In the figure above, segments PQ and QR are each parallel to
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We have to get the value of the ratio of the length of QR to PQ.
Let's take each statement one by one.
(1) The slope of the line that passes through points P and R is 0.75.
Say the coordinates of points P are (x1, y1) and the coordinates of points R are (x2, y2); thus, the coordinates of points Q would be (x1, y2).
Thus, QR = (x2  x1) and QP = (y2  y1)
Thus, QR/QP = (x2  x1)/(y2  y1)
It is given that the slope of the line that passes through points P and R is 0.75; thus, QP/QR = slope = (y2  y1)/(x2  x1) = 0.75
Thus, QR/QP = 1/0.75 = 4/3. Sufficient.
(2) The coordinates of point P are (4,2).
We do not have information about x2 and y2. Insufficient.
The correct answer: A
Hope this helps!
Jay
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It is generally true in Data Sufficiency that if the question asks you for a PROPORTION, you do not need concrete values to answer it.
In this case, we're asked for a RATIO of horizontal and vertical lengths, but not for actual values. We can think of any slope as a RATIO of vertical to horizontal (rise/run). For example, a slope of 2/3 means "rise 2, run 3." We could go from (0, 0) to (2, 3) to (4, 6) to (8, 9), etc. Between any two of these points, the ratio of the vertical distance to the horizontal distance will be 2 to 3.
Thus, when the question asks "What is the ratio of the length of QR to PQ?", we could rephrase as:
what is the slope of line PR?

(1) The slope of the line that passes through points P and R is 0.75.
This answers our target question. Sufficient.

(2) The coordinates of point P are (4,2).
A single set of values tells us nothing about proportions. Insufficient.

The answer is A.
In this case, we're asked for a RATIO of horizontal and vertical lengths, but not for actual values. We can think of any slope as a RATIO of vertical to horizontal (rise/run). For example, a slope of 2/3 means "rise 2, run 3." We could go from (0, 0) to (2, 3) to (4, 6) to (8, 9), etc. Between any two of these points, the ratio of the vertical distance to the horizontal distance will be 2 to 3.
Thus, when the question asks "What is the ratio of the length of QR to PQ?", we could rephrase as:
what is the slope of line PR?

(1) The slope of the line that passes through points P and R is 0.75.
This answers our target question. Sufficient.

(2) The coordinates of point P are (4,2).
A single set of values tells us nothing about proportions. Insufficient.

The answer is A.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education