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Welcome to our Math lesson on Summarizing the Properties of Variation, this is the sixth lesson of our suite of math lessons covering the topic of Variation. Types of Variation, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
In the previous paragraphs, we explained the three types of variation by illustrating them with examples. In this part, we will summarize the main properties of variation, so you will find it easier to take them as a reference when trying to solve examples.
If x and y vary directly as a and b, then the following formula is true.
20 oranges are divided between two children, where each share vary directly as 2 and 3. Calculate the number of oranges each child receives.
If we denote the children as x and y respectively, we obtain
Or
This means that
Giving that
we obtain
Hence,
(We have solved similar problems to this one in previous guides but using the proportion approach.)
If x and y vary inversely as a and b, then the following formula is true.
Four people can paint a building in 10 days. How many people are needed to paint the same building in 8 days?
This is an example of an inverse variation where the number of workers is inversely proportional to the time needed to paint the building. Thus, if expressing the workers by x and y and the number of days by a and b respectively, yields
In a joint variation where x and y vary inversely as a and b while z varies directly with either x or y as c with a or b, the following formula is true.
y varies directly as x2 + 1 and inversely as x + 2. Find the relation between x and y if y = 5 when x = 3.
The quantities that vary inversely are written in the same side of formula while those varying inversely on different sides. Thus, we have
where k is the usual constant of proportionality.
Substituting the known values yields
Thus, we obtain for the relationship between x and y:
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