Rates, Ratio, Proportion and Percentage
Numbers · Rates, Ratio, Proportion and Percentage
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 11 Rates, Ratio, Proportion and Percentage
Lesson objectives
By the end of this topic, you should be able to:
- Define and solve problems involving rates.
- Compare quantities using ratios and change quantities in a given ratio.
- Distinguish direct and inverse proportion and solve problems in each.
- Convert between fractions, decimals and percentages and solve percentage problems.
Rates, Ratio, Proportion and Percentage
These four ideas all compare quantities, and the same reasoning runs through them.
a) Rates
A rate compares two quantities of different kinds.
Speed is distance per time. Density is mass per volume. A wage is money per hour.
Rates carry units, and the units tell you what the number means. A rate of 60 means nothing; 60 km/h means something.
The word "per" signals a rate.
b) Ratio
A ratio compares quantities of the same kind, so it has no units.
Simplify by dividing each part by their common factor. So 12 : 18 becomes 2 : 3.
Order matters. 2 : 3 is not 3 : 2.
To compare two ratios, write them as fractions or bring them to a common form.
c) Dividing in a ratio
Add the ratio parts to find the total number of shares.
Divide the amount by that total to find one share.
Multiply out each portion.
Check by adding the portions back. They must give the original amount.
d) Increase and decrease in a ratio
To change a quantity in the ratio a : b, multiply by a/b.
To increase 200 in the ratio 5 : 4, multiply by 5/4 to get 250.
To decrease 200 in the ratio 3 : 4, multiply by 3/4 to get 150.
If the first number is larger, the quantity increases. If smaller, it decreases. Check your answer moved the right way.
e) Direct proportion
Two quantities are in direct proportion when one increases as the other increases, in the same ratio. Their quotient stays constant.
If 4 books cost KES 320, then 7 books cost KES 560.
The reliable method is to find the value of one unit first, then multiply.
f) Inverse proportion
Two quantities are in inverse proportion when one increases as the other decreases. Their product stays constant.
More workers means fewer days. Greater speed means less time.
Find the constant product first, then divide.
Before calculating, ask which way the answer should move. Then check it moved that way. That single habit catches most errors in this topic.
g) Percentage
A percentage is a fraction with denominator 100.
To convert a fraction to a percentage, multiply by 100. To go back, divide by 100 and simplify.
For percentage increase or decrease, always divide the change by the original amount.
To find the original after a percentage change, work backwards. If a price rose 20% to KES 600, then 120% is 600, so 100% is 500.
Dividing by 1.2 gives the original; taking 20% off 600 does not.
h) Where this is used
Currency conversion. Mixing concrete or fertiliser. Costing labour. Reading discounts, interest and inflation.
Words to know
- Rate -- a comparison of two quantities of different kinds, carrying units.
- Ratio -- a comparison of two quantities of the same kind, without units.
- Direct proportion -- a relationship where one quantity increases as the other increases.
- Inverse proportion -- a relationship where one quantity decreases as the other increases.
- Percentage -- a ratio expressed with denominator 100.
:::checkpoint Check yourself
- Divide KES 7 200 in the ratio 4 : 5.
- Increase 480 in the ratio 7 : 6.
- If 5 workers take 12 days, how long would 3 workers take?
- A price rose by 25% to KES 750. What was the original price? :::
Bridge to practice
The exercises begin with rates and ratios, move through sharing and changing in a given ratio, then direct and inverse proportion, and finish with percentages and percentage change. For every proportion question, state whether the answer should be larger or smaller before you calculate.