Acid Concentration, Heating Bill

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

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/