Direct variation is a relation that has the form
y = kx
where k is the constant of proportionality.
If you are told that a relation is a direct proportion, and you are given one data point, you can find k. The you can write the equation of the direct relation.
Here is an example.
The price of gasoline follows a direct variation.
John bought 5 gallons of gas and paid $15.
a) Write an equation for the relation.
b) Using the relation you found, how much do 13.8 gallons cost?
Solution:
Since the relation is a direct variation, it follows the general equation of a direct variation:
y = kx
We are given one data point, 5 gallons cost $15.
We plug in 5 for x and 15 for y and we find k.
y = kx
15 = k * 5
k = 3
Now that we know that k = 3, we rewrite the relation using our value of k.
y = 3x
This is the answer to part a).
Part b)
We use our relation, y = 3x, and we plug in 13.8 into x and find y.
y = 3x
y = 3 * 13.8
y = 41.4
The price of 15 gallons of gas is $41.40.
Let R and B be the number of red and blue marbles respectively.
Then,
R+B = 36
R:B = 5:4
Therefore,
Ratio of red marbles = 5/9
Ratio of blue marbles = 4/9
This means,
R = 5/9*36 = 20 marbles
B = 4/9*36 = 16 marbles
If one blue marble is removed from the bag, the new B= 16-1 = 15 blue marbles and the new total of marbles in the bag = 36-1 = 35 marbles.
The new ratios are
R = 20/35 =4/7
B = 15/35 = 3/7
That is, R:B = 4:3
The reasoning of the student is wrong. When a marble is removed, both the number of blue marbles changes as well as the total of the marbles in the bag. In other words, both the values in the ratio reduce by 1.
