Unitary Method
If 5 pencils cost Rs 40, how much do 8 pencils cost? To solve this, you first find the cost of 1 pencil (Rs 40 ÷ 5 = Rs 8), and then multiply by 8 (Rs 8 × 8 = Rs 64). This approach — finding the value of one unit first and then finding the value of the required number of units — is called the unitary method.
The unitary method is one of the most useful problem-solving techniques in mathematics. It works for a wide range of problems — cost and quantity, speed and distance, work and time, and many more.
The key idea is simple: find the value of 1, then multiply. This two-step approach can solve most ratio and proportion problems at this level.
In this chapter, you will learn the steps of the unitary method, practise with many word problems, and understand when to use it in daily life.
What is Unitary Method - Grade 6 Maths (Ratio and Proportion)?
Definition: The unitary method is a technique of solving a problem by first finding the value of a single unit (one item) and then finding the value of the required number of units by multiplication.
Two steps:
- Step 1: Find the value of 1 unit (divide the total value by the number of units).
- Step 2: Find the value of the required units (multiply the value of 1 unit by the required number).
Key terms:
- Unit: A single item, one piece, one kilogram, one metre, one hour, etc.
- Direct variation: When one quantity increases, the other also increases (more pencils cost more money).
Important:
- The unitary method works when quantities are in direct proportion — more items cost more, more workers do more work in the same time.
- Always state what you are finding: "Cost of 1 pencil = ...", "Distance in 1 hour = ..."
Unitary Method Formula
The Unitary Method Formula:
Value of 1 unit = Total value ÷ Number of units
Value of required units = Value of 1 unit × Required number
In one line:
Required value = (Total value ÷ Given units) × Required units
Where:
- Total value = the known amount (cost, distance, weight, etc.)
- Given units = the known number of items
- Required units = the number you want to find the value for
Derivation and Proof
Why the unitary method works:
The unitary method is based on the idea of direct proportion. When two quantities are in direct proportion, if one doubles, the other also doubles. If one triples, the other triples.
Example: If 3 kg of rice costs Rs 180, then:
- Cost of 1 kg = 180 ÷ 3 = Rs 60 (Step 1: find the unit value)
- Cost of 7 kg = 60 × 7 = Rs 420 (Step 2: multiply for required quantity)
Why divide first, then multiply?
Dividing gives us the rate (cost per unit, speed per hour, etc.). Once we know the rate, we can find the value for any number of units by multiplying.
Checking with common sense:
- 3 kg costs Rs 180
- 7 kg is more than 3 kg, so it should cost more than Rs 180
- Rs 420 > Rs 180 — makes sense!
Another check:
- If 3 kg = Rs 180, then 6 kg (double) should be Rs 360 (double).
- 6 × 60 = 360. Correct!
Types and Properties
Types of unitary method problems:
Type 1: Cost and Quantity
- Given the cost of some items, find the cost of more (or fewer) items.
- Example: 5 books cost Rs 150. Find the cost of 12 books.
Type 2: Work and Time
- Given that some workers complete a job in some time, find how long for a different number of workers.
- Note: More workers = less time (this is inverse, not direct — covered in later classes).
Type 3: Distance and Time
- Given speed (distance in some time), find distance in different time.
- Example: A car travels 240 km in 4 hours. How far in 7 hours?
Type 4: Weight and Cost
- Given the cost of some weight, find cost of different weight.
- Example: 2 kg of apples cost Rs 300. Find cost of 3.5 kg.
Type 5: Quantity from Cost
- Given a budget, find how many items you can buy.
- Example: Each pen costs Rs 12. How many pens can you buy with Rs 180?
Type 6: Mixed Real-Life Problems
- Fuel consumption, recipe scaling, map distances, etc.
Solved Examples
Example 1: Example 1: Cost of Items
Problem: If 6 notebooks cost Rs 180, find the cost of 10 notebooks.
Solution:
Step 1: Find the cost of 1 notebook.
- Cost of 1 notebook = 180 ÷ 6 = Rs 30
Step 2: Find the cost of 10 notebooks.
- Cost of 10 notebooks = 30 × 10 = Rs 300
Answer: 10 notebooks cost Rs 300.
Example 2: Example 2: Distance and Time
Problem: A bus travels 150 km in 3 hours. How far will it travel in 5 hours (at the same speed)?
Solution:
Step 1: Distance in 1 hour = 150 ÷ 3 = 50 km
Step 2: Distance in 5 hours = 50 × 5 = 250 km
Answer: The bus will travel 250 km in 5 hours.
Example 3: Example 3: Weight and Cost
Problem: 4 kg of mangoes cost Rs 320. Find the cost of 7 kg.
Solution:
Step 1: Cost of 1 kg = 320 ÷ 4 = Rs 80
Step 2: Cost of 7 kg = 80 × 7 = Rs 560
Answer: 7 kg of mangoes cost Rs 560.
Example 4: Example 4: Finding Number of Items
Problem: Each pen costs Rs 15. How many pens can Anu buy with Rs 225?
Solution:
- Number of pens = Total money ÷ Cost of 1 pen
- = 225 ÷ 15 = 15 pens
Answer: Anu can buy 15 pens.
Example 5: Example 5: Recipe Scaling
Problem: A recipe for 4 people needs 600 g of flour. How much flour is needed for 10 people?
Solution:
Step 1: Flour for 1 person = 600 ÷ 4 = 150 g
Step 2: Flour for 10 people = 150 × 10 = 1500 g = 1.5 kg
Answer: 1500 g (1.5 kg) of flour is needed for 10 people.
Example 6: Example 6: Fuel Consumption
Problem: A car uses 8 litres of petrol to travel 120 km. How much petrol is needed for 300 km?
Solution:
Step 1: Petrol for 1 km = 8 ÷ 120 = 1/15 litres
Step 2: Petrol for 300 km = (1/15) × 300 = 20 litres
Answer: 20 litres of petrol is needed.
Example 7: Example 7: Earning and Days
Problem: A worker earns Rs 4,500 in 6 days. How much will he earn in 15 days?
Solution:
Step 1: Earning per day = 4500 ÷ 6 = Rs 750
Step 2: Earning in 15 days = 750 × 15 = Rs 11,250
Answer: The worker earns Rs 11,250 in 15 days.
Example 8: Example 8: Map Scale
Problem: On a map, 2 cm represents 50 km. What distance does 7 cm represent?
Solution:
Step 1: 1 cm represents = 50 ÷ 2 = 25 km
Step 2: 7 cm represents = 25 × 7 = 175 km
Answer: 7 cm represents 175 km.
Example 9: Example 9: Finding Total Cost with Decimal Price
Problem: 3 metres of cloth cost Rs 337.50. Find the cost of 5 metres.
Solution:
Step 1: Cost of 1 metre = 337.50 ÷ 3 = Rs 112.50
Step 2: Cost of 5 metres = 112.50 × 5 = Rs 562.50
Answer: 5 metres cost Rs 562.50.
Example 10: Example 10: Exchange Rate
Problem: Rs 4,200 can be exchanged for 50 US dollars. How many dollars can you get for Rs 8,400?
Solution:
Step 1: Rs 1 = 50 ÷ 4200 dollars (but it is easier to see the pattern)
Notice: Rs 8,400 = 2 × Rs 4,200
So dollars = 2 × 50 = 100 dollars
Or using unitary method:
- Rs 4200 = 50 dollars
- Rs 1 = 50/4200 dollars
- Rs 8400 = (50/4200) × 8400 = 50 × 2 = 100 dollars
Answer: You can get 100 US dollars.
Real-World Applications
Unitary method in daily life:
- Shopping: If 3 kg of sugar costs Rs 135, the unitary method tells you the cost of any quantity.
- Cooking: Scaling a recipe up or down — if a recipe for 6 needs 3 cups of rice, how much for 10 people?
- Travel: If a car covers 60 km in 1 hour, how far in 4.5 hours? Distance = 60 × 4.5 = 270 km.
- Maps: Using map scales to find actual distances — if 1 cm = 25 km, then 4 cm = 100 km.
- Currency exchange: Converting between currencies using the exchange rate.
- Wages: Calculating daily, weekly, or monthly earnings from an hourly rate.
Key Points to Remember
- The unitary method finds the value of 1 unit first, then multiplies for the required number of units.
- Step 1: Value of 1 unit = Total value ÷ Number of units.
- Step 2: Required value = Value of 1 unit × Required number.
- This method works when quantities are in direct proportion (more items = more cost).
- Always label your units clearly: "Cost of 1 pen = Rs 12", "Distance in 1 hour = 50 km".
- Use common sense to check your answer — if you are buying more, the cost should be more.
- The unitary method is closely related to ratio and proportion.
- It can handle decimal values as well (Rs 112.50 per metre).
- For inverse proportion (more workers = less time), the method is slightly different — you will learn this in Grade 7.
- The unitary method is one of the most practical maths skills for everyday life.
Practice Problems
- If 8 oranges cost Rs 120, find the cost of 15 oranges.
- A train covers 360 km in 4 hours. How far will it go in 7 hours at the same speed?
- 5 kg of rice costs Rs 225. Find the cost of 12 kg.
- A car uses 6 litres of petrol for 90 km. How much petrol is needed for 210 km?
- Ravi earns Rs 6,000 in 8 days. How much will he earn in 20 days?
- On a map, 3 cm represents 75 km. What distance does 10 cm represent?
- 12 packets of chips cost Rs 480. How many packets can you buy for Rs 200?
- A factory produces 350 toys in 5 days. How many toys will it produce in 12 days?
Frequently Asked Questions
Q1. What is the unitary method?
The unitary method is a technique where you find the value of 1 unit (one item, one kg, one hour) first, and then use that to find the value of any number of units by multiplying.
Q2. What are the two steps of the unitary method?
Step 1: Find the value of 1 unit by dividing. Step 2: Find the value of the required number of units by multiplying. For example, if 5 pens cost Rs 75: cost of 1 pen = 75 ÷ 5 = Rs 15, cost of 8 pens = 15 × 8 = Rs 120.
Q3. When does the unitary method work?
The unitary method works when two quantities are in direct proportion — when one increases, the other increases at the same rate. For example, more items cost more money. It does not directly work for inverse proportion (more workers = less time).
Q4. Can the unitary method be used with decimals?
Yes. If 3 metres of cloth cost Rs 337.50, cost of 1 metre = 337.50 ÷ 3 = Rs 112.50. Cost of 5 metres = 112.50 × 5 = Rs 562.50.
Q5. How is the unitary method related to ratio?
The unitary method is essentially solving a proportion. If 5 items cost Rs 100, then 8 items cost Rs x. The proportion is 5/100 = 8/x. The unitary method solves this by finding the cost of 1 item first.
Q6. How do I find the number of items I can buy with a given amount?
First find the cost of 1 item. Then divide your total budget by the cost of 1 item. For example, each chocolate costs Rs 20. With Rs 300, you can buy 300 ÷ 20 = 15 chocolates.
Q7. What if the value of 1 unit is a fraction or decimal?
That is fine. Work with the fraction or decimal. If 7 books cost Rs 350, cost of 1 book = 350/7 = Rs 50 (exact). If 3 items cost Rs 100, cost of 1 = 100/3 = Rs 33.33 (approximately). Keep the fraction for exact answers.
Q8. Can I skip Step 1 and solve directly?
You can set up a proportion and solve directly: if 5 items = Rs 100, then 8 items = (100/5) × 8 = Rs 160. This combines both steps into one calculation, but the logic is the same.
Q9. What is the difference between unitary method and cross multiplication?
They are closely related. The unitary method finds the value of 1 unit and multiplies. Cross multiplication uses proportions (a/b = c/d → ad = bc). Both give the same answer. The unitary method is more intuitive for beginners.
Q10. Does the unitary method work for speed-distance-time problems?
Yes. If a car travels 240 km in 4 hours, speed (value of 1 hour) = 240 ÷ 4 = 60 km/h. Distance in 7 hours = 60 × 7 = 420 km. This is the unitary method applied to distance and time.
