Welcome to the World of Percentages!
In this chapter, we are going to master percentages. You see percentages everywhere: when there is a 50% sale at your favorite shop, when your phone battery is at 12%, or when a bank offers you interest on your savings. Percentages are just another way of talking about fractions and decimals, but they are often the most useful way to compare things in real life.
Don't worry if you've found this tricky before. We are going to break it down step-by-step until you're a percentage pro!
1. What is a Percentage?
The word percent comes from the Latin 'per centum', which literally means "per hundred".
Imagine a large square made of 100 smaller tiles. If you paint 25 of those tiles red, you have painted \( 25\% \) of the square. Every percentage is just a fraction with a denominator of 100.
Key Term: Percentage – A number of parts out of a total of 100.
Quick Review:
• \( 50\% \) means 50 out of 100 (or half).
• \( 10\% \) means 10 out of 100 (or one tenth).
• \( 100\% \) means the whole thing!
Key Takeaway: Always remember that "percent" means "out of 100". If you can remember that, you've already won half the battle!
2. The "Golden Trio": Converting Between FDP
To be successful in the J560 syllabus, you need to move comfortably between Fractions, Decimals, and Percentages (FDP). They are essentially the same value dressed in different "outfits".
From Decimal to Percentage
Multiply the decimal by 100 (move the decimal point two places to the right).
Example: \( 0.45 = 45\% \)
Example: \( 0.07 = 7\% \)
From Fraction to Percentage
Try to make the bottom number (denominator) 100, or simply divide the top by the bottom and multiply by 100.
Example: \( \frac{1}{4} = \frac{25}{100} = 25\% \)
From Percentage to Fraction
Put the percentage number over 100 and simplify the fraction.
Example: \( 60\% = \frac{60}{100} \). Dividing both by 20 gives \( \frac{3}{5} \).
Did you know? The symbol \( \% \) actually looks like the number 100 rearranged! The two circles are the zeros and the slash is the "1".
3. Calculating Percentages of Amounts
There are two main ways to do this: the "Building Block" method (great for non-calculator papers) and the "Multiplier" method (perfect for calculator papers).
The Building Block Method (Non-Calculator)
Most percentages can be found by finding \( 10\% \) or \( 1\% \) first:
• To find \( 10\% \): Divide by 10.
• To find \( 1\% \): Divide by 100.
Example: Find \( 15\% \) of £120.
1. Find \( 10\% \): \( 120 \div 10 = 12 \)
2. Find \( 5\% \) (half of 10%): \( 12 \div 2 = 6 \)
3. Add them together: \( 12 + 6 = 18 \). So, \( 15\% \) is £18.
The Multiplier Method (Calculator)
Turn the percentage into a decimal and multiply by the amount.
Example: Find \( 37\% \) of £450.
1. \( 37\% \) as a decimal is \( 0.37 \).
2. Calculation: \( 0.37 \times 450 = 166.5 \). So, £166.50.
Key Takeaway: For a quick "sanity check," ask yourself: "Is my answer sensible?" If you're finding \( 10\% \) of £100 and get £50, you know something went wrong!
4. Expressing One Quantity as a Percentage of Another
Often, you’ll be asked: "What percentage is \( A \) of \( B \)?" (e.g., your score on a test).
The Formula: \( \frac{\text{Part}}{\text{Whole}} \times 100 \)
Example: You scored 18 out of 25 in a quiz. What is your percentage?
\( \frac{18}{25} \times 100 = 72\% \)
5. Percentage Increase and Decrease
This is where we change an original value. You might add a pay rise (increase) or subtract a discount (decrease).
The Fast Way: Using Multipliers
This is a favorite for the Foundation and Higher tiers because it's fast and helps with harder problems later.
• For an increase: Add the percentage to \( 100\% \), then convert to a decimal.
• For a decrease: Subtract the percentage from \( 100\% \), then convert to a decimal.
Example 1 (Increase): Increase £50 by \( 12\% \).
New percentage = \( 100\% + 12\% = 112\% \). Multiplier = \( 1.12 \).
Calculation: \( 50 \times 1.12 = 56 \).
Example 2 (Decrease): A £150 coat is in a \( 20\% \) off sale.
New percentage = \( 100\% - 20\% = 80\% \). Multiplier = \( 0.8 \).
Calculation: \( 150 \times 0.8 = 120 \).
Common Mistake to Avoid: If a price increases by \( 10\% \) and then decreases by \( 10\% \), you do not end up back at the original price! The second \( 10\% \) is calculated on a new, larger number.
6. Reverse Percentages (Working Backwards)
These questions give you the new price and ask for the original price. They are often the hardest questions in this chapter.
The Rule: Never calculate the percentage of the new price. Instead, find out what percentage the new price represents.
Example: A phone costs £360 after a \( 10\% \) discount. What was the original price?
1. If there was a \( 10\% \) discount, the £360 must represent \( 90\% \) of the original price (\( 100 - 10 \)).
2. So: \( 90\% = £360 \).
3. Find \( 1\% \): \( 360 \div 90 = 4 \).
4. Find \( 100\% \) (Original): \( 4 \times 100 = 400 \). The original price was £400.
Memory Aid: In reverse percentage questions, you usually divide by the multiplier to go back in time to the original value.
7. Simple and Compound Interest
Banks use percentages to reward you for saving or charge you for borrowing.
Simple Interest
The interest is calculated only on the original amount. It stays the same every year.
Example: £1000 at \( 5\% \) simple interest for 3 years.
\( 5\% \) of £1000 is £50. You get £50 every year.
Total interest = \( £50 \times 3 = £150 \).
Compound Interest
The interest is calculated on the new balance each year. It’s "interest on interest."
The Formula (Higher Tier): \( \text{Total Amount} = P \times (\text{Multiplier})^n \)
Where \( P \) is the starting amount, and \( n \) is the number of years.
Example: £1000 at \( 5\% \) compound interest for 3 years.
Calculation: \( 1000 \times (1.05)^3 = 1157.63 \).
(Compare this to the £1150 you got with simple interest! Compound interest grows faster.)
Key Takeaway: Simple interest is like a flat addition; Compound interest is like a snowball rolling down a hill, getting bigger and bigger as it picks up its own interest.
Summary Checklist
Before you finish, make sure you can:
• Define percent as "parts per hundred".
• Convert between fractions, decimals, and percentages.
• Find a percentage of an amount (with and without a calculator).
• Use multipliers for percentage increase and decrease.
• Solve reverse percentage problems by finding what \( 1\% \) is worth.
• Distinguish between simple and compound interest.