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### Acid Concentration, Heating Bill

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Date: 7/6/96 at 12:58:17
From: Alfred T Chu
Subject: Acid Concentration, Heating Bill

Hello Math wizard!

Well, I have two algebra word problems:

1. A pharmacist has 8 liters of a 15 percent solution of acid.  How
much distilled water must she add to reduce the concentration of acid
to 10 percent?

2. By installing a \$120 thermostat that reduces the temperature
setting at night, a family hopes to cut its annual bill for heating
oil by 8 percent, and thereby recover the cost of the thermostat in
fuel savings after 2 years. What was the family's annual fuel bill
before installing the thermostat?

Can you show me how to do these and the logic behind them?
I want be able to do 'em later on myself.

Thank you very much.

Take care,
Alfred
```

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Date: 7/6/96 at 15:8:45
From: Doctor Paul
Subject: Re: Acid Concentration, Heating Bill

Number 1:

You begin with 8 liters of a 15 percent solution.  That means you have
15 percent acid and 85 percent water, right?  Let's figure out how
many liters of acid and water you have to begin with. 15 percent of 8
is 1.2 (.15*8) liters.  So you've got 1.2 liters of acid and 6.8
liters of water (8-1.2 = 6.8). Now you want to add water until the
concentration is down to 10 percent. Note that the amount of acid will
remain constant..you're only changing the amount of water. You want
1.2 liters of acid to be 10 percent of the entire quantity. Set up a
proportion to find out how much is 100 percent of the quantity.

10 percent            100 percent
----------     =      -----------
1.2 liters              x liters

now cross multiply:

1.2 = .1*x

divide both sides by .1 and you get:

12 = x

So 12 liters is the entire amount of solution that makes 1.2 liters
10% of the concentration. Now read the problem. It asks how much
diltilled water must be added to make the concentration 10 percent.
You go from 8 liters to 12 liters so that's 4 liters of water.

Number 2:
The family wants to save \$120 over a period of two years.  That means
they want to save \$60 a year, right? The problem also states they want
to cut their yearly bill by 8 percent.  Well, the two paragraphs above
tell you what you need to know...that \$60 is 8 percent of the total
bill.  Let's set up a proportation and solve for the total bill:

60            x
--------- = -----------
8 percent   100 percent

cross multiply:
60 = .08*x

divide both sides by .08
750 = x

The total yearly bill is \$750.
Let's check the answer. The family wants to save \$120 over two years
by saving 8 percent each year.  Well, 8 percent of \$750 is \$60 so the
new bill will be \$690 per year.  Over two years, that adds up to \$120
in savings.

-Doctor Paul,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
Middle School Algebra
Middle School Word Problems

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