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About Graphs — Class 8 Olympiad

Read and interpret line graphs, linear graphs, and coordinate plots; solve graph-based reasoning problems. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 13). On this page you can practice 47 questions across three difficulty levels — 17 easy, 20 medium, and 10 hard — each with a visual step-by-step solution, plus a timed 27-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Graphs — solved examples for Class 8 Olympiad

Example 1easy

A point P(a, b) is such that 'a' is an even negative integer and 'b' is a prime positive integer. In which quadrant does P lie?
  1. A)Quadrant I
  2. B)Quadrant II
  3. C)Quadrant III
  4. D)Quadrant IV

Step-by-step solution

  1. In Quadrant I, x > 0 and y > 0.
  2. In Quadrant II, x < 0 and y > 0.
  3. In Quadrant III, x < 0 and y < 0.
  4. In Quadrant IV, x > 0 and y < 0.
  5. Given that 'a' is an even negative integer, it means a < 0. Given that 'b' is a prime positive integer, it means b > 0.
  6. Since a < 0 and b > 0, the point P(a, b) lies in Quadrant II.

Answer: Quadrant II

Example 2medium

A cyclist travels from City P to City Q, 100 km away. The journey is represented by a distance-time graph. The cyclist starts at 8:00 AM. For the first 2 hours, they travel at 20 km/h. They then stop for 30 minutes. After the break, they increase their speed to reach City Q. If they arrive at City Q at 12:00 PM (noon), what was their average speed in km/h during the last segment of their journey?
  1. A)30 km/h
  2. B)40 km/h
  3. C)50 km/h
  4. D)60 km/h

Step-by-step solution

  1. Calculate distance and time for the first segment: Distance = 2 hours × 20 km/h = 40 km. Time elapsed = 2 hours. Current time = 8:00 AM + 2 hours = 10:00 AM.
  2. Account for the break: The cyclist stops for 30 minutes. Current time = 10:00 AM + 30 minutes = 10:30 AM.
  3. Determine remaining distance and time for the last segment: Remaining distance = Total distance - Distance covered = 100 km - 40 km = 60 km. Remaining time = Arrival time - Time after break = 12:00 PM - 10:30 AM = 1 hour 30 minutes = 1.5 hours.
  4. Calculate average speed for the last segment: Speed = Remaining distance / Remaining time.
    Speed=60km/1.5hours=40km/hSpeed = 60 km / 1.5 hours = 40 km/h

Answer: 40 km/h

Example 3hard

A robot, Alpha, moves along a straight line. Its speed-time graph is as follows: it moves at 10 m/s for the first 2 seconds, then remains stationary for the next 3 seconds, and finally accelerates to 15 m/s for the last 3 seconds. Another robot, Beta, starts at the same point and moves at a constant speed 'k' m/s for the entire 8-second duration. If both robots cover the exact same total distance in 8 seconds, what is the value of 'k'?
  1. A)7.5 m/s
  2. B)8.125 m/s
  3. C)8.75 m/s
  4. D)9.25 m/s

Step-by-step solution

  1. For Robot Alpha, calculate the distance covered in each phase: Phase 1 (0-2s): Distance = Speed × Time = 10 m/s × 2 s = 20 m.
  2. Phase 2 (2-5s): Distance = 0 m/s × 3 s = 0 m (stationary).
  3. Phase 3 (5-8s): Distance = 15 m/s × 3 s = 45 m. Total distance for Alpha = 20 + 0 + 45 = 65 m.
  4. For Robot Beta, the distance covered is k m/s × 8 s = 8k m. Since both distances are equal, 8k = 65, which gives k = 65/8 = 8.125 m/s.

Answer: 8.125 m/s

Practice questions on Graphs

  1. Q1.easy

    A distance-time graph shows a cyclist's journey divided into four segments. Segment P shows 20 km covered in 1 hour. Segment Q shows 15 km in 0.5 hours. Segment R shows 30 km in 2 hours. Segment S shows 10 km in 0.25 hours. Which segment represents the fastest average speed?
    1. A)P
    2. B)Q
    3. C)R
    4. D)S
    Show answer

    Answer: S

    Hint: Speed is calculated as the distance covered divided by the time taken. Compare the calculated speeds for each segment.

  2. Q2.easy

    Which set of points lies on the same straight line (are collinear)?
    1. A)(1, 1), (2, 3), (3, 4)
    2. B)(0, 0), (1, 2), (2, 4)
    3. C)(-1, 0), (0, 1), (1, 2)
    4. D)(2, 0), (2, 3), (3, 2)
    Show answer

    Answer: (0, 0), (1, 2), (2, 4)

    Hint: Points are collinear if the slope between any two pairs of points is the same. Alternatively, try plotting them.

  3. Q3.easy

    A line graph passes through the points (1, 4), (2, 7), and (3, 10). Which of the following points also lies on this line?
    1. A)(4, 12)
    2. B)(5, 15)
    3. C)(0, 2)
    4. D)(6, 19)
    Show answer

    Answer: (6, 19)

    Hint: Look for a pattern or a rule that relates the x-coordinate to the y-coordinate for the given points.

  4. Q4.medium

    Four points A(1,1), B(6,1), C(5,5), and D(2,5) are plotted on a coordinate plane. What is the area of the quadrilateral ABCD in square units?
    1. A)10
    2. B)12
    3. C)14
    4. D)16
    Show answer

    Answer: 16

    Hint: Observe the parallel lines formed by the points. This shape can be identified as a trapezoid.

  5. Q5.medium

    A local library charges a late fee for overdue books. The fee structure is shown by the graph: ₹5 for the first 3 days, and then ₹2 per day for each additional day. If a book is returned 8 days late, what is the total late fee?
    1. A)₹15
    2. B)₹18
    3. C)₹20
    4. D)₹22
    Show answer

    Answer: ₹15

    Hint: Identify the fixed fee for the initial period and then the variable fee for the additional days. Break down the 8 days into these two parts.

  6. Q6.medium

    Two cars, A and B, start their journeys from two towns 300 km apart at the same time. Car A travels from Town X towards Town Y, and Car B travels from Town Y towards Town X. Their distance-time graphs are plotted on the same axes. Car A travels at a constant speed of 60 km/h. Car B travels at a constant speed of 40 km/h. How far from Town X will they meet?
    1. A)120 km
    2. B)150 km
    3. C)180 km
    4. D)200 km
    Show answer

    Answer: 180 km

    Hint: Think about the combined speed when objects move towards each other. The meeting point is where their total covered distance equals the initial separation.

  7. Q7.hard

    Four points A(1,2), B(5,4), C(4,8), and D(0,6) are the vertices of a quadrilateral in the Cartesian plane. What is the area of this quadrilateral?
    1. A)16 square units
    2. B)18 square units
    3. C)20 square units
    4. D)22 square units
    Show answer

    Answer: 18 square units

    Hint: You can divide the quadrilateral into two triangles and sum their areas, or use the shoelace formula if you're familiar with it.

  8. Q8.hard

    A straight line passes through the point P(3, 2). It intersects the positive x-axis at point A and the positive y-axis at point B. If the area of triangle OAB, where O is the origin, is 12 square units, what is the equation of the line?
    1. A)2x + 3y = 12
    2. B)3x + 2y = 13
    3. C)x + y = 5
    4. D)x/4 + y/6 = 1
    Show answer

    Answer: 2x + 3y = 12

    Hint: Let the x-intercept be 'a' and the y-intercept be 'b'. Use the intercept form of a line and the area formula for triangle OAB.

  9. Q9.hard

    Two friends, Rohan and Sameer, live 100 km apart. Rohan starts from his home towards Sameer's at 9:00 AM, cycling at 20 km/h. Sameer starts from his home towards Rohan's at 9:00 AM, cycling at 15 km/h. Sameer stops for a 30-minute break after covering 30 km. At what time do they meet?
    1. A)11:45 AM
    2. B)12:00 PM
    3. C)12:05 PM
    4. D)12:04 PM
    Show answer

    Answer: 12:04 PM

    Hint: Break the journey into stages. First, determine when Sameer takes his break and where both are at that point. Then calculate their positions after Sameer's break and solve for their meeting time.

These are 9 of the 47 questions available for Graphs. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.