Unitary Method Calculator
If you know what one quantity is worth, this works out any other. It shows both steps — find one, then multiply.
Answer
$96.00
- One costs
- $12.00
- 8 cost
- $96.00
- You knew
- 5 for $60.00
- Step one: 5 cost $60.00, so one costs $60.00 ÷ 5 = $12.00.
- Step two: if one costs $12.00 then 8 cost $12.00 × 8 = $96.00.
- Always find one first. That is the whole method, and it is why it is called unitary — once you know what a single one is worth, any number of them is a multiplication.
- It only works when the relationship is proportional. Twice as many pens cost twice as much; two people do not dig a hole in twice the time, they do it in half.
About the unitary method
The unitary method solves a whole family of questions with one idea: work out the value of a single unit, then multiply by however many you want. If five pens cost 60, one pen costs 12, so eight pens cost 96.
It is deliberately two steps rather than one. A cross multiplication gets the same answer in a single line, but the unit value is the thing worth having — it is what you compare with, and it is what makes the second step obvious rather than mechanical.
It works whenever the relationship is proportional: twice as many pens cost twice as much, and half as many cost half as much. Most price, quantity, speed and recipe questions are like this.
Not everything is. Some relationships are inversely proportional — if one person digs a hole in six hours, two people take three, not twelve — and there the unit is "person-hours" rather than hours. Others are not proportional at all: a taxi with a flag fall, or a phone plan with an allowance included. Checking that doubling the input really doubles the output is the test worth applying before reaching for this.
What it works out
- The value of one unit
- The value of any number of units
- Both steps written out
- Any proportional relationship
The formula
One = Value ÷ Quantity Answer = One × Wanted
Step one: divide to find what a single unit is worth. Five pens cost $60, so one pen costs 60 ÷ 5 = $12.
Step two: multiply to find what you want. Eight pens at $12 each is 8 × 12 = $96.
Always find one first. That is the whole method, and it is where the name comes from — once a single unit is priced, any number of them is a multiplication you can do without thinking.
The unit value is useful in its own right. Knowing that pens are $12 each lets you compare against any other shop's offer immediately, whatever quantity they are selling in — which is the same reasoning behind unit pricing on a supermarket shelf.
Check the relationship is proportional before you start. Doubling the quantity should double the value. If there is a fixed charge on top, or if more of one thing means less of another, this method does not apply unchanged.
- Quantity
- How many you already know the value of. It cannot be zero.
- Value
- What that quantity is worth in total.
- Wanted
- How many you want the value of.
A worked example
Five pens cost $60. What do eight cost?
That works out to $96.00 .
- One costs
- $12.00
- 8 cost
- $96.00
- You knew
- 5 for $60.00
Questions
What is the unitary method?
Finding the value of one unit and then multiplying. If five pens cost $60, one costs $12, so eight cost $96.
Why find the value of one first?
Because it turns every other quantity into a single multiplication, and because the unit value is what you compare with. It is also the step a marker is looking for in the working.
When does the unitary method not work?
When the relationship is not proportional. A taxi fare with a fixed flag fall, or a phone plan with an allowance included, does not double when the distance or the usage doubles.
What about inverse proportion?
Then the unit is different. If one person digs a hole in six hours, the job is six person-hours, so two people take three hours rather than twelve. Find the total work first, then divide.
How is this different from cross multiplication?
It gets the same answer in two steps instead of one. Cross multiplication is quicker; the unitary method shows the unit value, which is usually the more useful number and the one the question is teaching.
Can I use it for speed and distance?
Yes. If a car goes 240 km in three hours, one hour is 80 km, so five hours is 400 km. The unit is one hour rather than one item, and the method is identical.
Can I use it for recipes?
That is one of its most common uses. Work out the quantity per person, then multiply by the number you are cooking for. Only the ingredients scale — the cooking time does not.
What if the numbers do not divide exactly?
The unit value can be a fraction or a decimal and the method still works. Three items for $10 makes one item $3.33, and seven of them $23.33.