a:b · Ratios
Not a size but a relationship — the ratio.
Juice from 1 cup concentrate + 2 cups water tasted great. Double everything — does the taste change?
Distinguish a quantity from a ratio.
A fair way to weigh two things of wildly different sizes.
Open the immersive version →A reading path containing this lesson: Parts of one whole · 3/3
Why does this question matter?
Whether 3 cups or 6 cups is more, counting answers. But 'which juice is stronger' or 'which store is cheaper' can't be answered by size — the bases differ.
The solution is to look at the relationship: write the 'so many to so many' (the ratio), divide by the base amount to get one number (the rate), and things of different sizes become fairly comparable.
Gain this eye and percent, exchange rates, unit prices, map scales, and trig ratios all reveal themselves as one tool in different clothes — the ratio is mathematics' most reused part.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — juice from 1 cup concentrate + 2 cups water tasted great. Make it with 2 cups + 4 cups — the taste?
Scale the juice up as much as you like
We multiply the 1 : 2 recipe. Keep your eye on the 'strength' number.
1/3 = 33.3%
strength (concentrate ÷ total)
Read the idea from another angle
A ratio is a number with the size erased and only the relationship left.
1:2 and 100:200 differ in size a hundredfold yet are the same ratio. Because size is erased, ratios handle everything that survives scaling — recipes, maps, models.
To compare, divide first.
Price by quantity, concentrate by total, opposite by hypotenuse — dividing by the base puts different-sized things on one scale. That single division is all a ratio is.
Now read the definitions with their conditions
What the juice experiment confirmed, in mathematical language.
Ratio
The 'so many to so many' relationship. Multiply (or divide) both sides by the same number and it's unchanged — why scaled-up juice tastes the same.
Rate (the value of a ratio)
One division turns a ratio into a single number — and single numbers can be compared with anything.
Four outfits of one relationship
Ratio, fraction, decimal, percent are four notations for one relationship. Percent is simply the rate with the base set to 100.
Make one more distinction
The juice with more concentrate is stronger.
3 cups concentrate + 5 water (37.5%) is weaker than 2 + 3 (40%). Strength is set not by the amount of concentrate but by its share of the whole — comparison is always a division.
Adding the same number to both sides keeps the ratio.
Add 1 to each side of 1:2 and you get 2:3 — strength jumps from 33% to 40%. Multiplication preserves ratios; addition breaks them. 1:2 = 2:4 = 3:6 works because you multiply.
Where else can this idea help?
Percent
A rate whose base is set to 100. Discounts, poll numbers, returns — most numbers in the world wear this outfit of the ratio.
Unit price
Divide price by quantity ($/g) and big and small packs become fairly comparable — the small print on shelf tags is exactly this value.
Exchange rates
An exchange rate is the ratio between two currencies. '1 dollar = 1,400 won' is the ratio 1:1400 — currency sense IS ratio sense.
Map scales and trig ratios
A 1:1000 map shrinks reality by that ratio. And 'equal angles give equal side ratios' (trigonometry) measures rivers and mountains without crossing or climbing.
Explain it in your own words
Try another question
Q1Juice A: 2 cups concentrate + 6 water. Juice B: 1 cup concentrate + 3 water. Same taste?Show answer ▾
Yes. Divide both sides of 2:6 by 2 and you get 1:3 — the same ratio. Strength is 2/8 = 1/4 = 25% for both. Different sizes, same relationship, same taste.
Q2A batter with 3 hits in 10 at-bats vs 5 hits in 20 — who's hitting better?Show answer ▾
Compare rates, not hit counts (3 < 5): 3/10 = .300 vs 5/20 = .250 — the first batter leads. A batting average is exactly 'hits divided by the base (at-bats)'.
Q3On a 1:50,000 map, two points are 4cm apart. Actual distance?Show answer ▾
4cm × 50,000 = 200,000cm = 2km. A scale is the map:reality ratio — one ratio converts any map length into real distance.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Parts of one whole · 3/3
Leads to next
Functions
Generalize 'double the water, double the concentrate' — relationships that change while preserving a rule — and you get functions.
Go to the Functions lab →Related lab
Related lab
Trigonometric Ratios
Equal angles, equal side ratios — ratio sense at its most spectacular.
Go to the Trigonometric Ratios lab →