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A number line approach for this question will be easiest.

Distance is always positive and thus can be represented with absolute value signs.
Translate the information given:
1. |R - H| = 3 |R - S|
2. |M - R| = 4|M - H|

Now, the question asks for the ratio of |M - S| to |H - S|.

Rewrite the distances in terms of a common variable so it cancels out at the end.

|H - S| will be the sum of the distance between |H - R| and |R - S| which is given, so take 4x for the denominator.

For the numerator, since the distance between |H - R| is 3x, we can get |M - S| by taking |M - R| + |R - S| |M - R| is 4*3x/5 (using (2)) and |R - S| is just x. Their sum is 17x/5.

So the final ratio is 17x/5 / 4x, which simplifies to 17/20 (D).
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Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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Let distance from Ridge to Summit be x then distance from Harbor to Ridge will be 3x.
Let distance from Harbor to Maple be y then distance from Maple to Ridge will be 4y.

Now Maple to Ridge will be y+4y=5y which is also equal to 3x.
3x=5y
x= (5/3)y

Total distance from Mapple to Summit is 4y+x=4y+(5/3)y=(17/3)y
Total distance from Harbor to Summit is 3x+x=4x=4(5/3)y=(20/3)y

The ratio : ((17/3)y)/((20/3)y)=17/20

Answer: D. 17/20
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Answer is E.
Y 4Y
<--------> <------>
H----------M---------R-----------S
<-----------------> <--------->
3X X

TO FIND : harbor to summit/maple to summit

from above diagram we can find
5Y=3X
Y=3X/5


HENCE:

Harbor to Summit / Maple to Summit = 3X+X/4Y+X
4X/(17X/5)
20X/17X = 20/17
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Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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it is given that H, M, R, S are in a straight line.
lets say distance from H->M is x, M->R is y and R->S is z
according to given info.
x+y = 3z and y = 4x

on solving we have 5x = 3z, now we have ratios as x:y = 1:4 and x:z = 3:5
their combined ratio will be -> x:y:z = 3:12:5

we need to find the fraction (y+z)/(x+y+z) = 17/20.
i have taken 17/20 and not 20/17, because we commonly miscalculate sometimes, they have asked what fraction of A is B? (in easy form) => fraction = B/A
Therefore, 17/20 is the correct option D
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The sequence is HMRS
From H to S the total distance suppose is 4x
And from H to R the total distance is 5y
And HR = 3x and HR = 5y
So we could say that: 3x=5y
And x = 5y/3

The total fraction asked is = MS/HS
the total distance HS in terms of y is = 4*5y/3 = 20y/3
and simply subtract distance of HM from HS now, 20y/3 - y = 17y/3
Then our fraction would be: (17y/3)/(20y/3)
Understand this: When we say fraction of 4 is 1 then it translates to = Part/Whole = 1/4
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To get a better understanding of the question, we can plot the stops on a straight line with the first stop as Harbor then second as Maple third as Ridge and the last one being Summit.
Let dist between Ridge and Summit be x,
Thereefore, dist between Ridge and Harbor e would be 3x.
Total dist between Harbor and Summit wil be x+3x = 4x.

Similarly, let dist between Maple and Harbor be y,
Therefore, dist between Maple and Ridge would be 4y.

Now, as dist between Ridge and Harbor = Harbor and Maple + Maple and Ridge, this will gice us a relationship between x and y.
3x = y + 4y
3x = 5y
y = 3x/5

Now, we need to find the fraction of dist Maple to Summit from the overall dist from Harbor to Summit,
Dist Maple to Summit = Dist Maple to Ridge + Dist Ridge to Summit
Dist Maple to Summit = 4y + x
= 4*3x/5 + x
= 12x/5 + x
= 17x/5

Total dist from Harbor to Summit = 4x
Therefore fraction = (17x/5)/4x = 17/20

Option D 17/20.
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Let dist(Ridge-Summit) be x
then dist(Harbor-Ridge) = 3x

Let dist(Harbor-Maple) be y
then dist(Maple-Ridge) = 4y

So the dist(Harbor-Summit) = dist(Harbor-Ridge)+dist(Ridge-Summit) = x+3x = 4x and,
dist(Maple-Summit) = dist(Maple-Ridge) + dist(Ridge-Summit) = 4y + x
dist(Harbor-Ridge) = 3x = dist(Harbor-Maple) + dist(Maple-Ridge) = y + 4y =5y

Therefore, 3x = 5y, thus y=3x/5

so dist(Harbor-Summit)/dist(Maple-Summit) = 4x / 4y + x = 4x/ (4(3x/5) + x)
Thus dist(Harbor-Summit)/dist(Maple-Summit) = 20/17

Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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From question, draw a line placing H M R S with spaces.
dist from MR = 4y then HM=y
and dist from RS=x and HMR=3x
we need MS/HS
So 3x=4y+y
x=5y/3

MS/HS = 4y+(5y/3) / y+4y+(5y/3)
solving this gives 17/20 . option D
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I assumed the distance from Harbor to Ridge = 100 units.
As the question says Ridge is 3 times as far from Harbor as it is from Summit.
So, Ridge -> Summit = 100/3 units.

Therefore, Harbor -> Summit = 100 + (100/3) = 400/3 units.

Maple is 4 times as far from Ridge as it is from Harbor.
If we say Harbor - Maple = x.
Then Maple to Ridge = 4x.

Since Harbor -> Ridge = 100:
x + 4x = 100
5x = 100 x = 20
So: Harbor -> Maple = 20 units.
Maple to Ridge = 80 units.
Therefore, Maple - Summit = (Maple -> Ridge) + (Ridge -> Summit) = 80 + 100/3 = 340/3 units.

Required fraction:
(Maple -> Sumit)/(Harbour -> Sumit) = (340/3)/(400/3) = 17/20
IMO the answer is D.
Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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3 times the distance equals to fract
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Lets draw a line from point H to S. From given information.

Let distance between
Harbor and Maple be = y
Maple to Ridge = 4y

Distance between Ripple to Summit = x
Harbor to Ripple = 3x

H<--(y)-->M<--(4y)-->R<--(x)-->S

However we also know,

H<--(3x)-->R<--(x)-->S

Therefore,

y+4y= 3x , 5y=3x y=3/5x (eq 1)

Distance from M to S
= 4y+x
Distance from H to S
= 3x+x

M to S/ H to S
= 4y+x/4x (eq 2)

Insert eq 1 in eq 2

=17x/20x
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Ratio of HR:RS = 3:1
Ratio of MR:MH = 4:1

Consider, distance HR = 3x and distance RS = x
Similarly, distance HM = a and distance MR = 4a

We need to find ratio MS:HS

HS = 3x + x

MS = MR + RS
= 4/5(3x) + x ........ (since MR = 4/5 of HR)

So, MS/HS = ( (12x/5) + x ) / (4x) = 17x/20x = 17/20
Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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Attachments

File comment: Lets draw a like and marl the points in the order.
Now as per the question,

if the distance from R to S is y then dis from H to R is 3y

similarly,
distance from H to M is k then distance from M to R is 4K

now logically, if we see the number lines diagram which we have drawn

Dis from H to R is equal to dis from H to M plus M to R
3y = k + 4k
Solving K = 3y/5

Now we need to fond the fraction
(H to S )/ (M to S) = 4y / (4K + y)

solving and substituting k= 3y/5 we get the ratio 20/17
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<--3/20-><--------17/20--------->
H---------M--------------------------R--------------S
<----------------3/4----------------><----1/4---->
let's call the fraction t.
t*HS = MS, therefore, t = MS/HS.
Now MS/HS = 1- HM/HS (I am doing this to simplify the parts we need to separately calculate)
As we need only fraction, Considering HS = 1
HR = 3/4
HM = 1/5*3/4 = 3/20
therefore, HS/MS = 3/20
and MS/HS = 17/20
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H------M------R-----S

HR = 3x____(1)
RS = x also,

HM = y
MR = 4y so HR = 5y = 3x (From 1)

Therefore, y = [3][/5]x

We are asked to Find MS/HS

MS = MR + RS = 4y+x = 4([3][/5]x) + x = [17][/5]x
and
HS = HR + RS = 3x+ x = 4x

So MS/HS = [17][/20]x

Drawing it out on paper would really help to visualise it!
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Using H as point zero we can arrange this as H<M<R<S
Let R distance from H be r and R to S be 1-r
So r=3(1-r) = r=3/4
Let M be at position m So M to R is r-m = 3/4-m = 4m
Hence m=3/20
Thus M to S equals 1-m= 1-3/20 = 17/20
Ans D
Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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I did the following.

H-M-R-S

H-R = 3x
R-S = x

H-M = y
M-R = 4y

MS/HS = 4y+x/4x

5y=3x rearrange and sub above

net result 17x/20x
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i am going with option d.
there are 4 stops namely h,m,r and s.
let the total distance thus be hm+mr+rs. we are given that hm+mr = 3rs and mr = 4hm
we can subsitute mr=4hm in first equation, hm+4hm = 3rs = 5hm =3rs
therefore rs = 5/3hm. we are asked to find ms and hs that is 17/3 hm and 20/3 hm.
ms/hs = 17/20.
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As per Q, H-----M-----R-----S in a straight line in this order. (H= harbor, M= Maple, R= Ridge, S= Summit)
Assuming RS=x, RH = 3x (since Ridge is 3 times from harbor as it is from Summit)
similarly, Assuming, HM=y, MR=4y
now we need to find relation between x & y
--> HR = 3x = HM+MR= 4y, hence y = 3x/5------(1)
Q asked let f be the req fraction,
then: MS= f x HS
or, f= MS/HS = (4y+x)/(4x)
Substituting y from (1), we get f = 17/20
Hope this helps!!
Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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HM+MR=3X
RS=X

MR=4Y
HM=Y

SO 3X = 5Y
ie X= 5Y/3

Question is MS/HS =?


MS/HS = (MR+RS)/(HM+MR+RS) = (4Y+X)/(Y+4Y+X)= (4Y+5Y/3)/(5Y+5Y/3) = (17Y/3)/(20Y/3) = 17/20 Ans

ie answer is D
Bunuel
Along a straight coastal road, four stops, Harbor, Maple, Ridge, and Summit, appear in that order. Ridge is 3 times as far from Harbor as it is from Summit. Maple is 4 times as far from Ridge as it is from Harbor. What fraction of the distance from Harbor to Summit is the distance from Maple to Summit?

A. 3/20
B. 1/4
C. 2/5
D. 17/20
E. 20/17


 


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