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Direct and Inverse Proportions

    Direct Proportion

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8th class > Direct and Inverse Proportions > Direct Proportion

Direct Proportion

If the cost of 1 kg of sugar is ₹ 36, then what would be the cost of 3 kg sugar? It is ₹ .

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 litres (x)Distance in km (y)
460
8
12180
15
20
25
We find that as the value of x increases, value of y also increases in such a way that the ratio xy does not change; it remains constant (say k). In this case, it is 115 (check it!).
We say that x and y are in direct proportion, if xy = k or x = ky.
In this example, 460 = 12180 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 x1y1 = x2y2. (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.

Try these

  1. Observe the following tables and find if x and y are directly proportional.
xy
2040
1734
1428
1122
816
510
24

Solution:

2040 = , and 1734 =,
1428=, and 1122 = ,
816 = , and 510 = , and 24 =
Hence, x and y are .

(ii)

xy
64
108
1412
1816
2220
2624
3028

Solution:

64 = , and 108 =,
1412=, and 1816 = ,
2220 = , and 2624 = , and 3028 =
Hence, x and y are .

Example 1

The cost of 5 metres of a particular quality of cloth is ₹ 210. Tabulate the cost of 2, 4, 10 and 13 metres of cloth of the same type.

Solution

Suppose the length of cloth is x metres and its cost, in ₹ , is y.

x (in ₹ )y
2y2
4y3
5210
10y4
13y5

As the length of cloth increases, cost of the cloth also increases in the same ratio. It is a case of direct proportion.

(i) Here x1 = 5, y1 = 210 and x2 = 2

x1y1=x2y2

  • Here, x1y1= x2y2 gives
  • or 5y2 =
  • or y2=
  • Therefore,the result is 84.

(ii) If x3 = 4, then 5210 = 4y3

5210=4y3

  • Here, x3 = 4
  • So, y3 =
  • Therefore, the result is 168.

(iii) If x4 = 10, then 5210=10y4 or

5210=10y4

  • Here, x4 = 10
  • So, y4 =
  • Therefore, the result is 420.

(iv) If x5 = 13, then 5210=13y5 or

5210=13y5

  • Here, x5 = 13
  • So, y5 =
  • Therefore, the result is 546.

Example 2

If the weight of 12 sheets of thick paper is 40 grams, how many sheets of the same paper would weigh 212 kilograms?

Solution:

Let the number of sheets which weigh 212 kg be x. We put the above information in the form of a table as shown below:

Number of sheetsWeight of sheets (in grams)
1240
x2500

Finding number of sheets

  • 1 kilogram = grams
  • 212 = grams
  • So, 1240 = x2500
  • Therefore x =
  • Thus, the required number of sheets of paper = 750

Alternate method:

Two quantities x and y which vary in direct proportion have the relation x = ky or xy = k
Here, k = number of sheetsweight of sheets in grams= 1240 =
Now x is the number of sheets of the paper which weigh 212 kg = g.
Using the relation x = ky, x =310 x 2500 =
Thus, 750 sheets of paper would weigh 212k.g

Example 3

The scale of a map is given as 1:30000000. Two cities are 4 cm apart on the map. Find the actual distance between them.

Solution:

Let the map distance be x cm and actual distance be y cm, then 1:30000000 = x : y
or 13x107 = xy
Since x = 4 so, 13x107 = 4y
or y = 4 × 3 × 107 = 12 × 107cm = km
Thus, two cities, which are 4 cm apart on the map, are actually 1200 km away from each other.