The probability of randomly
selecting a red ball from a jar that contains only red, blue, and orange balls is
one-quarter. The probability of selecting a blue
ball from the same jar is one-third. If the jar contains 10 orange
balls, find the total number of balls in the jar.
There are only three possible
outcomes for the color of the ball selected. They are red, blue, and orange. We know that the sum of all the
probabilities of all possible outcomes is one. In this case, the probability of
choosing a red ball plus the probability of choosing a blue ball plus the
probability of choosing an orange ball is equal to one.
We know that the probability of
choosing a red ball is one-quarter and the probability of choosing a blue ball is
one-third. So we can rewrite our equation as
one-quarter plus one-third plus the probability of choosing an orange ball is equal
In order to be able to add
one-quarter to one-third, we need to find the common denominator. The lowest common multiple of four
and three is 12, so the lowest common denominator here is 12. To change one-quarter into
twelfths, we multiply both the numerator and the denominator by three. That gives us three twelfths. Similarly, we multiply both the
numerator and the denominator of one-third by four, and we get four twelfths. Three twelfths plus four twelfths
is seven twelfths. So our equation becomes seven
twelfths plus the probability of choosing an orange ball is equal to one.
We can solve this equation to find
the probability of choosing an orange ball by subtracting seven twelfths from both
sides. We know that one whole is the same
as twelve twelfths, so the probability of choosing an orange ball is twelve twelfths
minus seven twelfths. The probability of choosing an
orange ball is therefore five twelfths.
Now at this point, it’s important
to remember that all balls are equally likely to be chosen. This means that the probability of
picking any ball is directly proportional to the number of balls of that color. There are 10 orange balls. If we call the total number of
balls in the jar 𝑥, we can use this information to form an equation.
Our equation is five twelfths of 𝑥
is equal to 10. We need to solve this equation for
𝑥. We’ll first multiply both sides by
12. That gives us five 𝑥 is equal to
120. Next, we’ll divide both sides by
five. 120 divided by five is 24, so 𝑥 is
equal to 24. Since we said that the number of
balls in the jar was equal to 𝑥, we can infer that there are 24 balls in the