Angry Question

[mathslug]

The first question on the 08/09 Study List is giving me fits.

"Mary uses a faulty ruler to measure a distance. She measured 45 ft, 4.77in, but corrected her measurement after realizing that the 1 yd ruler was actually 1 yd, .11in. What is the % difference between the corrected length Mary measured and Yolanda's independent measurement of 45 ft, 5.36in?"

Step 1: Convert everything to inches

Mary's Measurement: 12*45 + 4.77 = 544.77 in
What Mary thought the length of the ruler was: 12*3 = 36 in
The ruler's actual length: 36+.11 = 36.11 in
Yolanda's Measurement: 12*45+5.36 = 545.36 in


Step 2: Figure out Mary's corrected measurement

We know that she thought that she was using a 36 in ruler, but actually she was using a 36.11 in ruler. She calculated the distance to the object using the following formula: Total distance = (36 in)(# of full ruler lengths) + Remaining measurement. But she should have done: Total distance = (36.11 in)(# of ruler lengths) + Remaining measurement. We can set up two equalities:

Original formula: 544.77 in = (36 in)(# of ruler lengths) + Remaining measurement
Corrected formula: Corrected Distance = (36.11 in)(# of ruler lengths) + Remaining measurement

Using elimination we get: (Corrected Distance) = (.11 in)(# of ruler lengths) + 544.77 in

Since the number of ruler lengths and the Remaining measurement must be the same in both equations, we can use the original formula to determine the amount of full lengths of the yard stick used. 544.77 in / 36 in = 15 full lengths and change. Therefore, our # of ruler lengths = 15. Plugging this into our new equation, we obtain:

Corrected Distance = (.11 in)(15) + 544.77 in = 546.42 in


Step 3 : use the % difference formula and solve the answer

The formula for percent difference (already in percent) according to wikipedia given two values x and y is the following:
[200|x-y|]/[x+y]

Letting the Mary's corrected measurement be x, and Yolanda's measurement be y, we get: 200|546.42-545.36|/[546.42+545.36] = 200(1.06)/(1091.78) = .19%

Answer = .19%

This is WRONG!!!!!!!!!!!! AND I DON'T KNOW WHY! Somebody please help me correct this mess!

Zack

[AuSmith]

Maybe Mary also had some error in the remaining measurement. An unstated but reasonable assumption would be that Mary's ruler is only stretched a bit.... keeping the spacing equal. The question didn't state it, and so it's a little unclear. I like your lucid thinking - aren't you into programming?

[calcguy]

I will show you how my math team does this and you will be able to figure out your mistakes.
The percent difference between A and B is (B - A)/A x 100. To make sense of this, pick easy numbers: % diff between 2 and 3 is (3 - 2)/2 x 100 = 50%. That is, 3 is 50% larger than 2. Don't use absolute value. A negative % difference means the second number is smaller than the first. Now when you are interested in significant digits, use (B/A - 1) x 100 so you don't repeat any numbers.

Here's the procedure:
(45 x 12) + 5.36 = 545.36 this is Yolanda's measurement.
[(45 x 12) + 4.77] x 36.11 / 36 = 546.43... Mary's corrected measurement only good to 4 SD but don't round off here.
Now divide 545.36 / 546.43... = 0.9980 note: this is 4 SD
Now subtract 0.9980 - 1 = -0.0020 which is 2 SD, so the answer is -0.0020 x 100 = -0.20%

[AuSmith]

...which assumes that Mary's ruler is some percentage off on everything, and she doesn't have a .11 inch piece of extra wood on the end of a good yardstick.

[mathslug]

Man... it always seems like the most general approach in calculator is the correct one. I still don't think that the remainder should get that (36.11)/36 boost, though.

[AuSmith]

Yea, it's hard to say. Calc apps problems are often ambiguous like this. I think you should have won that argument if you made it to the verification period... sometimes they schedule other tests during verification period... like that region test with 8 errors :(

[calcguy]

In the real world, rulers do expand with increasing temperature. To help you make sense of this 36.11/36 boost, imagine Mary is going to measure a distance that is actually 36.11 inches, but her faulty ruler will tell her the distance is 36 inches because the ruler is 0.11 inch too long. So you must multiply whatever the faulty ruler says by 36.11/36 to get the true distance.

[SumitG]

[AuSmith]

Or, maybe the ruler was made out of marshmallow... no. There do exist metal rulers, and if heat can loosen up my wedding band, it could probably stretch a gold ruler. Wait, I never actually got married.

[SumitG]

I don't know about gold, but I suppose polyethylene could with only around 50 degrees F or so. Yeah that's definitely it. It must have been a polyethylene ruler.

[AuSmith]

That's one of those Sherlock Holmes deductions... I noticed with my keen eye that your ruler is a mite longer. Then, I immediately knew that it could only be made of polyethylene. But, it lustered so much for polyethylene that you had to have glazed it with wax. And, that is how I knew you were the one that left the wax on the postage box containing a bomb. At least you put the correct amount of stamps!

[SumitG]

I lol'd.