick for Struggling Students: Dividing by 100 just means moving the decimal point two places to the left.
Example: Convert 45% to a decimal.
\(45\% = 45 \div 100 = 0.45\)
Example: Convert 3.5% to a decimal.
\(3.5\% = 3.5 \div 100 = 0.035\)
2.2 Decimal to Percentage (D \(\to\) P)
To convert a decimal to a percentage, you must multiply by 100.
Trick: Move the decimal point two places to the right.
Example: Convert 0.72 to a percentage.
\(0.72 \times 100 = 72\%\)
2.3 Percentage to Fraction (P \(\to\) F)
Since percentages are always out of 100, write the percentage value over 100, and then simplify the fraction.
Example: Convert 60% to a fraction.
\(60\% = \frac{60}{100}\)
(Divide the numerator and denominator by 20 to simplify)
\(\frac{60 \div 20}{100 \div 20} = \frac{3}{5}\)
Quick Review: Essential Equivalents
- 50% = 0.5 = 1/2
- 25% = 0.25 = 1/4
- 10% = 0.1 = 1/10
- 1% = 0.01 = 1/100
- 33.33...% or \(33\frac{1}{3}\%\) = 1/3
Section 3: Calculating Percentages of an Amount
Calculating a percentage of a quantity is one of the most common applications in maths. This is how you work out discounts!
3.1 The Standard Method (Using Decimals)
The most reliable way to find a percentage of an amount is to convert the percentage to a decimal first, and then multiply.
Formula:
\(\text{Amount} \times \text{Decimal Equivalent of Percentage}\)
Example: Find 35% of 800 kg.
- Convert 35% to a decimal: \(35\% = 0.35\)
- Multiply the decimal by the amount: \(800 \times 0.35\)
- Result: \(280 \text{ kg}\)
3.2 Non-Calculator Method (Finding 10% and 1%)
If you don't have a calculator, you can break down the percentage into simpler parts using 10% and 1%.
- To find 10%, divide the amount by 10 (move the decimal one place left).
- To find 1%, divide the amount by 100 (move the decimal two places left).
Example: Find 42% of $500.
- Find 10%: \(500 \div 10 = 50\)
- Find 1%: \(500 \div 100 = 5\)
- Build 42%:
- \(40\% = 4 \times 10\% = 4 \times 50 = 200\)
- \(2\% = 2 \times 1\% = 2 \times 5 = 10\)
- Total: \(200 + 10 = 210\)
- Result: 42% of \$500 is \$210.
3.3 Expressing One Quantity as a Percentage of Another
Sometimes you are given two amounts and asked what percentage the first amount is of the second amount (the total).
Formula:
\(\frac{\text{Part}}{\text{Whole}} \times 100\)
Example: In a class of 40 students, 18 wear glasses. What percentage of the class wears glasses?
\(\frac{18}{40} \times 100\)
\(\frac{9}{20} \times 100 = 9 \times 5 = 45\%\)
Key Takeaway: Always remember to multiply by 100 at the end of the calculation when finding the percentage of the whole!
Section 4: Percentage Change and Multipliers
4.1 The Power of Multipliers
When dealing with percentage increases or decreases, mathematicians use a shortcut called a Multiplier. This is the decimal equivalent of the percentage you end up with.
Using multipliers is essential for complex problems and is the fastest way to work on a calculator.
4.2 Percentage Increase
If an amount increases, the original amount (100%) plus the increase is the new percentage.
Process:
- Add the percentage increase to 100%.
- Convert this new total percentage into a decimal multiplier.
Example: Increase $750 by 20%.
- New Percentage: \(100\% + 20\% = 120\%\)
- Multiplier: \(120\% \div 100 = 1.20\)
- Calculation: \(750 \times 1.20 = 900\)
- Result: \$900
4.3 Percentage Decrease (Discounts)
If an amount decreases, you subtract the percentage decrease from 100%.
Process:
- Subtract the percentage decrease from 100%.
- Convert the remaining percentage into a decimal multiplier.
Example: Decrease 450 km by 8%.
- New Percentage: \(100\% - 8\% = 92\%\)
- Multiplier: \(92\% \div 100 = 0.92\)
- Calculation: \(450 \times 0.92 = 414\)
- Result: 414 km
Common Mistake to Avoid:
When decreasing by a small percentage (e.g., 2%), students often write the multiplier as 0.2 or 0.8. Remember: \(100\% - 2\% = 98\%\), so the multiplier is 0.98.
Section 5: Reverse Percentages (Finding the Original Amount)
This is where things get tricky, but don't worry! Reverse percentage problems ask you to find the original price before the increase or decrease was applied.
You cannot simply reverse the operation (e.g., if a price increased by 10%, you CANNOT decrease the new price by 10% to find the original!).
The Key Concept:
The new amount you have is the result of the original amount multiplied by the multiplier.
Therefore, to find the original amount, you must divide the new amount by the multiplier.
Formula:
\(\text{Original Amount} = \frac{\text{New Amount}}{\text{Multiplier}}\)
5.1 Reverse Percentage Increase
Example: A coat is sold for $180 after a 20% increase in price. What was the original price?
- Identify the new percentage: The original price (100%) + the increase (20%) = 120%.
- Find the multiplier: \(120\% = 1.20\)
- Divide the new amount by the multiplier: \(\text{Original Price} = \frac{180}{1.20}\)
- Calculation: \(180 \div 1.20 = 150\)
- Result: The original price was \$150.
5.2 Reverse Percentage Decrease
Example: A TV is on sale for $475. This is after a 5% discount. What was the price before the discount?
- Identify the new percentage: The original price (100%) – the discount (5%) = 95%.
- Find the multiplier: \(95\% = 0.95\)
- Divide the new amount by the multiplier: \(\text{Original Price} = \frac{475}{0.95}\)
- Calculation: \(475 \div 0.95 = 500\)
- Result: The original price was \$500.
Key Takeaway for Reversing: If the question asks for the "original" or "price before the sale," you must DIVIDE by the Multiplier!
Section 6: Financial Mathematics – Compound Change
Percentages are fundamental in finance, particularly when calculating interest on savings or depreciation on assets.
6.1 Understanding Simple vs. Compound
Simple Interest: The interest is calculated only on the original amount each time. The amount of interest earned is the same every year.
Compound Interest: The interest is calculated on the original amount PLUS any accumulated interest from previous periods. This is often called "interest on interest."
Analogy: Compound interest is like a snowball rolling down a hill—it gets bigger and picks up speed (interest) faster over time!
6.2 The Compound Change Formula
For compound growth (interest) or compound decay (depreciation), we use repeated multiplication with the multiplier:
Formula:
\(\text{Final Amount} = \text{Initial Amount} \times (\text{Multiplier})^{\text{Number of Periods (n)}}\)
Where \(n\) is the number of years (or compounding periods).
Compound Interest (Growth)
Example: $2000 is invested for 3 years at 4% compound interest per year. Find the final value.
- Calculate the Multiplier (Increase): \(100\% + 4\% = 104\%\). Multiplier = 1.04.
- Apply the formula: \(\text{Final Amount} = 2000 \times (1.04)^3\)
- Calculation: \(2000 \times 1.124864 = 2249.73\)
- Result: The final amount is \$2249.73 (rounded to 2 decimal places for money).
Depreciation (Decay)
Depreciation is the loss in value of an item over time. This is a compound decrease.
Example: A car bought for $15,000 depreciates by 10% per year. What is its value after 5 years?
- Calculate the Multiplier (Decrease): \(100\% - 10\% = 90\%\). Multiplier = 0.90.
- Apply the formula: \(\text{Final Value} = 15000 \times (0.90)^5\)
- Calculation: \(15000 \times 0.59049 \approx 8857.35\)
- Result: The car is worth \$8857.35.
Key Takeaway: Compound problems require finding the multiplier once and raising it to the power of the number of years or periods.
Final Quick Review: Key Terms
- Percentage: A fraction out of 100.
- Decimal Equivalent: Percentage divided by 100. Used for calculation.
- Multiplier: A single decimal value used to find the final result after a percentage increase or decrease.
- Reverse Percentage: Finding the original amount by dividing the new amount by the multiplier.
- Compound Change: Applying a multiplier repeatedly over multiple time periods using a power (\(n\)).
You've mastered the building blocks of percentage calculations! Keep practicing those conversions and multipliers, and you’ll find these questions straightforward in the exam!