Grade 8 Math Practice Test: Slope, Unit Rate and Proportional Relationships
- Slope and Proportional Relationships
- 10 questions
- 15 minutes
This test connects slope and unit rate for lines through the origin, written as y = mx, and compares proportional relationships shown in different ways. Tip: on the graph of a proportional relationship, the point (1, m) shows the unit rate m.
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Question 1
The graph of the distance d (in miles) a train travels in t hours is a straight line through the origin and the point (3, 165). What is the slope of the line, and what does it mean?
Correct answer: A
Slope = 165 ÷ 3 = 55, and for a distance-time graph through the origin the slope is the speed: 55 miles per hour. 165 is the distance after 3 hours, not per hour.
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Question 2
Runner A's distance is d = 6.5t, where d is in kilometers and t in hours. Runner B's distances are shown in the table.
Time t (h) 2 5 Distance d (km) 14 35 Which runner is faster, and by how much?
Correct answer: C
Runner B's unit rate is 14 ÷ 2 = 35 ÷ 5 = 7 km per hour, and Runner A's is 6.5 km per hour. 7 − 6.5 = 0.5, so B is faster by 0.5 km per hour.
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Question 3
Which equation describes a line through the origin with a slope of 34?
Correct answer: C
A line through the origin has the form y = mx, and here m = 3/4. y = x + 3/4 has slope 1 and crosses the y-axis at 3/4, so it misses the origin.
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Question 4
Ty draws two different slope triangles between points on the same non-vertical line. Why do both triangles give the same slope?
Correct answer: D
Slope triangles on the same line have equal corresponding angles, so they are similar, and similar triangles have equal ratios of corresponding sides. They can have different sizes, so they are not always congruent.
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Question 5
A proportional relationship has a graph that passes through the origin and the point (−4, 10). Which equation represents it?
Correct answer: D
m = y ÷ x = 10 ÷ (−4) = −5/2, so y = −5/2 x. −2/5 divides x by y, and +5/2 ignores the negative x-coordinate.
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Question 6
Music service A charges c = 1.25n dollars for n songs. The graph of service B's cost is a line through the origin and the point (4, 4.60). Which service costs less per song?
Correct answer: B
Service B's unit rate is $4.60 ÷ 4 = $1.15 per song, which is less than service A's $1.25. $4.60 is the cost of 4 songs, not of one.
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Question 7
A tank fills at a constant rate. After 4 minutes it holds 30 gallons, and after 10 minutes it holds 75 gallons. It was empty at the start. What is the unit rate?
Correct answer: B
30 ÷ 4 = 7.5 and 75 ÷ 10 = 7.5, so the tank fills 7.5 gallons each minute. 0.13 is about 4 ÷ 30, minutes per gallon, and 18.75 divides 75 by 4, mixing up the two rows.
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Question 8
The graph of the cost y, in dollars, of x pounds of grapes is a line through the origin and the point (1, 2.80). What does the point (1, 2.80) mean?
Correct answer: B
The x-coordinate is pounds and the y-coordinate is dollars, so 1 pound costs $2.80. This is the unit rate and the slope. There is no fixed fee, because the line passes through the origin.
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Question 9
Which line is the steepest?
Correct answer: A
Steepness depends on the absolute value of the slope, and |−4| = 4 is the largest. A negative slope makes the line fall from left to right, but it can still be the steepest.
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Question 10
What is the equation of the line through the points (−3, 6) and (2, −4)?
Correct answer: C
Slope = (−4 − 6) ÷ (2 − (−3)) = −10 ÷ 5 = −2. Check (2, −4): −2 · 2 = −4, so the line passes through the origin and y = −2x. −1/2 divides the run by the rise.
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