A Container Holds 5/8 Gallon...

Date: 11/05/97 at 21:51:34
From: Brandi
Subject: Middle school math story problem

Hi. I am working on a content exam for my secondary math education 
class in college. This exam is a review for the teacher certification 
test. Can you help?

Question 1:

A full container holds 5/8 gallon of liquid. If the container is 
4/5 full and then 25 percent of the liquid is lost due to evaporation, 
how much liquid is left in the container? 

  (a) 1/4 gallon
  (b) 3/8 gallon
  (c) 1/2 gallon
  (d) 5/8 gallon
  (e) 3/4 gallon

I took (5/8) * (4/5) and got 1/2. Then I took 25% of 1/2 and got 1/8
What am I doing wrong?

Question 2:

An empty fuel tank is filled with brand Z gasoline. When the tank is 
half empty, it is filled with brand Y gasoline. When the tank is half 
empty again, it is filled with brand Z gasoline. When the tank is half 
empty again it is filled with brand Y gasoline. At this time, what 
percent of the gasoline in the tank is brand Z?

  (a) 50 percent
  (b) 40 percent
  (c) 33 1/3 percent
  (d) 25 percent
  (e) none of these

I'm not quite sure how to do this one at all.  

Thank you.

Date: 11/06/97 at 11:59:50
From: Doctor Pipe
Subject: Re: Middle school math story problem

Brandi,

Where you went wrong on Question 1 is what you did with the result of 
your 25 percent of 1/2 calculation.

You correctly calculated that the container, when 4/5 full, holds 
1/2 gallon. The 25 percent that evaporates is subtracted from the 
original contents of the container to give the amount remaining in the 
container:

     1/2 - (1/4 x 1/2)
   = 1/2 - 1/8
   = 4/8 - 1/8
   = 3/8

So, after 25 percent of the original 1/2 gallon evaporates, there is 
3/8 gallon remaining.


For Question 2, here is one way:

The tank is filled four times. When the empty tank is first filled, it 
contains 100 percent Brand Z gasoline.

Gas is used until the tank is half empty (which I prefer to look at as 
half full) and then the tank is filled with Brand Y. Now the tank is 
50 percent Brand Z and 50 percent Brand Y.

This 50/50 mix is used until the tank is half empty (half full) and 
the tank is filled again with Brand Z. Before the fill, the tank was 
1/4 Brand Z, 1/4 Brand Y and 1/2 empty. After filling the empty half 
with Brand Z, the tank is (1/4 + 1/2) Brand Z and 1/4 Brand Y, or 
3/4 Brand Z and 1/4 Brand Y.

This 75/25 mix is used until the tank is again half empty and the 
tank is filled with Brand Y. Before the fill, the tank is 3/8 Brand Z 
(75 percent of half the tank is 3/4 x 1/2) and 1/8 Brand Y (25 percent 
of half the tank is 1/4 x 1/2) and 1/2 empty. After filling the empty 
half with Brand Y, the tank is 3/8 Brand Z and 5/8 Brand Y.

So the correct answer is the percentage equivalent of 3/8 of the tank, 
which is 37.5 percent.

I repeatedly applied the process of determining the percentage of each 
brand of gas in the tank when full, then used that same percentage to 
determine the portion of the entire tank occupied by each brand when 
the tank was half empty, and then added to the appropriate brand the 
new 1/2 tank that was added.

Here is another way:

1Z + 0Y          - first time the tank is filled
0.5 (1Z + 0Y)    - first time the tank is 1/2 empty
0.5 (1Z + 0Y) + 0.5Y          - 2nd time the tank is filled
0.5 (0.5 (1Z + 0Y) + 0.5Y)    - 2nd time the tank is 1/2 empty
0.5 (0.5 (1Z + 0Y) + 0.5Y) + 0.5Z         - 3rd time filled
0.5 (0.5 (0.5 (1Z + 0Y) + 0.5Y) + 0.5Z)   - 3rd time 1/2 empty
0.5 (0.5 (0.5 (1Z + 0Y) + 0.5Y) + 0.5Z) + 0.5Y    - 4th time filled; 

now we simplify this to get the answer

0.5 (0.5 (0.5 (1Z + 0Y) + 0.5Y) + 0.5Z) + 0.5Y
= 0.5 (0.5 (0.5Z + 0.5Y) + 0.5Z) + 0.5Y
= 0.5 (0.25Z + 0.25Y + 0.5Z) + 0.5Y
= 0.5 (0.75Z + 0.25Y) + 0.5Y
= 0.375Z + 0.125Y + 0.5Y
= 0.375Z + 0.625Y

Since 0.375 + 0.625 = 1, which is 1 tank of gas, the tank contains 
0.375 of a tank of Brand Z or is 37.5 percent full of Brand Z.

Good luck on your teacher certification test!!

-Doctor Pipe,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/