Problem Solving Practice Set

Try the following Problem Solving questions using the Kaplan Method for Problem Solving. If you’re up to the challenge, time yourself; on Test Day, you’ll want to spend only about 2 minutes on each question.

  1. If r = 3s, s = 5t, t = 2u, and u ≠ 0, what is the value of

    1. 30
    2. 60
    3. 150
    4. 300
    5. 600

  2. In the diagram, ℓ1 is parallel to ℓ2. The measure of ∠a is 40 degrees. What is the sum of the measures of angles d and w less the sum of the measures of angles y and c?

    Parallel lines l1 and l2 are cut by a transversal line, forming eight angles. Four angles are on l1, with angle a degrees on the upper left of l1, followed by angles b, c, and d degrees in a clockwise direction. The other four angles are on l2, with angle z degrees on the upper left of l2, followed by angles w, x, and y degrees in a clockwise direction.

    Note: Figure not drawn to scale.

      degrees

  3. At Central Park Zoo, the ratio of sea lions to penguins is 4:11. If there are 84 more penguins than sea lions, how many sea lions are there?

    1. 24
    2. 36
    3. 48
    4. 72
    5. 121

  4. Which of the following numbers has more than two distinct prime factors? Indicate all such numbers.

    1. 20
    2. 30
    3. 100
    4. 200
    5. 210
  5.  
    Four squares touching sides, with the two top squares offset by one square from the two bottom squares. The lower left corner is labeled A, and the intersection of the two top squares is labeled B.

    The figure above is made up of 4 squares. If a straight line segment were to be drawn from A to B, it would have a length of units. What is the perimeter of the entire figure in units?

    1. 20
    2. 32
    3. 40
    4. 64
    5. 80
  6.  
    A rectangle has corners labeled, clockwise from bottom left, A, B, C, and D. Point F is located above the center of rectangle ABCD. Lines connect F to points A and D to create triangle ADF. Triangle ADF crosses the top of rectangle ABCD at points labeled E and G, left to right.

    The figure above consists of triangle ADF and rectangle ABCD. BE = CG = 1.5 and EG = 2. If the area of triangle EFG = x, what is the area of rectangle ABCD?


  7. Before last night’s game, a basketball player had scored an average (arithmetic mean) of 20 points per game. She scored 25 points in last night’s game, raising her average to 21 points per game.

    How many games did she play before last night’s game?

    1. 3
    2. 4
    3. 5
    4. 6
    5. 7

  8. Set A consists of the values 1, 2, and 3. Set B consists of the values 5, 6, and 7. One number is selected at random from each set. The two selected numbers are then added together.

    The probability that the sum is even is how much greater than the probability that the sum is a prime number?

  9. How many positive odd factors does 768 have?

    1. 0
    2. 1
    3. 2
    4. 3
    5. 4

  10. Evan trains for running on a circular path with a radius of  km.

    If Evan starts at one point and runs continuously in one direction for a total of 5 km, how many times does Evan complete a full lap around the entire circle? 

    1. 2
    2. 3
    3. 4
    4. 5
    5. 6