Unitary Method
ICSE Class 6 · Mathematics · Chapter 13
What this chapter covers
- The value of one unit
- Direct variation
- Unitary method, ratio and proportion
- Word problems using the unitary method
Key definitions
The unitary method is a way of solving a problem in which you work out the value of one unit first, and then use it for the number of units you need.
The value of one unit is the amount that goes with exactly one item, such as the cost of one pen or the distance covered in one hour.
Direct variation is the link between two quantities in which, when one quantity increases, the other increases in the same way, and when one decreases, the other decreases in the same way.
A rate is the value of one unit of a quantity, such as the cost of one kilogram or the distance travelled in one hour.
Key points
- The unitary method solves a problem by first finding the value of one unit.
- Step 1 is division: go down to one. Step 2 is multiplication: go up to many.
- Write the quantity you want to find at the end of each line of working.
- The unitary method works only when two quantities are in direct variation.
- In direct variation, more goes with more and less goes with less, at a steady rate.
- When two quantities are in direct variation, their matching ratios form a proportion.
- A rate such as "₹40 per kg" is already the value of one unit.
- Comparing the value of one unit tells you which offer is cheaper.
- Always write the unit in the final answer.
Questions answered
What is meant by the value of one unit? Give one example from a shop.
The value of one unit is the amount that goes with exactly one item. For example, if 1 kg of onions costs ₹30, then ₹30 is the value of one unit.
8 oranges cost ₹64. Find the cost of 5 oranges.
Cost of 1 orange = ₹64 ÷ 8 = ₹8. Cost of 5 oranges = ₹8 × 5 = ₹40.
A tap fills 45 litres of water in 9 minutes. How much water will it fill in 14 minutes at the same rate?
Water in 1 minute = 45 ÷ 9 = 5 litres. Water in 14 minutes = 5 × 14 = 70 litres.
Are the number of cricket balls bought and the money paid in direct variation? Give a reason.
Yes. If you buy more balls, you pay more money, and if you buy double the number of balls, you pay double the money.
A cyclist rides 36 km in 3 hours at a steady speed. How long will she take to ride 60 km?
Time for 36 km = 3 hours, so 1 km takes (3)/(36) = (1)/(12) hour. Time for 60 km = (1)/(12) × 60 = 5 hours.
Farida pays ₹120 for 6 m of ribbon. How many metres of ribbon can she buy for ₹200?
Cost of 1 m = ₹120 ÷ 6 = ₹20. Length for ₹200 = 200 ÷ 20 = 10 m.
Solve by proportion and by the unitary method: 7 kg of potatoes cost ₹245. Find the cost of 3 kg.
Unitary method: 1 kg costs ₹245 ÷ 7 = ₹35, so 3 kg cost ₹105. Proportion: 7 : 3 :: 245 : x, so 7 × x = 3 × 245 = 735 and x = 105. Both give ₹105.
A 500 g packet of tea costs ₹220 and a 250 g packet costs ₹120. Which is the better buy?
For the 500 g packet, 250 g costs ₹220 ÷ 2 = ₹110. The 250 g packet costs ₹120. So the 500 g packet is the better buy.