Unitary Method (Including Time and Work)
ICSE Class 7 · Mathematics · Chapter 7
What this chapter covers
- The unitary method: one unit first
- Finding the number of units from a value
- The general expression for the unitary method
- Direct variation
- Solving direct variation problems
- Time and work: work done in one day
- Two people working together
- Sharing wages by work done
- More workers, fewer days
Key points
- The unitary method finds the value of one unit first and then the value of the required number of units.
- Step 1 divides to reach one unit; step 2 multiplies to reach the number of units you need.
- To find how many units a sum of money buys, divide the money by the value of one unit.
- General expression: value of b units = (value of a units)/(a) × b.
- Two quantities vary directly when doubling or halving one doubles or halves the other; their quotient stays fixed.
- A direct variation problem can be solved by the unitary method or by writing a proportion.
- In time and work, the whole job is counted as 1.
- A person who finishes a job in n days does (1)/(n) of it in one day, and a person who does (1)/(n) of it in a day takes n days.
- When people work together, add their one day's work, never their days.
- Wages for a shared job are divided in the ratio of the work each person does.
- More workers finish a job in fewer days: find the time for one worker by multiplying, then divide by the new number of workers.
Questions answered
Nine bananas cost ₹54. Find the cost of 15 bananas.
One banana costs 54 ÷ 9 = ₹6, so 15 bananas cost 15 × 6 = ₹90.
A scooter uses 5 litres of petrol to travel 75 km. How much petrol does it need for 120 km?
It travels 75 ÷ 5 = 15 km on 1 litre, so 120 km needs 120 ÷ 15 = 8 litres.
Which of these pairs vary directly? (a) The number of bus tickets bought at a fixed fare and the total fare paid. (b) A child's age and the child's weight.
(a) varies directly, since doubling the tickets doubles the fare. (b) does not, since a child twice as old is not exactly twice as heavy.
Four metres of ribbon cost ₹38. Using the general expression, find the cost of 10 m.
(38)/(4) × 10 = 9.5 × 10 = ₹95.
With ₹1,350, how many chairs can be bought if 6 chairs cost ₹1,620?
One chair costs 1620 ÷ 6 = ₹270, so 1350 ÷ 270 = 5 chairs.
Arjun can finish a clay model in 9 days. What part of it does he finish in 1 day, and in 6 days?
(1)/(9) in 1 day and 6 × (1)/(9) = (2)/(3) in 6 days.
Pooja can weed a vegetable patch in 12 days and Qasim can weed it in 6 days. How long will they take working together?
(1)/(12) + (1)/(6) = (1)/(12) + (2)/(12) = (3)/(12) = (1)/(4) of the patch each day, so 4 days.
Two friends together can paint a room in 8 days. One of them alone takes 24 days. How long would the other take alone?
(1)/(8) - (1)/(24) = (3)/(24) - (1)/(24) = (2)/(24) = (1)/(12), so 12 days.