Direct Proportion

If the cost of 1 litre of milk is Rs 9, then what would be the cost of 5 litres of milk? It is Rs45.

Similarly we can find the cost of 7lts or 9lts of milk.

 Litre of milk (ltrs) 1 3 5 7 9 11 13 Cost (in Rs) 9 27 45 63 81 ..... .....

Note that as litre of milk increases, cost also increases is such a way that their ratio remains constant.

Example

Suppose a jeep uses 4 litres of petrol to travel a distance of 50km. How far will it travel using 8litres?

Solution

The answer is 100km. We calculate the above answer in the following way

4 litres of petrol to travel 50km distance

8 litres of petrol is twice 4 litres. Therefore . Let the consumption of petrol be litres and the corresponding distance travelled be km. Then, complete the following table

 Petrol in litres (x) 4 8 12 15 20 25 Distance in km (y) 50 100

Direct proportion :-

As increases, value of also increases in such a way that the ratio does not change. It remains constant (say k)

are in direct proportion, if
In the above example where 4 and 8 are the quantities of petrol consumed in litres and 50 and 100 are the distance in km. When are in direct proportion we can write ( are values of corresponding to the values of respectively).

Example 1

The cost of 6 metres of a particular quality of lace is Rs.180. Tabulate the cost of 2, 4, 10 and 12 metres of lace of the same type.

Solution

Suppose the length of lace is metres and its cost in Rs is

 x 2 4 6 10 12 y y2 y3 180 y4 y5

As the length of lace increases the cost also increases in the same ratio. Therefore it is a direct proportion

We use the relation,

(i) Here

Therefore

(ii) If

(iii) If

(iv) If

Example 2

A tower of 50 metres high, casts a shadow of 35 metres. Find the height of a light house that costs a shadow of 45metres under similar conditions.

Solution

Let the height of the light house be metres. We form a table as shown below

 Height (in metres) 50 x Length of the shadow (in metres) 35 45

More the height of an object, the more would be the length of its shadow. Therefore it is a direct proportion. We use the following relation

The height of the light house is 64.2 metres

or
This relation can be written as

Example 3

If the weight of 20 beads is 15grams. How many beads of the same would weight kilograms.

Solution

Let the number of beads which weigh kg be

 Number of beads 20 x Weight of bead (in grams) 15 4500

As the weight of the beads increases the number of beads also increases. Therefore it is in direct proportion.

Alternate method: Two quantities and which vary in direct proportion have the relation

is the number of beads of weight kg(4500g)

6000 beads of would weigh kg

Example 4

An aeroplane flying at a uniform speed of 520km/hour

i. How far will it travel in 45 minutes?

ii. Find the time required to cover a distance of 500km

Solution

Let the distance travelled (in km) in 45 minutes be and time taken (in minutes) to cover 500km be

 Distance travelled (in km) 520 x 500 Time taken (in min) 60 45 y

(i) Since the speed is uniform, the distance covered would be directly proportional to time.

An aeroplane cover a distance 390km in 45minutes.

(ii) Since the speed is uniform, the distance covered would be directly proportional to time.

Time taken by the aeroplane = 58 minutes.

Example 5

Map is a miniature representation of very large region. 1cm on the map represents 8km of actual distance

i.e, 1cm : 8km or 1:800,000 then

2cm:16km or 2 : 1600,000 and so on

The scale of map is based in the concept of direct proportion

The scale of map is given as 1 : 20000000. Two cities are 5cm apart on the map find the actual distance between then

Solution

Let the map distance be cm and actual distance be cm

Thus two cities, which are 5cm apart on the map all actually 1000km away from each other.