Unitary Method
Describe the concept of unitary method
Unitary means “of one”. In unitary method the cost of several objects is given and then by finding the cost of one object we can calculate the cost of many objects.
Calculate the value of many objects of the same kind when the value of one of these objects is given.
If the value of one object is known we can find the value of many objects of the same kind by multiplication. Following example illustrates this process
Example Cost of 1 book is Rs. 20. What is the cost of 10 such books?
Solution: Cost of 1 book = Rs. 20
Cost of 10 books = 20 × 10 = Rs. 200
Cost of 10 books is Rs. 200.
Calculate the value of one object when the value of many objects of the same type is given
Let us explain this method with the help of a few examples.
Example 1 If 4 oranges cost Rs.12, how much do 9 oranges cost?
Solution: Cost of 4 oranges = Rs. 12
Cost of 1 orange = \(12 \over 4\) = Rs. 3
Cost of 9 oranges = 9 × 3 = Rs. 27
Cost of 9 oranges is Rs. 27
Example 2 Cost of 5 pens is Rs 125. What is the cost of 10 pens?
Solution: Cost of 5 pens = Rs. 125
Cost of 1 pen = \(125 \over 5\) = Rs. 25
Cost of 10 pens = 10 × 25 = Rs. 250
Cost of 10 pens is Rs. 250
Example 3 5 workers can complete a work in 22 days. In how many days same work will be completed if 11 workers are employed?
Solution: 5 workers can complete a work in = 22 days
1 worker can complete the work in = 5 × 22 = 110 days
11 workers can complete the work in = \(110 \over 11\) = 10 days
11 workers can complete the work in 10 days.
Direct and Inverse Proportion
Define ratio of two quantities
A ratio is a relation between two quantities of the same kind. It can be expressed as a fraction. The symbol for ratio is a colon (:). Ratio shows how much of one quantity there is as compared to another quantity. Ratios are used to make comparisons between quantities. In general, the ratio of a to b is written as a : b = \(a \over b\)
Examples
Define and identify direct and inverse proportion.
Direct Proportion: It is a relationship between two quantities such that if one increases, other also increases. If one decreases, the other also decreases.
Inverse Proportion: It is a relationship between two quantities such that if one increases, other decreases. If one decreases, the other increases
Some situations of Direct Proportion:
• More articles, more money is required to purchase. Fewer articles, less money is required to purchase. • More men at work, more work is done. Fewer men at work, lesser work is done.
• More money borrowed, more interest is to be paid. Less money borrowed, less interest is to be paid.
• More speed, more distance is covered in fixed time. Less speed, less distance is covered in fixed time. • More working hours, more work will be done. Less working hours, less work will be done.
Some situations of Inverse Proportion.
• More men at work, less time is taken to finish the same work.
• More speed, less time is taken to cover the same distance.
• More men in the camp and less number of days for food stock to last.
• More is the cost less you could buy with the same amount of money.
Solve real life problems involving direct and inverse proportion (by unitary method).
Unitary method can be used to solve real life problems involving direct and inverse proportions. Following examples illustrate the process of solving problems
(a)Direct Proportion by Unitary Method
Example 1 If 12 flowers cost Rs.156, what do 28 flowers cost?
Solution This is the situation of direct proportion as: More flowers result in more cost.
Cost of 12 flowers = Rs. 156
Cost of 1 flower = \(156 \over 12\) = Rs. 13
Cost of 28 flowers = 13 × 28 = Rs. 364
Example 2 A car travels 240 km in 40 litres of petrol. How much distance will it cover in 9 litres of petrol?
Solution This is the situation of direct proportion as: Less quantity of petrol, less distance is to be covered.
In 40 litres of petrol, distance covered = 240 km
In 1 litres of petrol, distance covered = \(240 \over 40\) = 6 km
In 9 litres of petrol, distance covered = 6 × 9=54 km
Example 3 A labourer gets Rs.9800 for 14 days work. How many days should he work to get Rs.21,000?
Solution This is a situation of direct proportion as: more money will be received for working more days.
Number of days to earn Rs. 9,800 = 14 days
Number of days to earn Rs. 1 =\(14 \over 9,800\) days
Number of days to earn Rs. 21,000 = \(14 \over 9,800\)× 21,000 = 30 days.
Therefore, Rs.21,000 can be earned by a labourer in 30 days.
(b) Inverse Proportion by Unitary Method
Real life problems involving inverse proportion can be solved using unitary method. This is illustrated by the following examples.
Example 1 16 men can build a wall in 56 hours. How many men will be required to do the same work in 32 hours?
Solution This is a situation of inverse proportion as: More the number of men, then faster they will build the wall i.e., less number of days needed.
Number of men who build the wall in 56 hours = 16 men
Number of men who build the wall in 1 hour = 16 × 56 men
Number of men who can build the wall in 32 hours =\(\frac{16\times 56}{32}\) = 28 men
Therefore, in 32 hours, wall is built by 28 men.
Example 2 12 typists can type a book in 18 days. In how many days 4 typists will type the same book?
Solution This is a situation of indirect proportion as: Less number of typists will take more days
Number of days in which 12 typists can type a book = 18 days
Number of days in which 1 typist can type a book = 18 × 12
Number of days in which 4 typists can type a book = \(\frac{18\times 12}{4}\) = 54 days
Therefore, 4 typists will type a book in 54 days
Example 3 If 72 workers can do a piece of work in 40 days. How many more days are required to complete the same work if 8 workers left the job?
Solution This is a situation of indirect proportion as: Less workers will require more days to complete the work.
Number of workers left the job = 8
Number of remaining workers to complete work = 72 — 8 = 64
Number of days to complete work by 72 workers = 40 days
Number of days to complete work by 1 worker = 72 × 40
Number of days to complete work by 64 workers =\(\frac{72\times 40}{64}\)= 45 days
More number of days = 45 — 40 = 5 days
Therefore, 64 workers will require 5 more days to complete the same work