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A certain quantity is measured on two different scales, the

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A certain quantity is measured on two different scales, the  [#permalink]

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Updated on: 02 Oct 2012, 00:23
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Question Stats:

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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

Originally posted by appy001 on 26 Jun 2010, 12:40.
Last edited by Bunuel on 02 Oct 2012, 00:23, edited 1 time in total.
Renamed the topic and edited the question.
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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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18 Feb 2013, 13:13
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mun23 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 the 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

As R increases by 18 (from 6 to 24), S increases by 30 (from 30 to 60). Thus increase of 18 in R corresponds to increase of 30 in S.

Therefore change of 70 in S from 30 (when R is 6) to 100 must correspond to change of 70/30*18=42 in R. Hence, 100 in S corresponds to 6+42=48 in R.

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27 Jun 2010, 16:30
23
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appy001 wrote:
Two measure standards R and S. 24 and 30 measured with R are 42 and 60 when they are measured with S, respectively. If 100 is acquired with S, what would its value be measured with R?

Please don't paraphrase questions - by not using the actual wording of the question, a LOT gets lost in translation. As written, the question is completely unanswerable.

Also, please provide answer choices, so we can discuss not only pure algebra, but the key alternative strategies that will get you a great score on test day.

Using my magical powers, I'll post the actual question:

Quote:
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

First, we have to understand what "linearly" means. It's not a straight ratio (since 6:30 does NOT equal 24:60). We need to look at the increases in each measurement to see what the scalar actually is.

From 6 to 24 we have an increase of 18. From 30 to 60 we have an increase of 30. Therefore, the increase ratio is 18:30 or 3:5. In other words, for every 3 that R increases, S increases by 5.

We know that S is 100. To get from 60 to 100, we went up by 40, or 8 "jumps" of 5; therefore, R will go up by 8 "jumps" of 3.

24 + 8(3) = 24 + 24 = 48: choose (c).

Note that (a) makes no sense, since if S=60 corresponds to R=24, how could S=100 correspond to a lower value for R?
General Discussion
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27 Jun 2010, 12:26
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appy001 wrote:
Two measure standards R and S. 24 and 30 measured with R are 42 and 60 when they are measured with S, respectively. If 100 is acquired with S, what would its value be measured with R?

This not a good question. In order to calculate R for S=100 we must know how are the scales of R ans S related.

If they are related linearly, then we would have: $$42=24m+b$$ and $$60=30m+b$$. The question: if $$100=Rm+b$$, then $$R=?$$

Solving system of equations for $$m$$ and $$b$$ --> $$m=3$$ and $$b=-30$$ --> substituting these values in $$100=Rm+b$$ --> $$R=\frac{130}{3}$$.

But if they are not related linearly and for example are related like: $$S=\frac{m}{R}+b$$, then we would have $$42=\frac{m}{24}+b$$ and $$60=\frac{m}{30}+b$$. The question: if $$100=\frac{m}{R}+b$$, then $$R=?$$

Solving system of equations for $$m$$ and $$b$$ --> $$m=-2160$$ and $$b=132$$ --> substituting these values in $$100=\frac{m}{R}+b$$ --> $$R=\frac{135}{2}$$.

Hope it's clear.
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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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18 Feb 2013, 20:13
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Consider a line built on R scale.

--------------------------|-----------------------------------|-------------------------------------------
6 24

The increase in value in R scale -=24-6=18 points
Now consider a line built on scale S. For the same increase in value, the increase in S scale is from 30-60 i.e30 points(60-30)

So, we know that an increase of 18 points on R scale corresponds to 30 points on S scale. We need to find our what measurement on R scale corresponds to a value of 100 on S scale.

an increase from 30 to 100 value on S scale= 70 points

R scale S scale
18 point increase= 30 point increase
? = 70 point increase

cross multiply

?=70*18/30=42

Thus, an increase of 70 points on S scale equals an increase of 42 points on R scale. So, the actual value=6+42=48

shanmugamgsn wrote:
Bunuel wrote:
mun23 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 the 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

As R increases by 18 (from 6 to 24), S increases by 30 (from 30 to 60). Thus increase of 18 in R corresponds to increase of 30 in S.

Therefore change of 70 in S from 30 (when R is 6) to 100 must correspond to change of 70/30*18=42 in R. Hence, 100 in S corresponds to 6+42=48 in R.

Sorry Bunuel, i didnt understand both question and concept of this!
How u moved on with this?
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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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19 Feb 2013, 10:17
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mun23 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 the 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 there is a linear relationship between R and S....we can assume R =aS + b (or any other linear equation in terms of R and S)
Now, when R= 6, S= 30....so the equation becomes 6 = 30a + b---(i)
When R = 24, S = 60...so the equation becomes 24 = 60a + b----(ii)
Solving the above two equations we get a = 3/5, b= -12...so the equation becomes R = (3/5)S - 12....now put the value of S =100 in the equation...u'll get the value of R as 48...
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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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07 Mar 2013, 22:00
On scale R when we measure 6 and 24 we measure 30 and 60 respectively on scale S. Therefore an increase in 18 on scale R results in an increase of 30 on scale S.

90 on S (60 + 30) is equivalent to 42 (24 + 18) on R. To get 100 on S we add 10, an increase in 10 on S results in an increase of 6 on R (18*10/30).

42 + 6 = 48

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Re: A certain quantity is measured on two different scales, the  [#permalink]

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02 Aug 2016, 13:29
2
Linear equations so:
30= 6x+b (x variable, b contant)
60= 24x+b

solving these two-- b= 20, x= 5/3
100= 5/3 (x)+ 20
80= 5/3 (x)
x= 48
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Re: A certain quantity is measured on two different scales, the  [#permalink]

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02 Aug 2016, 23:10
3
2
appy001 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 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

Think of the graphical approach here.

Related linearly means that their relation represents a line. e.g. x and y co-ordinates are related linearly on a line segment. We know how to deal with lines.

So think of it as two points (6, 30) and (24, 60) lying on a line. So what will be (r, 100) on the same line?
We see that an increase of 18 in x co-ordinate causes an increase of 30 in y co-ordinate. So y increases by 30/18 = 5/3 for every 1 point increase in x co-ordinate.
From 60 to 100, the increase in y co-ordinate is 40. So x co-ordinate will increase from 24 to 24 + 40*(3/5) = 48.

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Re: A certain quantity is measured on two different scales, the  [#permalink]

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22 Aug 2016, 22:50
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appy001 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 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

R scale and S are linearly related i.e, y = mx + c, and y is the S scale and x is the R scale.
Given,
30 = m*6 + C
60 = m*24 + C
Solving these 2 equation for m and C gives, m =$$\frac{5}{3}$$ and C = 20

so solve for S scales of 100

100 = $$\frac{5}{3}$$ * x + 20
x = 48

+1 for kudos
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A certain quantity is measured on two different scales,the R  [#permalink]

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20 Mar 2017, 11:42
2
mun23 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 the 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

6 > 24 > x
30 > 60 > 100

Through concept of gradient we can solve in less than 30 seconds.
(24-6)/(60-30)=(x-24)/(100-60)
==>x=48
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A certain quantity is measured on two different scales,the R  [#permalink]

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16 Aug 2017, 00:49
when
R increased from 6 to 24 ie an increase of 18 points
Than S increased from 30 to 60 ie 30 points.
There is a relationship between increase in R and than subsequent increase in S.
This relationship can be represented as increase in S/R=30/18=5/3
Which means that for every increase in the point of R their will be 5/3 increase in the points of S.
The question asks if S is 100 than what is R.
S has moved from 60 to 100 a increase of 40 points
When S has increased by 40 than let R increase=X
we now have a relationship X*5/3=40
or X=24
Original R=24. Therefore we have 24+24=48
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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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06 Jan 2018, 05:33
mun23 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 the 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 scales R & S are linearly related we can use y=mx+c, which can be use S=m*R+c
substitute the values (R,S) (6,30) (24,60) to get values of m = 5/3 & c=20
now for S = 100, find R in this equation , 100=(5/3)*R + 20
=> R=48
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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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25 Feb 2018, 16:31
Hi All,

The answer choices to this question are 'spread out' enough that you can use a bit of logic to estimate the correct answer.

We're told that the relationship between the values on the R-Scale and S-Scale are LINEAR, which means that as one value increases, the other value will increase by a fixed amount. We're then told the relationship between the R-Scale and S-Scale for two sets of values (6 and 30; 24 and 60). Notice how that when the R-scale value increases from 6 to 24 (an increase of 18), the S-scale value increases from 30 to 60 (an increase of 30). The question asks for the relative R-scale value when the S-scale value is 100.

Since an increase of 30 on the S-scale = an increase of 18 on the R-scale, when we go from 60 to 100 on the S-scale, we're increasing by 40 (a little more than 30)....so the increase on the R-scale should be a little more than 18....

24 + (a bit more than 18)..... = a bit more than 42....

There's only one answer that matches:

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Re: A certain quantity is measured on two different scales,the R  [#permalink]

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07 Mar 2018, 11:12
Let linear relation be S = mR + C, where m is slope, C is S-intercept (when R = 0)
Given: 30 = 6m + C (when R = 6, S = 30)
60 = 24m + C (when R = 24, S = 60)
solving for m and C, we get m = 30/18, C = 20
S = (30/18)R + 20
when S = 100, R = 48
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Re: A certain quantity is measured on two different scales, the  [#permalink]

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04 Jun 2018, 14:22
equation of straight line passing through points (6,30), (24,60) and (r,100) is needed.

equation is:
(y-y1)/(y2-y1) = (x-x1)/(x2-x1)

substituting x1,y1 = (6,30) and (x2,y2) = 24,60
equation is 5x-3y+60 = 0

now substitue the point (r,100) where y = 100. we get x value as 48

appy001 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 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
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Re: A certain quantity is measured on two different scales, the  [#permalink]

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24 May 2019, 00:32
Bunuel

Your solution using Coordinate Geometry is what clicked first on seeing the problem..
But logically it should be
slopes 4 2 1 ?

Leading to an answer of 60 or 84 (approx) rather than 48?
the real answer being 2.5 times of 30 75?
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A certain quantity is measured on two different scales, the  [#permalink]

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16 Jun 2019, 10:03
appy001 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 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

TL;DR

Think of the two scales R and S as x­- and y­-coordinates. Think of points on R as x and think of points on S as y. Point 1: (6, 30) & Point 2: (24, 60). Need x in Point 3: (x, 100)

2 ways to solve equations:

Method 1: Find slope and then intercept
m = (60-30)/(24-6) = 30/18 = 5/3
y = (5/3)x + c => 30 = (5/3)6 + c => c = 20

Method 2: Use 2 line equations
We can get two equations for the line that depicts their relationship:
30 = 6m + c => Eq 1
60 = 24m + c => Eq 2
Subtracting Eq 2 from Eq 1
-30 = -18m => m = 5/3
30 = 6*(5/3) + c => c = 20

100 = (5/3)x + 20 => x = (300 - 60)/5 = 240/5 = 48 => ANS: C

Veritas Prep Official Solution

Let’s think of the two scales R and S as x- and y-coordinates. We can get two equations for the line that depicts their relationship:

30 = 6m + c ……. (I)

60 = 24m + c ……(II)

(II) – (I)

30 = 18m

m = 30/18 = 5/3

Plugging m = 5/3 in (I), we get:

30 = 6*(5/3) + c

c = 20

Therefore, the equation is S = (5/3)R + 20. Let’s plug in S = 100 to get the value of R:

100 = (5/3)R + 20

R = 48

Alternatively, we have discussed the concept of slope and how to deal with it without any equations in this post. Think of each corresponding pair of R and S as points lying on a line – (6, 30) and (24, 60) are points on a line, so what will (r, 100) be on the same line?

We see that an increase of 18 in the x-coordinate (from 6 to 24) causes an increase of 30 in the y-coordinate (from 30 to 60).

So, the y-coordinate increases by 30/18 = 5/3 for every 1 point increase in the x-coordinate (this is the concept of slope).

From 60 to 100, the increase in the y-coordinate is 40, so the x-coordinate will also increase from 24 to 24 + 40*(3/5) = 48. Again, C is our answer.
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Re: A certain quantity is measured on two different scales, the  [#permalink]

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24 Nov 2019, 07:18
appy001 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 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

This question is very simple but the GMAT has cleverly disguised it to make this a harder question. The explanation is long, but the solution is intuitive.

The question relates to the concept of lines in geometry. We know that we are dealing with geometrical lines because the question uses the words "related linearly". If we can recognize these words and properly translate into algebraic forms, then the question is easy to solve.

The simplest algebraic representation of a line is of the following form:

Y = mX + B

where Y = y-intercept, X = x-intercept and B = constant that represents the value of the y-intercept when X = 0.

From the above equation, we can also derive the slope of the line. The slope will be the value of m when the constant B equals zero. Hence:

Slope = m = Y/X

Coming to the problem, we should recognize that we are given two points: (6, 30) and (24, 60), if we represent R as X and S as Y. Sunstituting these values of the two points in the equation of the line will provide us with two equations in the two unknowns, m and B.

Once we obtain the equation of the line, we can calculate the X-intercept when S. i.e., Y equals 100.

In summary, there are multiple ways of looking at this problem:

1. Since we are given two points that lie on a line, we can calculate the slope of the line. From this, we can calculate the slope when we are given the third point on the line, i.e., (X, 100).

2. We can set up algebraic equations, calculate the slope and the intercept and finally, use the equation of the line to find out the x-intercept when the y-intercept equals 100.

Thus:

Y = mX + B

=> 30 = 6X + B, and
60 = 24X + B

Solving:

m = 30/18 and B = 20.

So, the equation of the line is:

Y = (30/18)X + 20

The question asks us to calculate the value of X when Y = 100.

100 = (30/18)X + 20 or X = 48

The key to the question is to recognize that we are dealing with a geometrical line and that we are provided with two equations in two unknowns.
Re: A certain quantity is measured on two different scales, the   [#permalink] 24 Nov 2019, 07:18
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