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the R-scale and the S-scale

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by Needgmat » Sat Aug 27, 2016 9:36 am
A certain quantity is measured on two different scale, 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 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?

A) 20

B) 36

C) 48

D) 60

E) 84

OAC

Please explain. I don't understand what to do.

Many thanks in advance.

Kavin
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by GMATGuruNY » Sat Aug 27, 2016 9:42 am
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 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?

A. 20
B. 36
C. 48
D. 60
E. 84
Since the relationship between R and S is linear, any pair of points (R,S) must yield the same slope.

Given points are (6,30), and (24,60).
Slope = (S₂ - S�)/(R₂ - R�) = (60-30)/(24-6) = 30/18 = 5/3.

(6,30) and (R,100) must yield the same slope.
(100-30)/(R-6) = 5/3.
70/(R-6) = 5/3.
Cross-mulitplying, we get:
5R-30 = 210.
R = 48.

The correct answer is C.
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by Needgmat » Sat Aug 27, 2016 8:33 pm
Since the relationship between R and S is linear, any pair of points (R,S) must yield the same slope.

Given points are (6,30), and (24,60).
Slope = (S₂ - S�)/(R₂ - R�) = (60-30)/(24-6) = 30/18 = 5/3.

(6,30) and (R,100) must yield the same slope.
(100-30)/(R-6) = 5/3.
70/(R-6) = 5/3.
Cross-mulitplying, we get:
5R-30 = 210.
R = 48.

The correct answer is C.
Hi GMATGuruNY ,

Thank you so much for your reply.

But how do we come to know that we have to find slope in this question?

Fist time I saw this question. I got blank totally.

Please explain.

Many thanks in advance.

Kavin
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by GMATGuruNY » Mon Aug 29, 2016 11:06 pm
Needgmat wrote:
Hi GMATGuruNY ,

Thank you so much for your reply.

But how do we come to know that we have to find slope in this question?

Fist time I saw this question. I got blank totally.

Please explain.

Many thanks in advance.

Kavin
The problem above is the same as the following:

A certain quantity is measured on two different axes, the x-axis and the y-axis, that are related linearly. Measurements on the x-axis of 6 and 24 correspond to measurements on the y-axis of 30 and 60, respectively. What measurement on the x-axis corresponds to a measurement of 100 on the y-axis?

To be related linearly = to lie on the same line.
Thus, all three points -- (6,30), (24,60) and (x,100) -- lie on the same line.
Slope = (y₂ - y�)/(x₂ - x�).
The equation above must hold for any ANY TWO POINTS on the line.
Thus, (6,30) and (x,100) must yield the same slope as (6,30) and (24,60).
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by Matt@VeritasPrep » Thu Sep 01, 2016 5:41 pm
Needgmat wrote: But how do we come to know that we have to find slope in this question?
If we say that r and s are related linearly, we're really saying that r and s have the relationship s = r*m + b ... which is just our boring old y = mx + b equation with r and s replacing x and y.

Since we have s = r*m + b, we then plug in the points we're given, and solve for m and b:

if r = 6, then s = 30, so 30 = 6m + b
if r = 24, then s = 60, so 60 = 24m + b

From there, you have two variables, two equations, so you can solve for m and b, then use the equation of the line.
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by Matt@VeritasPrep » Thu Sep 01, 2016 5:43 pm
There's also a nice way of cheating this problem. Since we're told that for one set of points, S doubles when R quadruples, we can roughly guess that R will approximately double if S goes up by 50%.

Since S is going from 60 to 100, it's going up by a little more than 50%. That means R should more or less double, so our answer should be about 24*2, or 48. Lo and behold, it's exactly 48, and we're good to go.

This isn't the most reliable method in the world, but it's sound enough here if you only have 10 seconds left to try the problem.
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