Direct proportionality indicates a situation in which, when one term is multiplied by a certain amount, the same exact thing happens to the second term.
In the same way, when one term is divided by a certain amount, the same exact thing happens to the second term.
The ratio between both magnitudes remains constant.
Let's observe an example that illustrates this concept.
Diana's credit card company charges a monthly fee of 2$, plus 1$ for each bank transaction.
Is the ratio of the amount Diana has to pay to the number of transactions she made during the month directly proportional?
Solution:
To answer this kind of question, it is convenient to draw a table:
X represents the number of transactions Diana made
Y represents the amount Diana has to pay
Notice, the question says that the credit card company applies a cost of 2$ each month, that is, even if Diana does not make any transactions, she will have to pay 2$.
Let's draw a table:
Now let's see:
Does the X multiply by a certain number and also the Y increase multiplied by the same number?
The answer is no.
We can see that when the X doubles and goes from 1 to 2
the Y does not double! From 3 to 4 what it does is 34.
Therefore, we can determine that the ratio of the amount Diana has to pay to the number of transactions she made during the month is not directly proportional.
Examples and exercises with solutions on direct proportion
Exercise #1
There are two circles.
One circle has a radius of 4 cm, while the other circle has a radius of 10 cm.
How many times greater is the area of the second circle than the area of the first circle?
Video Solution
Step-by-Step Solution
The area of a circle is calculated using the following formula:
where r represents the radius.
Using the formula, we calculate the areas of the circles:
Circle 1:
π*4² =
π16
Circle 2:
π*10² =
π100
To calculate how much larger one circle is than the other (in other words - what is the ratio between them)
All we need to do is divide one area by the other.
100/16 =
6.25
Therefore the answer is 6 and a quarter!
Answer
641
Exercise #2
The rectangle ABCD is shown below.
AB = X
The ratio between AB and BC is 2x.
The length of diagonal AC is labelled m.
Choose the correct answer.
Video Solution
Step-by-Step Solution
We know that:
BCAB=2x
We also know that AB equals X.
First, we will substitute the given data into the formula accordingly:
BCx=2x
x2=BCx
xx2=BC
xx×x×2=BC
x×2=BC
Now let's look at triangle ABC and use the Pythagorean theorem:
AB2+BC2=AC2
We substitute in our known values:
x2+(x×2)2=m2
x2+x×2=m2
Finally, we will add 1 to both sides:
x2+2x+1=m2+1
(x+1)2=m2+1
Answer
m2+1=(x+1)2
Exercise #3
Given the rectangle ABCD
AB=X the ratio between AB and BC is equal to2x
We mark the length of the diagonal A with m
Check the correct argument:
Video Solution
Step-by-Step Solution
Let's find side BC
Based on what we're given:
BCAB=BCx=2x
BCx=2x
2x=xBC
Let's divide by square root x:
x2×x=BC
x2×x×x=BC
Let's reduce the numerator and denominator by square root x:
2x=BC
We'll use the Pythagorean theorem to calculate the area of triangle ABC:
AB2+BC2=AC2
Let's substitute what we're given:
x2+(2x)2=m2
x2+2x=m2
Answer
x2+2x=m2
Exercise #4
There are 18 balls in a box, 32 of which are white.
How many white balls are there in the box?
Video Solution
Answer
12
Exercise #5
In a box there are 28 balls, 41 of which are orange.
How many orange balls are there in the box?
Video Solution
Answer
7
Check your understanding
Question 1
How many times longer is the radius of the red circle than the radius of the blue circle?