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Quality scale

by alex.gellatly » Thu Jun 28, 2012 5:29 pm
A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurments on the S-scale of 20 and 60, respectively. What measurement on the R-scale corresponds to aa measurement of 100 on the S-scale?

A. 20
B. 36
C. 48
D. 60
E. 84

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by Anurag@Gurome » Thu Jun 28, 2012 6:06 pm
alex.gellatly wrote:A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 20 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?
Algebraic Approach:

Say, the measurement of R-scale is r and the same on S-scale is s.
As they are related linearly, we can assume s = ar + b, where a and b are some constants.

Now, for r = 6, s = 20 ---> 20 = 6a + b ................. (A)
And, for r = 24, s = 60 ---> 60 = 24a + b ..............(B)

Solving (A) and (B),
a = (60 - 20)/18 = 40/18 = 20/9
b = (20 - 6*20/9) = (9*20 - 6*20)/9 = 3*20/9 = 20/3

Hence, s = (20/9)r + 20/3 ---> 9s = 20r + 60 ---> r = (9s - 60)/20

Therefore, when s = 100 ---> r = (9*100 - 60)/20 = (9*5 - 3) = 42

The correct answer is not in options.
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by eagleeye » Thu Jun 28, 2012 6:12 pm
alex.gellatly wrote:A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurments on the S-scale of 20 and 60, respectively. What measurement on the R-scale corresponds to aa measurement of 100 on the S-scale?

A. 20
B. 36
C. 48
D. 60
E. 84
F. 42
Thanks
There are a number of ways of doing this. I think this might be the most efficient.

Since there is a LINEar relationship between R and S, the relationship can be modeled as the equation of a LINE. Since we need to find R, Let R be the Y-axis, and S be the x-axis.

Then we have equation of the line is (R-R1) = ((R2-R1)/(S2-S1)) * (S-S1)

Here we have: R1 = 6, R2 = 24, S1=20, S2 = 60, and S = 100.

So the equation becomes R-6 = ((24-6)/(60-20)) * (100-20) = 18*80/20 = 36.

So R-6 = 36 => R = 42.

Hence none of the options are correct.

Let me know if this helps :)
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by Anurag@Gurome » Thu Jun 28, 2012 6:14 pm
Tricky Approach:

As the measurements are related linearly, change in measurements in any of the scale by same amount will correspond to equivalent change in the other.

In S-scale,
  • From 20 to 60 ---> Change is (60 - 20) = 40
    From 60 to 100 ---> Change is (100- 60) = 40
In R-scale,
  • From 6 to 24 ---> Change is (24 - 6) = 18
Hence, for a increase of 40 in S-scale the measurement on R-scale will increase by 18.

Therefore, when S-scale shows 100, R-scale will show (24 + 18) = 42
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by GMATGuruNY » Thu Jun 28, 2012 6:44 pm
The question as posted has a typo. I offered a solution to the correct version here:

https://www.beatthegmat.com/s-and-r-scale-t112642.html
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