Monday, January 28, 2008

Tolerance Problem

The “Tolerance Problem.”
Monday January 28, 2008’s SAT Question of the day, goes something like this:
The weights of 12 sacks of potatoes range from 14.75 pounds to 15.15 pounds. If P is the weight, in pounds, of one of these sacks, which of the following must be true?
A. P – 14.75≤ 0.2
B. P – 14.95≤ 0.2
C. P + 14.95≤ 0.2
D. P – 0.2≤ 0.2
E. P – 12≤ 0.2

**** SORRY ****** For some reason, Blogger will not recognize the absolute value sign. For a better view of the problem, please download the pdf (link at bottom)

Solution:
Sorry, there is no “solution technique,” that I know of, to solve this medium level SAT algebra problem. Less than half of the SAT students have selected the right answer to this medium level difficulty question. You have to know the specific content of what is being asked in the Tolerance Problem. The answer is an algebraic expression that defines the acceptable tolerances or limits for something that is produced or sold. I happen to like this problem because I studied manufacturing as an undergrad and despite never working in the field I have retained this specific content (that and the reason why McDonald’s chairs used to be bright orange; oh my professors would be so proud).

Here is how the Tolerance Problem works. You are given the range of two units of measurement, in this case weight in pounds, of multiple items, produce like potatoes or apples or something being manufactured like the size of a bottle opening or the width of a nail. Rarely does the number of items have anything to do with the problem; it is just there as a statistical sample size to show you that there is more than a couple of the thing being measured. The combined weight of the 12 sacks is not mentioned, you are working to solve the acceptable tolerance limit for an individual sack, defined by an algebraic expression using absolute value ( P )and a less than or equal to inequality (≤).

When you buy a bag or box of anything at the store that is sold by weight not by volume, you do not get the EXACT amount. Instead, you purchase an amount that is ABOUT the weight listed. If it is produce like potatoes or apples in a bag, the store sets a limit to what the bag can weigh and still be sold. This RANGE of weight is the acceptable lower limit (any less and the consumer does not get enough) to the upper limit (any more and the store is giving away too much).

To find the RANGE, take the UPPER LIMIT (larger measurement, here: 15.15) and SUBTRACT the LOWER LIMIT (smaller measurement, here: 14.75)
· RANGE = UPPER LIMIT – LOWER LIMIT
· RANGE = 15.15 – 14.75 = 0.04 pounds
· The TOLERANCE is expressed as the AVERAGE of the LIMITS plus or minus
one-half of RANGE.
· The AVERAGE of the LIMITS = (UPPER LIMIT + LOWER LIMIT) ÷ 2
AVERAGE = (15.15 + 14.75) ÷ 2 = 14.95 pounds
· Expressed in algebra, using absolute value, an individual potato sack would be:
P – (AVERAGE of the LIMITS)≤ RANGE/2

So our solution today is: B. P – 14.95≤ 0.2
The TOLERANCE is 14.95 ± 0.2 pounds