TIME AND WORK FORMULA & CONCEPTS

COMPOUND VARIATION:

1) If two quantities are such that an increase or decrease in one quantity makes a corresponding increase or decrease (same effect) in the other quantity, then they are said to be in direct proportion or said to vary directly.

2) x and y are said to vary directly if y = kx always, where k is called the proportionality constant and k > 0 assuming that y depends on x and so k = y/x.

3) Examples of Direct Proportion:

1. Distance –Time (under constant speed): If the distance increases, then the time taken to reach that distance will also increase and vice- versa.

2. Purchase – Spending: If the purchase on utilities for a family during the festival time increases, then the spending limit also increases and vice versa.

3. Work Time – Earnings: If the number of hours worked is less, then the pay earned will also be less and vice-versa.

4) If two quantities are such that an increase or decrease in one quantity makes a corresponding decrease or increase (opposite effect) in the other quantity, then they are said to be in inverse (indirect) proportion or said to vary inversely.

5) x and y are said to vary inversely, if xy = k always, where k is called proportionality constant and k > 0.

6) Examples of Inverse Proportion:

1. Price – Consumption: If the price of consumable products increases, then naturally its consumption will decrease and vice-versa.

2. Workers – Time: If more workers are employed to complete a work, then the time taken to complete the work will be less and vice-versa.

3. Speed – Time (Fixed Distance): If we travel with less speed, then the time taken to cover a given distance will be more and vice-versa.

7) There will be problems which may involve a chain of two or more variations in them. This is called as Compound Variation.

8) The different possibilities of two variations are:

Direct-Direct, Direct-Inverse, Inverse-Direct, Inverse-Inverse.

9) Proportion Method: In this method, we shall compare the given data and find whether they are in direct or inverse proportion. By finding the proportion, we can use the fact that

The product of the extremes = The product of the means

and get the value of the unknown (x).

10) Formula Method:

Identify the data from the given statement as Persons (P), Days (D), Hours (H) and Work (W) and use the formula,

where the suffix 1 contains the complete data from the first statement of the given problem and the suffix 2 contains the unknown data to be found out in the second statement of the problem.

That is, this formula says, P₁ men doing W₁ units of work in D₁ days working H₁ hours per day is the same as P₂ men doing W₂ units of work in D₂ days working H₂ hours per day. Identifying the work W₁ and W₂ correctly is more important in these problems. This method will be easy for finding the unknown (x) quickly.

TIME AND WORK:

11) Work to be done is usually considered as one unit. Work can be in any form like building a wall, making a road, filling or emptying a tank or even eating a certain amount of food.

12) Time is measured in hours, days etc., Certain assumptions are made that the work so done is uniform and each person shares the same work time in case of group work in completing the work.

13) Unitary Method:

If two persons X and Y can complete some work individually in a and b days, then their one day’s work will be 1/a and 1/b respectively.

Working together, their one day’s work = 1/a + 1/b = (a + b)/ab and so,

X and Y together can complete the work in ab/(a + b) days.

14) The time taken to complete a work or task depends on various factors such as number of  persons, their capacity to do the work, the amount of work and the time spent per day for the completion of work.

15) If A is a/b times as good a worker as B, then A will only b/a of the time taken by B to complete the work.

16) Sharing of the money for work:

When a group of people do some work together, based on their individual work they get a share of money themselves. In general, money earned is shared by people, who worked together, in the ratio of the total work done by each of them.

17) If the ratio of the time taken by A and B in doing a work is x : y, then the ratio of work done by A and B is 1/x : 1/y = y : x. This is the ratio for their separate wages too.

18) If three persons A, B and C can do a work in x, y and z days respectively, then the ratio in which their wages will be distributed to them is 1/x : 1/y : 1/z.