Working Backward

Date: 01/22/2001 at 22:49:31
From: Elaien Chesebro
Subject: Working Backward

This problem was given to my son in 7th grade. He was told to use a 
working backward strategy to solve it. I cannot figure it out using 
that method. Can you explain it that way??

Thanks.

Ann, Barry, Chip, Doug, Evelyn and Fred went to lunch to celebrate 
Evelyn's and Fred's birthdays. Each person's meal cost the same 
amount. Evelyn and Fred were being treated for their meals, but each 
was to chip in equally for the his or her share of the other's meal. 
If the bill came to $54.00, HOW MUCH SHOULD EACH PERSON PAY? (Evelyn 
is being treated, but she is still paying a share of Fred's meal, and 
the same goes for Fred.)

Thanks for any help you can offer.

Date: 01/23/2001 at 08:52:08
From: Doctor Rick
Subject: Re: Working Backward

Hi, Elaine.

Let's see if we can re-state the situation in a way that makes it 
clearer what is happening. 

There are 6 people. Evelyn pays nothing for her meal, but the other 
five share the cost of her meal. That is, each of those five 
(including Fred) pays for 1/5 of Evelyn's meal.

Likewise, Fred does not pay for his meal, but each of the other five 
(including Evelyn) pays for 1/5 of his meal.

The four who are not being treated each pay for their own meal, plus 
1/5 of each of Ev's and Fred's meals.

We can now make a table showing how many meals each person pays for:

Ann     1 2/5
Barry   1 2/5
Chip    1 2/5
Doug    1 2/5
Evelyn    1/5
Fred      1/5

Add up these fractions, and what do we get? Exactly 6. That's no 
surprise: 6 meals were eaten, and they must all be paid for somehow. 
This is a check that we've understood the math correctly so far.

If we only knew how much one meal cost, we could figure out how much 
each person paid, couldn't we? We just multiply the cost of a meal by 
the number of meals. Ann, for instance, paid 1 2/5 times the cost of 
one meal.

Here, at last, is where "working backward" comes in. We know the total 
cost was $54. We know that there were 6 meals. How much did each meal 
cost? If we knew the cost per meal, we would multiply it by 6 to get 
the total cost. Working backward, starting from the total cost, we 
must DIVIDE by 6 to get the cost per meal:

  [Cost per meal] * 6 = $54

  $54 / 6 = [Cost per meal]

We've already figured out how to get from this figure to the amount 
that each person paid. You can finish the work.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/