Learning Objectives
Distinguish a part-to-part ratio from a part-to-whole fraction.
Find and interpret a unit rate.
Calculate a percentage of an amount and a percentage change.
Prerequisites: equivalent fractions, decimal multiplication, and division. A ratio compares quantities. A rate compares quantities with different units, such as kilometres per hour or dollars per kilogram.
1. Read the Order of a Ratio
If a mixture has 2 cups of concentrate and 3 cups of water, the concentrate-to-water ratio is 2:3. Reversing the order gives the water-to-concentrate ratio 3:2. The labels are part of the meaning.
Worked Example: Parts and the Whole
A bag contains 4 red counters and 6 blue counters. The red-to-blue ratio is 4:6, which simplifies to 2:3.
There are 10 counters altogether. The fraction that are red is 4/10 = 2/5, not 2/3.
The ratio 2:3 describes five total equal-sized parts: two red parts and three blue parts.
2. Scale Both Quantities Together
Equivalent ratios multiply or divide both quantities by the same positive factor. If a recipe uses flour and milk in a ratio of 3:2, doubling the batch gives 6:4. Adding the same amount to both numbers generally changes the ratio.
Scale factor | Flour amount | Milk amount |
|---|---|---|
1 | 3 | 2 |
2 | 6 | 4 |
3 | 9 | 6 |
1/2 | 1.5 | 1 |
A proportional relationship has a constant ratio between corresponding quantities. A fixed extra charge can break proportionality. For example, a delivery fee plus a cost per item does not usually give a constant total cost per item.
3. A Unit Rate Means per One
Worked Example: Comparing Prices
A 3 kg bag costs $7.50 and a 5 kg bag costs $13.00. Find each price per kilogram.
The first costs 7.50 ÷ 3 = $2.50/kg. The second costs 13.00 ÷ 5 = $2.60/kg.
The 3 kg bag has the lower unit price. Dividing in the opposite order would give kilograms per dollar, which is a different but usable comparison if interpreted consistently.
4. Percent Means per Hundred
A percentage is a fraction with denominator 100. Thus 35% = 35/100 = 0.35. To find 35% of 80, calculate 0.35 × 80 = 28. To express 18 out of 24 as a percentage, calculate (18/24) × 100% = 75%.
Worked Example: A Discount
A $60 item is reduced by 15%. The discount is 0.15 × 60 = $9.
The sale price is 60 − 9 = $51. Equivalently, 85% of the original price remains, so 0.85 × 60 = $51.
The percentage uses the original price as its base. It is not 15% of the final price.
Percentage change compares a change with the original amount: percentage change = [(new − original)/original] × 100%, provided the original is nonzero. A rise from 40 to 50 is a change of 10 out of 40, so it is a 25% increase.
Common Mistakes
For a part-to-whole fraction, include every part in the denominator.
Keep rate units attached so you know what each division means.
Equal percentage increases and decreases do not usually cancel because the base changes.
5. Practice
1. A class has 12 students wearing blue and 8 wearing green. Find the blue-to-green ratio and the fraction wearing blue.
2. A cyclist travels 42 km in 3 hours at a constant speed. Find the unit rate and the distance in 5 hours at that rate.
3. Find 12% of 250.
4. A quantity falls from 80 to 68. Find the percentage decrease.
Worked Solutions
1. The ratio is 12:8 = 3:2. The blue fraction is 12/20 = 3/5.
2. The rate is 42/3 = 14 km/h. In 5 hours the distance is 14 × 5 = 70 km.
3. 0.12 × 250 = 30.
4. The decrease is 12, and 12/80 = 0.15. The percentage decrease is 15%.
6. Summary
- Label a ratio and keep its order consistent.
- A unit rate compares amounts per one unit.
- Convert percentages to fractions or decimals before calculating.
- For a percentage change, identify the original amount used as the base.