### Video Transcript

A bag contains 24 white balls and
an unknown number of red balls. The probability of choosing at
random a red ball is seven over 31. How many balls are there in the
bag?

We are told that this bag contains
24 white balls and an unknown number of red balls. We are also told that the
probability of choosing a red ball from the bag is seven over 31.

We can recall that, in general, the
probability of a particular event occurring can be found by dividing the number of
successful outcomes by the total number of outcomes. This means that the probability of
choosing a red ball from the bag would’ve been calculated by dividing the number of
red balls in the bag by the total number of balls in the bag. We don’t know either of these
values though. So we need to introduce some
algebra to help us solve the problem.

We know the bag contains 24 white
balls. We don’t know the number of red
balls, so we can use the letter 𝑛 to represent this. The total number of balls in the
bag can then be represented by the expression 𝑛 plus 24. We can then form an equation by
substituting the given probability of seven over 31, 𝑛 for the number of red balls
and 𝑛 plus 24 for the total number of balls, giving seven over 31 equals 𝑛 over 𝑛
plus 24.

To solve this equation for 𝑛, we
begin by cross multiplying, giving seven multiplied by 𝑛 plus 24 is equal to
31𝑛. Distributing the parentheses on the
left-hand side gives seven 𝑛 plus 168 equals 31𝑛. And then we can collect the like
terms on the right-hand side of the equation by subtracting seven 𝑛 from each side
to give 168 equals 24𝑛. Finally, we can divide both sides
of the equation by 24 to give 𝑛 equals 168 over 24, which is seven. We now know that there are seven
red balls in the bag, and so there are seven plus 24 balls in total. There are therefore 31 balls in the
bag.

Now, if you’re wondering why we
couldn’t have just equated the numerators and denominators of the two fractions,
which would have given the same answer, it’s important to realize that whilst this
would have worked for this specific set of numbers, it won’t work in every case. If the fraction for the probability
had been simplified from its initial form, then equating the numerators and
denominators would give inconsistent equations that we wouldn’t be able to solve for
𝑛. Following the formal method of
solving the equation ensures we obtain the correct answer, which is 31.