Unitary Method

Unitary Method

The unitary method is a technique used to find the value of a different number of identical items when the value (or quantity) for some other number is given. The method works by first finding the value for a single unit and then scaling that value to the required number of units. The unitary method is based on the concept of proportion, which occurs in two standard types.

Direct proportion

Two quantities are in direct proportion when an increase (or decrease) in one causes an increase (or decrease) in the other. In direct proportion, as one quantity grows, the other grows by the same factor.

Example: If the number of workers increases, the total wages to be paid (for the same period and rate) will increase. If the number of workers decreases, the wages decrease.

Indirect (inverse) proportion

Two quantities are in indirect or inverse proportion when an increase in one causes a decrease in the other, or vice versa. In inverse proportion, if one quantity is multiplied by a factor, the other is divided by the same factor.

Examples:

• If more workers are employed to do a task, the time taken to finish the task decreases (for the same total work).
• If the speed of a vehicle decreases, the time to travel a fixed distance increases, and vice versa (more speed, less time).

 SOLVED EXAMPLES

 1. Some stock of fodder lasts for 36 days for 20 cows. How long will it last if no. of cows decreases to 15. 
Solutions: For 20 cows the stock lasts for = 36 days

For 1 cows the stock lasts for = 36 x 20

For 15 cows the stock last for =  = 48 days

2. A Canteen requires of 525 kg of wheat every week. How many kilograms of wheat is required in the month of Sept. 
Solutions: Wheat required in 7 days = 525 kg

Wheat required in 1 day = 525 / 7

 Wheat required in 30 days =  =  2250 kg

3) The cost of 8 diaries and 12 calendars is Rs. 3138. Find the cost of 12 diaries and 18 calendars. 
Solutions: 8 diaries and 12 calendars cost Rs. 3138

12 diaries and 18 calendars are 3/2 times of 8 diaries and 12 calendars. So the cost is also 3/2 times the cost of Rs. 3138.

So  = Rs. 4707

4. Total cost of 5 pens and 6 pencils is Rs. 84. The cost of pencil is equal to one third of the cost of a pen. Find the cost of 4 pens and 5 pencils. 
Solutions: It is given that cost of one pencil = 1/3 the cost of a pen

So 6 pencils = 6/3 pens = 2 pens

Cost of 5 pens and 6 pencils = cost of 5 pens + 2 pens

So Cost of 7 pens = Rs. 84

So cost of one pen is 84/7 = Rs. 12

And cost of one pencil = 12/3 = Rs. 4

Cost of 4 pens and 5 pencils = 4 x 12 + 5 x 4 = Rs. 68

 
5. 300 apples are equally distributed amongst certain number of students. If there were 10 more students, each will get one apple less. Find total number of students. 
Solution: Let the number of students be x

Each will get (300 / x) apples.

If no. of students is x + 10 each will get (300 / x) - 1 apples

or - x = 10

or 3000 - x² = 10x

or x² + 10x - 3000 = 0

or x² + 60x - 50x - 3000 = 0

or x (x + 60) - 50 (x + 60) = 0

(x - 50) (x + 60) = 0

X = 50 or - 60

So x = 50

6. One seventh of a number is 51. Find 64% of that number

Solutions: One seventh of a member is 51

So the number is 51 x 7 = 357

64% of 357 = 357 x (64 / 100) = 5712 / 25 = 228.48

7. A certain number of sweets were distributed among 56 children, equally. Each child got 8 sweets and 17 sweets were left. Find the total number of sweets?  

Solutions: Total sweets = 56 x 8 + 17

= 448 + 17 = 465

 
8. A person takes 3 minutes to write a letter. During 10 am to 12 noon 1960 letters are to be written. How many persons should be employed to complete this job in time.
Solutions: No. of person to be employed =

9. Sweets were to be distributed equally among 200 children. 40 children were absent, so each child got 2 sweets extra. How many sweets were distributed? 

Solution: Let the sweets be x

Each child were to get (x / 200) sweets

Children presence = 200 - 40 = 160

and each child got x / 200 + 2 sweets

or 40x / 200 = 320 or x =  = 1600 sweets

Using the Unitary Method - General procedure

Follow these steps when using the unitary method:

• Identify whether the relation is direct or inverse proportion.
• Find the value corresponding to one unit (the 'unit value').
• Scale the unit value to find the value for the required number of units.
• For inverse proportion, remember to invert the factor when scaling (e.g., if quantity doubles, the related quantity halves).

Keywords and expressions to remember

• Unit - the single instance used to scale up or down.
• Direct proportion - multiply the unit value by the factor of change.
• Inverse proportion - divide the unit value by the factor of change (or multiply by reciprocal).
• Always check units (days, kg, rupees, minutes, etc.) and the direction of change before applying the method.