Direct or Indirect Proportion-RRB
Direct or Indirect Proportion
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:
(i) As the speed of a vehicle increases, the time taken to cover the same distance decreases
(ii) More apples cost more money
(iii) More interest earned for more money deposited.
(iv) More distance to travel, more petrol needed.
Direct Proportion
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 x/y=k= [k is a positive number] = Constant
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, x2 of x respectively then
Example:
You are paid $20 an hour
How much you earn is directly proportional to how many hours you work
Work more hours, get more pay; in direct proportion.
This could be written:
Earnings ∝ Hours worked
If you work 2 hours you get paid $40
If you work 3 hours you get paid $60
Inversely Proportional(Indirect Proportion)
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= Constant
Then x and y are said to vary inversely.
In this case if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively then
x1 y1 = x2 y2
Example: speed and travel time
Speed and travel time are Inversely Proportional because the faster we go the shorter the time.
As speed goes up, travel time goes down
And as speed goes down, travel time goes up
This: y is inversely proportional to x
Is the same thing as: y is directly proportional to 1/x, Which can be written: y = k/x
Example:
4 people can paint a fence in 3 hours. How long will it take 6 people to paint it? (Assume everyone works at the same rate)
It is an Inverse Proportion:
As the number of people goes up, the painting time goes down.
As the number of people goes down, the painting time goes up.
We can use: t = k/n
- Where:
- t = number of hours
- k = constant of proportionality
- n = number of people
“4 people can paint a fence in 3 hours” means that t = 3 when n = 4
So now we know: t = 12/n
And when n = 6:
So 6 people will take 2 hours to paint the fence.How many people are needed to complete the job in half an hour?
So it needs 24 people to complete the job in half an hour.
(Assuming they don’t all get in each other’s way!)
Constant of Proportionality
The “constant of proportionality” is the value that relates the two amounts
Example:
you are paid $20 an hour (continued)
The constant of proportionality is 20 because:
Earnings = 20 × Hours worked
This can be written: y = kx
Where k is the constant of proportionality
Example:
y is directly proportional to x, and when x=3 then y=15.
What is the constant of proportionality?
They are directly proportional, so: y = kx
Put in what we know (y=15 and x=3): 15 = k × 3
Solve (by dividing both sides by 3):
The constant of proportionality is 5: y = 5x
When we know the constant of proportionality we can then answer other questions
Example: (continued)
What is the value of y when x = 9?
What is the value of x when y = 2?