Averages and Sums Formulas

Everyone knows how to find an average, but the power of this formula is often underestimated. We know:

average=(sum of the items)(number of items)average = \dfrac{(sum \space of \space the \space items)}{(number \space of \space items)}

Notice, we can also write this as:

sum of items=(average)∗(number of items)sum \space of \space items = (average)*(number \space of \space items)

This latter form can be powerful. For example, if we add or subtract one item from a set, we can easily figure out how that changes the sum, and that can allow us to calculate the new average. Also, if we are combining two groups of different sizes, we can’t add averages, but we can add sums.

Practice Questions: Averages

Q1. There are 1717 students in a certain class. On the day the test was given, Taqeesha was absent. The other 1616 students took the test, and their average was 7777. The next day, Taqeesha took the test, and with her grade included, the new average is 7878. What is Taqeesha’s grade on the test?

(A) 7878

(B) 8080

(C) 8787

(D) 9191

(E) 9494

Q2. A company has 1515 managers and 7575 associates. The 1515 managers have an average salary of $120,000\${120,000}. The 7575 associates have an average salary of $30,000\${30,000}. What is the average salary for the company?

(A) $35,000\${35,000}

(B) $45,000\${45,000}

(C) $55,000\${55,000}

(D) $65,000\${65,000}

(E) $75,000\${75,000}

Answers and Explanations

Q2. The average of the first 1616 students is 7777. This means, the sum of these 1616 scores is

sum=sum = (average)∗(number of scores)=(average)*(number \space of \space scores) = 77∗16=123277*16 = 1232

Once Taqeesha takes her test, the average of all 1717 scores is 7878. This means, the sum of these 1717 scores is:

sum=sum = (average)∗(number of scores)=(average)*(number \space of \space scores) = 78∗17=132678*17 = 1326

Once we had the sum of the 1616 scores, all we had to do was add Taqeesha’s score to that total to get the sum of all 1717. Therefore, the difference in these two sums is Taqeesha’s score. 1326−1232=941326 - 1232 = 94.

Answer: E

Q2. The 1515 managers have an average salary of $120,000\${120,000}. The sum of their salaries is:

sum=sum = (average)∗(number of salaries)=(average)*(number \space of \space salaries) = $120,000∗15=\${120,000}*15 = $1,800,000\${1,800,000}

The 7575 associates have an average salary of $30,000\${30,000}. The sum of their salaries is:

sum=sum = (average)∗(number of salaries)=(average)*(number \space of \space salaries) = $30,000∗75=\${30,000}*75 = $2,250,000\${2,250,000}

When we add those two sums, we get the total payroll of all 9090 employees.

$1,800,000+$2,250,000\${1,800,000} + \${2,250,000} =$4,050,000= \${4,050,000}

So, we have 9090 employees, and together they earn $4,050,000\${4,050,000}, so the average is

average=$4,050,00090=$45,000average = \dfrac{\${4,050,000}}{90} = \${45,000}

Answer: B