# A Guide to the GRE/Averaging with Variables

## Averaging with Variables

If an average consists of a word problem, place the known values and variables into the average formula accordingly.

In some cases, the proportions can be calculated even if the exact amounts are not known. Suppose the average trips to the grocery store per month is 5 in Scranton but 7 in Dunmore. If the combined average of the two places is 5.5, what is the proportion of their populations? If the populations were equal, the average would be 6 - Scranton must have a greater population.

Let s and d equal the populations of Scranton and Dunmore, respectively.

```5s + 7d  = 5.5	(s + d)		Multiply both sides by (s + d).
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```5s + 7d  = 5.5	s + 5.5d	Expand the parentheses.
```

5s + 1.5d = 5.5s Subtract 5.5d from both sides.

1.5d = .5s Subtract 5.5s from both sides.

3d = s Divide both sides by .5.

Thus, 3d = s; in other words, Scranton has 3 times the population of Dunmore.

Practice

1. Debbie sold an average of 44 fur coats per month at her fur coat store in 2011. If Debbie's fur coat sales in November and December were three times as high as in the rest of the months, how many fur coats did she sell, on average, in November and December?

2. The average lifespan of the flamingos on Island A is 15 years while the average lifespan of flamingos on Island B 17 years. The combined average lifespan on the islands is 15.8 years. If 1,000 flamingos live on Island B, how many flamingos live on Island A?

3. The average height of the 12 office buildings in New Hartford is 32 feet. If the Union Bank building is excluded, the average height of the office buildings is 29 feet. How tall is the Union Bank building?

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1. 33

Let s equal Debbie's average fur coat sales in November and December and r equal her average fur coat sales for the other 10 months.

= 44 Take the initial equation.

= 44 Substitute the known value for s in terms of r.

= 44 Expand the parentheses.

= 44 Consolidate the variables.

= 528 Multiply both sides by 12.

``` r = 33				Divide both sides by 16. The average sales per month 				in the 10 months outside of November and December 				is 33.
```

Since the regular rate is 33, the average rate for November and December is 99 - three times this amount.

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2. 1500

Let a equal the flamingo population on island a.

= 15.8 Take the initial equation.

1,000(17) + a(15) = 15.8(1,000 + a) Multiply both sides by (1,000 + a).

17,000 + 15a = 15,800 + 15.8a Expand the parentheses.

1200 + 15a = 15.8a Subtract 15,800 from both sides.

1200 = .8a Subtract 15a from both sides.

1500 = a Divide both sides by .8. The flamingo population is 1500 on Island A.

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3. 65 feet

Let u equal the height of the union bank building.

= 32 Take the initial equation.

11(29) + u = 384 Multiply both sides by 12.

319 + u = 384 Work out the parentheses.

u = 65 Subtract 319 from both sides. The building is 65 feet tall.