Tuesday, March 22, 2016

Chapter 3.10 - Inverse proportions - Solved examples

In the previous sections we saw some solved examples on inverse proportions. In this section, we will see a few more.
Solved example 3.21 
A student cycles to school at an average speed of 8 km/hr. He reaches the school in 15 minutes. At what speed should he ride, if he has to reach the school in 10 minutes?
Solution: We will form the table directly from the data. It is shown below:


The distance from home to school remains the same. So if speed increases, the time decreases, and vice versa. Thus it is a case of inverse proportion. 
• The new time 10 minutes and the corresponding new speed x1 should be given appropriate places in the table. 
• Also, x1 and 10 will give the same constant 120. 
So we can write:
10 × x1 = 120 ⇒ x1 = 120 / 10 = 12 km/hr
Solved example 3.22
A work force of 32 men can complete a job in 90 days. 
(i) How many days will it take if the work force is reduced to 24. 
(ii) How many more men should join the present work force, if the job is to be completed in 75 days?
Solution:
The table is given below:

(i) We know that it is a case of inverse proportion because, when the work force increases, time decreases and vice versa. 
• So the new work force 25 and the corresponding new time y1 is given appropriate places in the table as shown above. 
• Also 25 and y1 should give the same constant 2880. 
So we can write:
24 × y1 = 2880 ⇒ y1 = 2880 / 24 = 120 days


(ii) This is an indirect question. They are asking how many 'more'?. But it should not cause any difficulty. We can find the total new number required. This we do by the usual method. Then subtracting the 'old total number of workers' from the 'new total number of workers' will give us the 'more' number of workers.
The new time 48 days and the corresponding new work force x1 is given appropriate places in the table as shown above. 
• Also 48 and x1 should give the same constant 2880. 
So we can write:
48 × x1 = 2880 ⇒ x1 = 2880 / 48 = 60 workers
So the number of workers needed more = 60 -32 = 28
Solved example 3.23
In a hostel, there are 75 people. The available provisions would last for 35 days. Some people left the hostel on vacation. Now it was found that the provisions would last for 105 days. How many people leave on vacation?
Solution:
This is an indirect question just as in part (ii) of the previous question. We will do it in the usual way to find the final number and then do the subtraction. The table is given below:

• The new time 105 days and the corresponding new no. of people x1 is given appropriate places in the table as shown above. 
• Also 105 and x1 should give the same constant 2625. 
So we can write:
105 × x1 = 2625 ⇒ x1 = 2625 / 105 = 25
So the number of people who left = 75 -25 = 50
Solved example 3.24

Two quantities x and y are inversely proportional to each other. Fill up the missing values in the table given below:

Solution:
• In the table, some columns have x value. They miss the corresponding y value. We have to calculate this missing y value in such columns.
• Some columns have y value. They miss the corresponding x value. We have to calculate this missing x value in such columns.
• We know the equation xy =k. If we have the value of k, and any one of x and y, the other missing value can be calculated ( x = k/y and y = k/x). For this method, we have to know the value of k. 
• But k is not given. Or is it?
• k is hidden in the table. If we look at column (vi), we will find that, in it, neither x nor y is missing. We can multiply those values to get k. Thus we get k = 45 × 8 = 360
• Once we have k, we can calculate any missing value. So we proceed as follows:

♦ column (ii):    y = k/x  y = 360/12 = 30
♦ column (iii):   x = k/y  x = 360/90 = 4
♦ column (iv):   y = k/x  y = 360/3 = 120
♦ column (v):    x = k/y  x = 360/45 = 8
♦ column (vii):  x = k/y  x = 360/9 = 40


In the next section, we will see inverse variation of angular measurements.

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