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Averages

How do we calculate averages?

Average is the result of adding all numbers in a set then dividing by amount of numbers.

Example 1

The average of 1 and 5 is 3, because $$ (1+5)/2= 3 $$

Example 2

The average of 1, 5, and 9 is 5, because $$ (1+5+9)/3=5 $$

Practice Problems

Problem 1

The average (arithmetic mean) of $$3$$ numbers is $$60$$. If two of the numbers are $$50$$ and $$60$$, what is the third number?

  1. $$50$$
  2. $$55$$
  3. $$60$$
  4. $$65$$
  5. $$70$$

E. $$70$$

$$3*60=180$$, which is the total number of points earned.

The two numbers we do know are $$50$$ and $$60$$ which add up to $$110$$.

The third number is $$180-110=70$$ since all three numbers must add up to $$180$$.

TEST METHOD: AVERAGE PIE

Use the average pie to chart out what you know and simplify this problem.

Problem 2

In the triangles below what is the average (arithmetic mean) of $$a$$, $$b$$, $$c$$, $$x$$, and $$y$$?

Average Triangles
  1. $$21$$
  2. $$45$$
  3. $$50$$
  4. $$52$$
  5. $$54$$

E. $$54$$

$$x+y=90$$ since the other angle is $$90°$$ and a triangle has $$180°$$ in total.

The two numbers we do know are $$50$$ and $$60$$ which add up to $$110$$.

The third number is $$180-110=70$$ since all three numbers must add up to $$180$$.

TEST METHOD: AVERAGE PIE

Tip: Total degrees = $$180+180=360$$, the total number of degrees in two triangles.

Problem 3

In a certain game, each of the $$5$$ players received a score between $$0$$ and $$100$$ inclusive. If their average (arithmetic mean) score was $$80$$, what is the greatest possible number of the $$5$$ players who could have received a score of $$50$$?

  1. None
  2. One
  3. Two
  4. Three
  5. Four

C. Two

METHOD: AVERAGE PIE. You know the total $$(5*80=400)$$. You know the number ($$5$$), and you know the average ($$80$$). You also know the answer is between $$0$$-$$4$$. By trial and error you should be able to answer this one.

KEYWORDS: "greatest possible", "inclusive"

If you're clueless, NO MATTER what you should be able to eliminate A and guess. If you couldn't eliminate A, review how to solve average problems.

Problem 4

The average (arithmetic mean) of $$a$$, $$b$$, and $$c$$ is equal to the median of $$a$$, $$b$$, and $$c$$. If $$0 < a < b < c$$, which of the following must be equal to $$b$$?

  1. $$(a+c)/2$$
  2. $$(a+c)/3$$
  3. $$(c-a)/2$$
  4. $$(c-a)/3$$

A. $$(a+c)/2$$

TEST KEYWORD: "MUST"

Plug in numbers for $$a$$, $$b$$, $$c$$ and see which answer 'must' work.

Problem 5

The average (arithmetic mean) of nine numbers is $$9$$. When a tenth number is added the average of the ten numbers is also $$9$$. What is the tenth number ?

  1. $$0$$
  2. $$9/10$$
  3. $$10/9$$
  4. $$9$$
  5. $$10$$

A. $$0$$

Eliminate some obvious wrong ones B, C, E. Remember to always estimate and have a good rough estimate of where the answers should be which is right around the original average of $$9$$; if you added a new number that was much higher or lower like say $$0$$ or $$1,000$$ the average could never remain $$9$$.

TEST TRAP: D is a trap and since you're not dealing with the first few problems you should not have gone on without really thinking about his one.

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