A : B → A ÷ B
Ratio to Decimal & Fraction Calculator
Turn a ratio into its decimal value, a simplified fraction, a mixed number and the 1 : n form — both directions.
Open → Ratio to Decimal & Fraction CalculatorUnit Ratio Calculator
Reduce any ratio to a per-one rate — price per kilogram, miles per hour, the 1 : n form used by maps and models. Both directions are given, units are labelled, and a comparison mode tells you which of two options is better value.
Formula A : B → A⁄B : 1
Try an example
Option 1
Option 2
Try an example
Unit rate
Method
One division, and the units tell you if it is right
Decide which quantity goes "per one"
This is the whole decision. "Price per kilogram" puts kilograms on the bottom; "kilograms per pound spent" puts money on the bottom. Both are unit rates for the same purchase, and they are reciprocals of each other.
Divide the other quantity by it
14.97 ÷ 3 = 4.99. That is the rate: $4.99 for one kilogram. The
division always goes numerator-of-the-rate over denominator-of-the-rate, matching
the word "per".
Say the units out loud
"Four ninety-nine dollars per kilogram." If that sounds like the answer to the question, you divided correctly. If it comes out as "kilograms per dollar", flip the division. This check catches the error more reliably than re-checking the arithmetic.
Use the rate to scale
Multiply by any quantity to get the total: 4.99 × 7 = $34.93 for 7 kg.
This is why the unit rate is worth finding even for a one-off question — it answers
every subsequent question about the same relationship in one multiplication.
Or use it to compare
Two packages of different sizes cannot be compared by price alone, but their unit rates can be compared directly. The lower price per unit is the better value, whatever the pack sizes.
unit rate = A ÷ B
The rate. A per one B. Its units are "A units per B unit", which is the phrase to check against your question.
A : B = 1 : B/A = A/B : 1
Both unit forms. Divide through by A for one, by B for the other. They are reciprocals, and both are correct — pick the one that reads naturally.
The reciprocal is not the same rate
$4.99 per kg and 0.2 kg per dollar describe one purchase, but they answer opposite questions and behave oppositely: a lower price per kg is better value, while a higher quantity per dollar is. Mixing them up inverts the conclusion.
Worked examples
Prices, speeds and scales
Price per unit
3 kg of coffee costs $14.97. What is the price per kilogram?
- Rate wanted
- dollars per kilogram
- So divide
- 14.97 ÷ 3
- Unit rate
- $4.99 per kg
- Reversed
- 0.2004 kg per dollar
- 7 kg would cost
- 4.99 × 7 = $34.93
$4.99 per kg
Speed
A train covers 150 miles in 3 hours. What is its speed?
- Rate wanted
- miles per hour
- So divide
- 150 ÷ 3
- Unit rate
- 50 mph
- Reversed
- 0.02 hours per mile
- 8 hours would cover
- 50 × 8 = 400 miles
50 miles per hour
Better value
A 500 g pack costs £3.20 and a 750 g pack costs £4.50. Which is better value?
- Option 1
- 3.20 ÷ 0.5 = £6.40 per kg
- Option 2
- 4.50 ÷ 0.75 = £6.00 per kg
- Compare
- £6.00 < £6.40
- Saving
- 40p per kg, 6.25% cheaper
- Note
- the dearer pack is the better deal
The 750 g pack, by 6.25%
Recurring rate
7 items cost $23. What is the price per item?
- Divide
- 23 ÷ 7
- Exactly
- 23/7
- As a decimal
- 3.285714…
- Rounded
- $3.29 per item
- 20 items, exact
- 460/7 = $65.71
23/7 ≈ $3.29 each
Reference
Rates you meet every day
Every one of these is a ratio with one side reduced to 1.
| Rate | Divide | By | Reciprocal rate |
|---|---|---|---|
| Price per kilogram | Total price | Kilograms | Kilograms per unit of currency |
| Speed (miles per hour) | Distance | Time | Pace — time per mile |
| Fuel economy (miles per gallon) | Distance | Fuel used | Litres per 100 km |
| Density (grams per cm³) | Mass | Volume | Specific volume |
| Hourly wage | Total pay | Hours worked | Hours per unit of pay |
| Interest rate per year | Interest | Principal | — |
| Map scale 1 : n | Real distance | Map distance | Map distance per real unit |
| Population density | People | Area | Area per person |
| Calories per 100 g | Calories | Grams ÷ 100 | Grams per calorie |
| Cost per click | Total spend | Clicks | Clicks per unit spent |
| Gear ratio 1 : n | Output teeth | Input teeth | Reverse gear ratio |
Background
Why reducing to one makes comparison possible
The reason unit rates are so useful is that they give every option a common denominator. Two packs of coffee at £3.20 for 500 g and £4.50 for 750 g cannot be compared as written — one is cheaper and one is bigger. Convert both to a price per kilogram and the comparison becomes a single subtraction. That is the same trick percentages use, and it is why supermarket shelf labels are legally required to show a unit price in many countries.
It also makes non-obvious results visible. The larger pack is not always cheaper per unit — "bigger is better value" is a heuristic, not a rule, and retailers do sometimes price the large size at a premium. The only way to know is to compute both rates, which takes a few seconds and occasionally saves a surprising amount.
Choosing which direction to use
For every rate there is a reciprocal, and for most questions one of the two reads far more naturally. Runners use pace — minutes per mile — rather than speed, because the number they care about is how long the next mile will take. Drivers use speed. Fuel economy is quoted as miles per gallon in some countries and litres per 100 km in others, and these rank cars in opposite directions: higher mpg is better, lower L/100 km is better. Neither convention is wrong, but mixing them produces exactly backwards conclusions.
Unit rates and proportionality
A unit rate only describes the whole relationship if that relationship is directly proportional — if doubling the quantity doubles the total. Grocery weight usually is. Postage often is not, because of fixed handling charges and weight bands, so "cost per kilogram" for a parcel can be misleading. Where there is a fixed component, the relationship is linear rather than proportional, and a single unit rate cannot capture it. The direct proportion calculator explains that distinction in detail.
Related tools
For the decimal and fraction forms of the underlying ratio, use the
ratio to decimal and fraction calculator. For
the proportional-relationship framing with a value table and graph, the
direct proportion calculator. And for the
1 : n form applied to physical scale, including area and volume factors, the
scale factor calculator.
Questions
Unit rates — questions
Which way round to divide, and why the reciprocal rate ranks options in the opposite order.
What is a unit ratio?
A unit ratio is a ratio rewritten so that one of the two terms is exactly 1.
4 : 10 becomes 1 : 2.5, and $14.97 : 3 kg becomes $4.99 : 1 kg. Because one side is a single unit, the other side reads directly as a rate — dollars per kilogram, miles per hour, cost per item.How do you find a unit rate?
Divide the quantity you want the rate for by the quantity you want it per. For "price per kilogram" divide the price by the kilograms:
14.97 ÷ 3 = 4.99. Getting it the wrong way round gives you kilograms per dollar, which is a valid number but not the one you asked for — reading the units out loud is the reliable check.Which unit rate should I use to compare prices?
Whichever puts the thing you are buying on the bottom. For groceries that is usually price per unit of weight or volume, because it lets you compare packages of different sizes directly. A 500 g pack at £3.20 is £6.40 per kg; a 750 g pack at £4.50 is £6.00 per kg, so the larger pack is better value even though it costs more in absolute terms.
Is a unit rate the same as the constant of proportionality?
Yes, for a directly proportional relationship they are the same number. If cost is proportional to weight, the constant of proportionality k in
cost = k × weight is the price per kilogram. The direct proportion calculator approaches it from the equation side; this one from the rate side.What does 1 : n mean on a map or a model?
That one unit of length on the model corresponds to n of the same units in reality. A 1:50 architectural drawing means 1 cm on paper is 50 cm in the building. It is a unit ratio where the model side has been reduced to 1, which is exactly why the form is used — you can multiply any measured length by n without further arithmetic. For the dimensional side of that, use the scale factor calculator.
Can a unit rate be less than 1?
Certainly. If 5 items cost $2, the rate is $0.40 per item. And a rate can be very large or very small depending on which way round you take it — $0.40 per item is the same relationship as 2.5 items per dollar. Both are unit rates; which reads better depends on what you are deciding.
What is the difference between a rate and a ratio?
A ratio compares two quantities of the same kind, so the units cancel and the result is a pure number —
3 : 5 boys to girls. A rate compares quantities of different kinds, so the result carries units — dollars per kilogram, miles per hour. A unit rate is a rate with the denominator reduced to 1. In everyday use "ratio" is often stretched to cover both.How do I handle a rate with awkward numbers?
Divide as normal and round only at the end. If 7 items cost $23, the unit rate is
23 ÷ 7 = $3.2857… — exactly 23/7. For a price you would report $3.29, but if you are then multiplying back up to a large quantity, use the exact fraction to avoid the rounding compounding. This calculator shows both.