The honor society at a local high school sponsors a blood drive. The given table indicates the frequency of each donor blood type after one day of the blood drive.

Blood Type	Frequency for One Day
O	$30$
A	$25$
B	$6$
AB	$2$

We are told that the blood drive is extended for another day and the frequency doubles for each blood type. We want to check if the relative frequencies for each blood type change. Recall that to calculate the relative frequencies we divide the frequency of the particular event by the total of the frequencies. Let's find the total after one day of the blood drive.

Total for one day : 30 + 25 + 6 + 2 = 63

Let's add a column to our table representing the relative frequencies for each blood type after one day of the blood drive.

Blood Type	Frequency for One Day	Relative Frequency
O	$30$	$\frac{3 0}{6 3} = \frac{1 0}{2 1}$
A	$25$	$\frac{2 5}{6 3}$
B	$6$	$\frac{6}{6 3} = \frac{2}{2 1}$
AB	$2$	$\frac{2}{6 3}$

Now let's consider the fact that after the second day of the blood drive, the frequencies for all blood types $double .$ We can use this fact to make the frequency table for the two-day long blood drive.

Blood Type	Frequency for Two Days
O	$30 \cdot 2 = 60$
A	$25 \cdot 2 = 50$
B	$6 \cdot 2 = 12$
AB	$2 \cdot 2 = 4$

To calculate the relative frequencies for this data, we need to calculate the total once again.

Total for two days : 60 + 50 + 12 + 4 = 126

Note that this is

twice

as much as the total after one day, because if we double all the frequencies then their sum also gets doubled. We can now calculate the relative frequencies for each blood type after two days of the blood drive.

Blood Type	Frequency for Two Days	Relative Frequency
O	$60$	$\frac{6 0}{1 2 6} = \frac{1 0}{2 1}$
A	$50$	$\frac{5 0}{1 2 6} = \frac{2 5}{6 3}$
B	$12$	$\frac{1 2}{1 2 6} = \frac{2}{2 1}$
AB	$4$	$\frac{4}{1 2 6} = \frac{2}{6 3}$

Finally, we can compare the obtained relative frequencies to see if they changed.

Blood Type	Relative Frequency for One Day	Relative Frequency for Two Days	Equal?
O	$\frac{1 0}{2 1}$	$\frac{1 0}{2 1}$	Yes
A	$\frac{2 5}{6 3}$	$\frac{2 5}{6 3}$	Yes
B	$\frac{2}{2 1}$	$\frac{2}{2 1}$	Yes
AB	$\frac{2}{6 3}$	$\frac{2}{6 3}$	Yes

The relative frequencies did not change. Note that when we multiplied each frequency by $2$ the total of the frequencies also were multiplied by $2 .$ Therefore, when we calculated the relative frequencies for two days we had doubled the numerator divided by the doubled denominator. This allowed us to reduce the fraction and gave the same result as before.

Exercise