Ratios And Percentages
A ratio is a way of comparing two or more quantities. A percentage is a ratio that compares a quantity to 100 (per cent = per hundred).
NB: Be sure to refresh your memory on conversions between fractions, decimals, and percentages before diving more into this topic.
Understanding Ratios
A ratio shows the relative sizes of two or more quantities. For example, if a class has 12 boys and 8 girls, the ratio of boys to girls is written as 12:8 (read as "12 to 8").
Example 1: Writing Ratios
In a basket, there are 15 apples and 10 oranges.
Ratio of apples to oranges = 15:10
Ratio of oranges to apples = 10:15
Answer: 15:10 and 10:15
Order matters in ratios. The ratio 15:10 is not the same as 10:15.
Equivalent Ratios and Simplification
Equivalent ratios are formed by multiplying or dividing both parts of a ratio by the same number. To simplify a ratio, divide both parts by their highest common factor (HCF).
Example 2: Simplifying Ratios
Simplify the ratio 15:10
Step 1: Find the HCF
The factors of 15 are: 1, 3, 5, 15
The factors of 10 are: 1, 2, 5, 10
Therefore, the HCF of 15 and 10 is 5
Step 2: Making sure to preserve the order of the numbers divide both of them by the HCF
15 ÷ 5 = 3
10 ÷ 5 = 2
Step 3: Write the answer
Answer: 15:10 = 3:2
Example 3: Forming Equivalent Ratios
Write three equivalent ratios for 2:3
We find equivalent ratios using the same procedure as finding equivalent fractions, like so:
Step 1: Choose any number to multiply by
We can multiply both parts of the ratio by any whole number to create an equivalent ratio. Let's use 2, 3, and 4.
Step 2: Multiply both parts by 2
2 × 2 = 4
3 × 2 = 6
First equivalent ratio: 4:6
Step 3: Multiply both parts by 3
2 × 3 = 6
3 × 3 = 9
Second equivalent ratio: 6:9
Step 4: Multiply both parts by 4
2 × 4 = 8
3 × 4 = 12
Third equivalent ratio: 8:12
Answer: 4:6, 6:9, 8:12
Note: Any whole number works; 5, 10, 100, etc. The ratios will always be equivalent because you are multiplying (or dividing) both parts by the same number.
Representing Ratios as Fractions
A ratio can be written as a fraction to show what part of the whole each quantity represents.
Example 4: Ratio to Fraction
In a class, the ratio of boys to girls is 3:2. Represent the ratio as fractions to find what fraction of the class are boys and what fraction are girls.
Step 1: Find the total number of parts
Add both numbers in the ratio: 3 + 2 = 5 parts
Step 2: Write the fraction for boys
Boys = 3⁄5 of the class
(The numerator is the boys' part of the ratio, the denominator is the total parts)
Step 3: Write the fraction for girls
Girls = 2⁄5 of the class
(The numerator is the girls' part of the ratio, the denominator is the total parts)
Step 4: State the answer
Therefore: Boys = ⅗ of the class, Girls = ⅖ of the class
Comparing Quantities Using Ratios
To compare two ratios, convert them to fractions and compare the fractions, or calculate the percentages they represent.
Example 5: Shooting Game Comparison
Team A scores 15 goals out of 20 attempts. Team B scores 12 goals out of 15 attempts. Which team performed better?
Method 1: Ratio as fraction
Team A
Step 1: Write the goals scored as a fraction of the total attempts
15 ⁄ 20
Step 2: Simplify the fraction by dividing numerator and denominator by their HCF (which is 5)
= ¾
Step 3: Convert the fraction to a decimal by dividing numerator by denominator
= 0.75
Team B
Step 1: Write the goals scored as a fraction of the total attempts
12 ⁄ 15
Step 2: Simplify the fraction by dividing numerator and denominator by their HCF (which is 3)
= ⅘
Step 3: Convert the fraction to a decimal by dividing numerator by denominator
= 0.8
Step 4: Compare the decimals
0.8 is greater than 0.75, so Team B performed better
Method 2: Percentage
Team A
Step 1: Write the fraction and multiply by 100% to convert to a percentage
15 ⁄ 20 × 100%
Step 2: Simplify: 15 ÷ 20 = 0.75, then 0.75 × 100% = 75%
= 75%
Team B
Step 1: Write the fraction and multiply by 100% to convert to a percentage
12 ⁄ 15 × 100%
Step 2: Simplify: 12 ÷ 15 = 0.8, then 0.8 × 100% = 80%
= 80%
Step 3: Compare the percentages
80% > 75%, so Team B is better
Answer: Team B performed better with 80% success rate
Key Rule: To find what percentage one quantity is of another, use: (part ÷ whole) × 100%
Percentage Increase and Decrease
Percentage increase and percentage decrease tell us how much a quantity has gone up or down relative to its original value.
Formula
Percentage Increase = (Increase ÷ Original) × 100%
Percentage Decrease = (Decrease ÷ Original) × 100%
Example 6: Percentage Increase
A shirt originally costs M40. The price increases to M50. Find the percentage increase.
Step 1: Find the increase by subtracting the original price from the new price
Increase = M50 - M40 = M10
Step 2: Divide the increase by the original price
10 ÷ 40 = 0.25
Step 3: Multiply by 100% to convert to a percentage
0.25 × 100% = 25%
Answer: 25% increase
Example 7: Percentage Decrease
A jacket originally costs M80. It is on sale for M60. Find the percentage decrease.
Step 1: Find the decrease by subtracting the new price from the original price
Decrease = M80 - M60 = M20
Step 2: Divide the decrease by the original price
20 ÷ 80 = 0.25
Step 3: Multiply by 100% to convert to a percentage
0.25 × 100% = 25%
Answer: 25% decrease
Key Rule: Always divide by the original amount, not the new amount. The original is the starting value before the change.
Reverse Percentages
Reverse percentages help us find the original amount when we know the final amount after a percentage increase or decrease.
Why do we need reverse percentages? Sometimes you know the final price AFTER a discount or AFTER a price increase, but you need to find the original price before the change happened.
Example 8: Finding the Original Price (Increase)
The problem: After a 20% increase, a laptop costs M600. What was the original price?
Step 1: Understand what happened to the original price
The original price is 100% of itself. A 20% increase means we ADD 20% to the original price.
New price = Original price + 20% of Original price
New price = 100% + 20% = 120% of the original price
Step 2: Write the relationship as an equation
M600 = 120% of the original price
Write 120% as a decimal by dividing by 100: 120 ÷ 100 = 1.2
So: M600 = 1.2 × (original price)
Step 3: Solve for the original price
If 1.2 × original = M600, then divide both sides by 1.2:
Original price = M600 ÷ 1.2
Multiply both numbers by 10 to make it easier: 6000 ÷ 12 = 500
Original price = M500
Step 4: Check your answer
Original price = M500
20% of M500 = M500 × 0.2 = M100
New price after 20% increase = M500 + M100 = M600
Summary: For a percentage INCREASE of r%, the multiplier is (100% + r%) = (1 + r/100). Then divide the final amount by this multiplier.
Example 9: Finding the Original Price (Decrease)
The problem: After a 30% discount, a phone costs M210. What was the original price?
Step 1: Understand what happened to the original price
The original price is 100% of itself. A 30% discount (decrease) means we SUBTRACT 30% from the original price.
Sale price = Original price - 30% of Original price
Sale price = 100% - 30% = 70% of the original price
Important: A 30% discount means you only pay 70% of the original price.
Step 2: Write the relationship as an equation
M210 = 70% of the original price
Write 70% as a decimal by dividing by 100: 70 ÷ 100 = 0.7
So: M210 = 0.7 × (original price)
Step 3: Solve for the original price
If 0.7 × original = M210, then divide both sides by 0.7:
Original price = M210 ÷ 0.7
Multiply both numbers by 10: 2100 ÷ 7 = 300
Original price = M300
Step 4: Check your answer
Original price = M300
30% discount = M300 × 0.3 = M90
Sale price after discount = M300 - M90 = M210
Summary: For a percentage DECREASE of r%, the multiplier is (100% - r%) = (1 - r/100). Then divide the final amount by this multiplier.
Key Rule for Reverse Percentages:
Original amount = Final amount ÷ Multiplier
where Multiplier = (100% + r%) for increase, or (100% - r%) for decrease, written as a decimal.
Dividing a Quantity in a Given Ratio
To divide a quantity in the ratio a:b:c, find the total number of parts (a + b + c), then find the value of one part, and multiply by each part of the ratio.
Example 10: Dividing in the form a:b:c
Divide M1200 among three people in the ratio 2:3:5.
Total parts = 2 + 3 + 5 = 10 parts
Value of one part = M1200 ÷ 10 = M120
Person 1: 2 × M120 = M240
Person 2: 3 × M120 = M360
Person 3: 5 × M120 = M600
Answer: M240, M360, M600
Example 11: Real-Life Savings Problem
Three friends invest money in a business in the ratio 3:4:5. If the total investment is M2400, how much did each invest?
Total parts = 3 + 4 + 5 = 12 parts
One part = M2400 ÷ 12 = M200
Friend 1: 3 × M200 = M600
Friend 2: 4 × M200 = M800
Friend 3: 5 × M200 = M1000
Answer: M600, M800, M1000
Summary Table: Ratios and Percentages
| Concept | Formula / Method | Example |
|---|---|---|
| Simplify Ratio | Divide by HCF | 15:10 = 3:2 |
| Percentage of a Quantity | (part ÷ whole) × 100% | 15/20 = 75% |
| Percentage Increase | (Increase ÷ Original) × 100% | (10÷40)×100% = 25% |
| Percentage Decrease | (Decrease ÷ Original) × 100% | (20÷80)×100% = 25% |
| Reverse Percentage | Original = Final ÷ Multiplier | 600 ÷ 1.2 = 500 |
| Dividing in Ratio a:b:c | Find total parts, one part value | 2:3:5 → M240, M360, M600 |
NB: Ratios and percentages are used in many daily activities such as shopping discounts and bank interest rates to sports statistics and business investments. Always simplify ratios to their lowest terms, and remember that percentage means per hundred. For reverse percentages, be careful to divide by the correct multiplier.