Question

What are some examples of direct variation in real life?

Answer 1

Hint: In this mathematical concept is not much required, what we need to do is just understand the situation and try to find out the variation and deal with it in a theoretical manner only. Much details are not majorly included in this section.

Complete step-by-step solution:
At the local farmer market, we see that someone is buying strawberries of 5 pounds and pay $\$12.50$ and you know that you too want to buy those strawberries but you have only $\$2$ with you. How much do you expect to pay?
This situation is an example of direct variation where you would expect that the strawberries are priced on a “per pound” basis and if you buy $\dfrac{2}{5}th$ of the amount of strawberries then you would pay $\dfrac{2}{5}th$ of $\$12.50$ for your strawberries. In the same way if you bought 10 pounds of strawberries which is twice the amount then you will pay $\$25$ that is twice of $\$12.50$ and if you don’t buy any strawberries then you won’t pay any amount.
Direct variation can be expressed as equation $y=(k)x$, where k is called the constant of variation.
Direct variation occurs when:
1) The fraction $\dfrac{rise}{run}$ is always the same and,
2) The ordered pair (0,0) is the solution of the situation that occurs.
For example: If y varies directly with x according to the relationship $y=k.x$ and $y=7.5$ when $x=2.5$ determine the constant of proportionality k?
We can solve for the constant of proportionality using substitution.
Substitute $x=2.5$, $y=7.5$ into the equation $y=kx$
$7.5=k (2.5) $
Now dividing both sides by 2.5 we get k=3.
So, the constant of proportionality that we get is 3 i.e., k=3.

Note: Statement of the variation must be very much clear. We must study the proper situation first and then do the analysis and then proceed forward to understand the exact concept rather than just completing it.