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# Points Q and R are on segment PS, which has length 24. If PQ is half

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Joined: 02 Sep 2009
Posts: 54496
Points Q and R are on segment PS, which has length 24. If PQ is half  [#permalink]

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12 Jul 2017, 23:43
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Difficulty:

45% (medium)

Question Stats:

68% (01:57) correct 32% (02:16) wrong based on 72 sessions

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Points Q and R are on segment PS, which has length 24. If PQ is half as long as QS, and QR is 3 times as long as RS, how long is RS?

(A) 2
(B) 3
(C) 4
(D) 8
(E) 12

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Re: Points Q and R are on segment PS, which has length 24. If PQ is half  [#permalink]

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13 Jul 2017, 00:38
Bunuel wrote:
Points Q and R are on segment PS, which has length 24. If PQ is half as long as QS, and QR is 3 times as long as RS, how long is RS?

(A) 2
(B) 3
(C) 4
(D) 8
(E) 12

Let PQ = x
QS = 2x
So, x + 2x = 24
-> x = 8

PQ = 8
QS = 16
P..............Q................................S

Let RS = y
QR = 3y
so y + 3y = 16
-> y = 4

RS = 4
QR = 12

P.......8.....Q.............12..........R.....4....S

Answer C.
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Re: Points Q and R are on segment PS, which has length 24. If PQ is half  [#permalink]

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13 Jul 2017, 04:29
Bunuel wrote:
Points Q and R are on segment PS, which has length 24. If PQ is half as long as QS, and QR is 3 times as long as RS, how long is RS?

(A) 2
(B) 3
(C) 4
(D) 8
(E) 12

PQ=X, QS=2X
X=2X= 24
SO, X=8.
P.........Q.......R.........S
X 3Y Y
QR=3Y,RS=Y
QR+RS=4Y=2X
SO,\frac{Y=X}{2}
Y=8/2=4.
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Joined: 22 May 2016
Posts: 2651
Points Q and R are on segment PS, which has length 24. If PQ is half  [#permalink]

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13 Jul 2017, 05:27
1
Bunuel wrote:
Points Q and R are on segment PS, which has length 24. If PQ is half as long as QS, and QR is 3 times as long as RS, how long is RS?

(A) 2
(B) 3
(C) 4
(D) 8
(E) 12

Sketch a number line, find a common term, and use ratios to move the variables around.

P________Q_______________R___S

1. Take the segments in both ratios and add them to get an idea of what might be a common term.

PQ + QS = PS

QR + RS = QS

2. Two ratios are given. Both contain QS (the second, indirectly).

Use the more restrictive condition for QS. Here, that's the ratio with more "parts."

3. QS = QR + RS, and $$\frac{QR}{RS}$$ = $$\frac{3}{1}$$

Let RS = x
QR = 3x
QS (total of ratio parts) = 4x

4. Set up the second proportion in terms of x, given that QS = 4x.

$$\frac{QS}{PQ}$$ = $$\frac{2}{1}$$ = $$\frac{4x}{2x}$$

PQ + QS = PS

2x + 4x = 6x

PS = 6x, and given that PS = 24:

6x = 24

x = 4

5. From above, RS = x, x = 4, hence RS = 4.

Check, starting with first segment PQ: If RS = 4, PQ = 8

That puts the number line values of the variables at
P = 0
Q = 8
R = 20 (QR = 3x = 12, 8+12=20)
S = 24 (RS = 4, 20+4=24)

Correct. RS is 4.

Answer C
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Re: Points Q and R are on segment PS, which has length 24. If PQ is half  [#permalink]

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15 Jul 2017, 01:34
let PQ= a QR = b ans RS = c and the question says that PQ=1/2 (QS) therefore a= 1/2 (b+c) QR = 3 (RS) b = 3(c) and PS = a+b+c=24 , 2a=b+c therefore a=8 , 2a=b+c 16 =b+c 16=4c c=4
Therefore the answer is C
Hope this method helps.
Re: Points Q and R are on segment PS, which has length 24. If PQ is half   [#permalink] 15 Jul 2017, 01:34
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# Points Q and R are on segment PS, which has length 24. If PQ is half

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