Ratio & Rate
“A ratio compares two amounts; a rate is that comparison divided into one number.”
The formula
a : b = a ÷ bHow to read it: The ratio 'a to b', written as one number, is a divided by b (the rate)
- a
- — the amount being compared (the front number)
- b
- — the amount used as the base (the back number)
The hook
2 red marbles to 3 blue — a ratio compares two amounts as 'this to that'.
In plain words
A ratio compares two amounts as 'this to that'; a rate turns that into one number by dividing.
The intuition
With 2 red and 3 blue, the ratio is 2 : 3. To turn it into a single number, compute 2 ÷ 3 — that is the rate. A ratio compares; a rate compresses that comparison into one number.
How it's built
In a ratio a : b, the front a is compared and the back b is the base. Multiplying or dividing both by the same number keeps the ratio the same — 4 : 2 equals 2 : 1.
Example
Walk 10 km in 2 hours and your rate is 10 ÷ 2 = 5 km per hour.
Common mistake
The ratio 2 : 3 does not mean '2/3 of the whole'. The whole is 2+3 = 5, so red is 2/5 of it. Don't confuse a ratio with a fraction of the total.
Where it's used
Recipe mixes, speed (distance to time), map scales, mixing paint — ratios and rates are everywhere.
Where it came from
Ratios were studied deeply in ancient Greek geometry as a way to compare lengths and areas.
Prerequisites
Quick check
There are 2 red marbles and 5 blue. What is the ratio of red to blue?
- 2 : 5✓
- 5 : 2
- 2 : 7
- 5 : 7
Practice
Write the ratio 4 : 2 in its simplest form.
Answer: undefined
- Divide both numbers by 2.
- 4÷2 : 2÷2
- → 2 : 1
Takeaway: Divide both sides by the same number and the ratio holds.
You walked 10 km in 2 hours. How many km per hour?
Answer: 5
- Per hour = distance ÷ time
- 10 ÷ 2 = 5
- → 5 km/h
Takeaway: A rate is one number you get by dividing.
Mix flour to sugar as 2 : 1. For 4 cups of flour, how many cups of sugar?
Answer: 2
- Flour doubled (2→4).
- So double the sugar: 1 × 2 = 2
- → 2 cups
Takeaway: Scale one side and scale the other by the same amount.