# Problem: 7 - 11*

* Problem from Professor Doug Brumbaugh, University of Central Florida.

A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is \$7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly \$7.11.

What are the exact costs of each item?

Additional note from Professor Jim Wilson, University of Georgia:

I interpret the "exactly" and "exact" to mean just that -- not "to the nearest penny." Is there a solution with this strict interpretation of "exactly"?

## Discussion/Solution? :

Let's begin by representing the problem with a pair of equations.

x y z w = 7.11
x + y + z + w = 7.11

If x,y,z,w represent the amounts of the four items in cents then we have

x y z w = 711000000
x + y + z + w = 711

What can we say so far?

The factors : 711 = (3)(3)(79)

If there is to be an exact solution then the product xyzw must be divisible by 3, 9, 79.
The exact sum of the 4 items is 7.11 so the cost of each item is less than \$7.11.

After exploring these two equations on scratch paper I came up with two scenerios.

Scenerio 1

Since the clerk did not know that total meant to add, he must not have known how to round to the nearest penny (or even that you should round). Suppose the customer selected four items which cost \$.79, \$2.00, \$1.76, and \$2.56. The total cost of these 4 items is exactly \$7.11. However, the product of the 4 items is 7.118848. Thus the clerk rounds incorrectly or doesn't round at all and quotes the amount \$7.11.

Scenerio 2

The clerk does know how to round to the nearest cent. Suppose the customer selected four items which cost \$.79, \$1.75, \$2.00, \$2.57. The total cost is \$7.11. However, the product is 7.10605.
The clerk rounds correctly and quotes the amount \$7.11.

Are there other amounts that would fit into one of the two scenerios mentioned?

Are there other scenerios?

... the total cost of the four items is \$7.11.
... simply multiplied the cost of each item and arrived at the total.

Neither of these phrases say anything about "exactly" or "exact". Thus the clerk could have rounded or truncated the product to arrive at his quote.

... then added the items together and informed the customer that the total was still exactly \$7.11.

When he added the 4 amounts he did get an exact total. Thus the customer was told he "still owed exactly \$7.11".

... exact costs of each item?

In both scenerios the cost of each item was "exact".

So--------

Is this response to Dr. Wilson's comment accurate?

Now let's seriously consider the question "is there an 'exact' solution"?