Class 5 Mathematics Chapter 6 Complete Guide Book — Unitary Method

The unitary method has one beautiful idea: first find the value of one thing, then you can find the value of anything. The word 'unitary' itself comes from 'unit' — one. Everything in this chapter starts with that first step.

To find the cost of one item, divide the total cost by the number of items. If 6 eggs cost Rs 90, one egg costs 90 ÷ 6 = Rs 15. If 4 kg of sugar costs Rs 480, one kg costs 480 ÷ 4 = Rs 120. And if a worker earns Rs 900 in 3 days, one day's pay is 900 ÷ 3 = Rs 300. Same step every time: total ÷ how many.

Why bother finding one first? Because one is the key that opens every other door. Once you know one egg costs Rs 15, you can price 2 eggs, 10 eggs or 100 eggs without any new information. Students who skip this step and guess usually go wrong; students who find one first almost never do.

Once you know the cost of one, the cost of many is just a multiplication. One notebook costs Rs 60, so 7 notebooks cost 60 × 7 = Rs 420. One metre of cloth costs Rs 250, so 6 metres cost 250 × 6 = Rs 1,500. The pattern never changes: unit price × how many = total price.

Notice the two steps together: divide first to find one, multiply second to find many. That divide-then-multiply pair is the whole unitary method. Writing both steps down, even when the numbers look easy, protects you from the commonest slip — multiplying the total price instead of the unit price.

Always check your answer against common sense. Seven notebooks at Rs 60 each must cost more than one notebook (Rs 60) and less than Rs 700 (which would be Rs 100 each). Rs 420 sits nicely in between. If your answer for many items is smaller than the price of one item, something has gone badly wrong.

Sometimes the question runs backwards: you know the money in your pocket and you want to know how many items it buys. The method is still the same, just turned around. First find the price of one, then divide your money by that price.

With Rs 500, how many Rs 50 toys can you buy? Here the unit price is already given (Rs 50), so one step is enough: 500 ÷ 50 = 10 toys. But with Rs 100 for toffees, you must first find the unit price: 12 toffees cost Rs 60, so one toffee costs 60 ÷ 12 = Rs 5, and then 100 ÷ 5 = 20 toffees. Two steps, same idea.

A remainder here means real money left over. If you had Rs 105 for Rs 5 toffees, you could buy 21 exactly; if you had Rs 103, you could still only buy 20, because the shop will not sell you part of a toffee. Always round down when buying whole items — you cannot buy more than your money allows.

The unitary method is not only about shopping. A car travelling 90 km in 1 hour travels 90 × 4 = 360 km in 4 hours at the same speed — the 'one hour' value does all the work. A pump filling 240 litres in 4 hours fills 240 ÷ 4 = 60 litres in 1 hour, so filling 900 litres takes 900 ÷ 60 = 15 hours. Find one hour first, then scale.

It also settles arguments about bargains. One shop sells 5 pencils for Rs 75 (Rs 15 each) and another sells 8 pencils for Rs 104 (Rs 13 each). The second shop looks dearer at first glance — Rs 104 against Rs 75 — but per pencil it is cheaper. Comparing totals misleads; comparing unit prices never does.

But beware one opposite case. If 10 workers build a wall in 6 days, 5 workers will not take fewer days — they will take more, because fewer hands do the same work more slowly. The total work is 10 × 6 = 60 worker-days, so 5 workers need 60 ÷ 5 = 12 days. Here more workers means fewer days, so the method flips: multiply first to find the total work, then divide. Spotting this flip is the mark of a careful student.