Direct and Inverse Proportions

Objective:

This topic gives an overview of; 

Direct Proportion

  • Direct Proportion
  • Inverse Proportion

Introduction

Mohan prepares tea for himself and his sister. He uses 300 mL of water, 2 spoons of sugar, 1 spoon of tea leaves and 50 mL of milk. How much quantity of each item will he need, if he has to make tea for five persons?

If two students take 20 minutes to arrange chairs for an assembly, then how much time would five students take to do the same job? We come across many such situations in our day-to-day life, where we need to see variation in one quantity bringing in variation in the other quantity.

For example:

  1. If the number of articles purchased increases, the total cost also increases.
  2. More the money deposited in a bank, more is the interest earned.
  3.  As the speed of a vehicle increases, the time taken to cover the same distance decreases.
  4. For a given job, more the number of workers, less will be the time taken to complete the work. Observe that change in one quantity leads to change in the other quantity. Write five more such situations where change in one quantity leads to change in another quantity.

How do we find out the quantity of each item needed by Mohan? Or, the time five students take to complete the job? To answer such questions, we now study some concepts of variation.

Direct Proportion

If the cost of 1 kg of sugar is Rs 18, then what would be the cost of 3 kg sugar? It is Rs 54.

Similarly, we can find the cost of 5 kg or 8 kg of sugar. Study the following table.

Observe that as weight of sugar increases, cost also increases in such a manner that their ratio remains constant.

Take one more example. Suppose a car uses 4 litres of petrol to travel a distance of 60 km. How far will it travel using 12 litres? The answer is 180 km. How did we calculate it? Since petrol consumed in the second instance is 12 litres, i.e., three times of 4 litres, the distance travelled will also be three times of 60 km. In other words, when the petrol consumption becomes three-fold, the distance travelled is also three fold the previous one. Let the consumption of petrol be x litres and the corresponding distance travelled be y km . Now, complete the following table:

Petrol in Litter (x) 4 8 12 15 20 25
Distance in KM (y) 60   --- 180   ---   ---    ---

We find that as the value of x increases, value of y also increases in such a way that the ratio «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»x«/mi»«mi»y«/mi»«/mfrac»«/math»does not change; it remains constant (say k). In this case, it is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»15«/mn»«/mfrac»«/math».

We say that x and y are in direct proportion, if  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»x«/mi»«mi»y«/mi»«/mfrac»«/math»= k or x = ky.

In this example, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»4«/mn»«mn»60«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»12«/mn»«mn»180«/mn»«/mfrac»«/math» ,where 4 and 12 are the quantities of petrol consumed in litres (x) and 60 and 180 are the distances (y) in km. So when x and y are in direct proportion, we can write«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«/mfrac»«/math» [y1,y2  are values of y corresponding to the values x1, x2 of x respectively].

The consumption of petrol and the distance travelled by a car is a case of direct proportion. Similarly, the total amount spent and the number of articles purchased is also an example of direct proportion.

Think of a few more examples for direct proportion. Check whether Mohan [in the initial example] will take 750 mL of water, 5 spoons of sugar,  2«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math» spoons of tea leaves and 125 mL of milk to prepare tea for five persons! 

Inverse Proportion

Two quantities may change in such a manner that if one quantity increases, the other quantity decreases and vice versa. For example, as the number of workers increases,time taken to finish the job decreases. Similarly, if we increase the speed, the time taken to cover a given distance decreases.

To understand this, let us look into the following situation.

Zaheeda can go to her school in four different ways. She can walk, run, cycle or go by car. Study the following table.

Observe that as the speed increases, time taken to cover the same distance decreases.

As Zaheeda doubles her speed by running, time reduces to half. As she increases her speed to three times by cycling, time decreases to one third. Similarly, as she increases her speed to 15 times, time decreases to one fifteenth. (Or, in other words the ratio by which time decreases is inverse of the ratio by which the corresponding speed increases). Can we say that speed and time change inversely in proportion?

Let us consider another example. A school wants to spend Rs 6000 on mathematics textbooks. How many books could be bought at Rs 40 each? Clearly 150 books can be bought. If the price of a textbook is more than Rs 40, then the number of books which could be purchased with the same amount of money would be less than 150. Observe thefollowing table.

Price of each book (in Rs) 40 50 60 75 80 100
Number of books that can be bought 150 120 100 80 75 60

What do you observe? You will appreciate that as the price of the books increases, the number of books that can be bought, keeping the fund constant, will decrease.

Ratio by which the price of books increases when going from 40 to 50 is 4 : 5, and the ratio by which the corresponding number of books decreases from 150 to 120 is 5 : 4. This means that the two ratios are inverses of each other. Notice that the product of the corresponding values of the two quantities is constant; that is, 40 × 150 = 50 × 120 = 6000.

If we represent the price of one book as x and the number of books bought as y, then as x increases y decreases and vice-versa. It is important to note that the product xy remains constant. We say that x varies inversely with y and y varies inversely with x. Thus two quantities x and y are said to vary in inverse proportion, if there exists a relation of the type xy = k between them, k being a constant. If y1, y2 are the values of y corresponding to the values x1, x2 of x respectively then x1y1 = x2y2 (= k), or «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mfrac»«/math».

We say that x and y are in inverse proportion.

Hence, in this example, cost of a book and number of books purchased in a fixed amount are inversely proportional. Similarly, speed of a vehicle and the time taken to cover a fixed distance changes in inverse proportion.

Thin of more such examples of pairs of quantities that vary in inverse proportion. You may now have a look at the furniture – arranging problem, stated in the introductory part of this chapter. 

Summary:

  • Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.
  • That is if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»x«/mi»«mi»y«/mi»«/mfrac»«mo»=«/mo»«mi»k«/mi»«/math»[ is a positive number], then x and y are said to vary directly. In such a case if y1, y2 are the values of  y corresponding to the values x1, xof x respectively then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«/mfrac»«/math» .
  • Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant.
  • That is, if xy = k, then x and y are said to vary inversely. In this case if y1,yare the values of y corresponding to the values x1,x2 of  respectively then x1y1=x2yor «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«/mfrac»«mo»=«/mo»«mfrac»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mfrac»«/math».

 

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