Math Kangaroo Practice Pack — Grade 8

Answer Key — Math Kangaroo Grade 8

1. 1. A — Let's denote the cost of the ball as x. Since the bat costs $1.00 more than the ball, the bat costs x + $1.00. The total cost of the bat and the ball together is $1.10, so we can set up the equation: x + (x + $1.00) = $1.10. Simplifying the equation, we get 2x + $1.00 = $1.10. Subtracting $1.00 from both sides gives us 2x = $0.10. Dividing both sides by 2 gives us x = $0.05. Therefore, the ball costs $0.05. 💡 Tip: This problem requires setting up an equation based on the given information and solving for the unknown cost of the ball. It's essential to carefully translate the problem statement into a mathematical equation and then solve it step by step.
2. 2. D — The volume of the whole prism is length * width * height = 8 * 5 * 3 = 120 cm^3. The smaller prism has the same length and width but a height of 2 cm (since it's sliced 2 cm from the top), so its volume is 8 * 5 * 2 = 80 cm^3. However, we are looking for the ratio of the smaller prism (which is actually the part cut off, not the remaining larger part) to the whole. The sliced part's volume is 8 * 5 * 1 = 40 cm^3 (since the slice is 1 cm high from the top). The ratio of this smaller part to the whole prism is 40/120 = 1/3. 💡 Tip: Calculate the volumes of the prisms and then find the ratio of the volumes as requested.
3. 3. B — The snail effectively climbs 1 foot per day (3 feet up during the day, then slips back 2 feet at night). However, on the final day of climbing, the snail will reach or exceed the top of the well and not slip back. Thus, we need to calculate the number of days it takes for the snail to climb 18 feet (since on the 18th foot, it will reach the top). This would take 18 days of climbing 1 foot per day effectively. 💡 Tip: Consider the net progress per day and adjust the final calculation based on whether the last step exceeds the target.
4. 4. D — First, calculate the total number of pens Tom has initially: 1+2+3+4+5+6+7+8+9+10+11+12 = 78. Then, identify the boxes with an odd number of pens: 1, 3, 5, 7, 9, 11. Calculate the total number of pens in these boxes: 1+3+5+7+9+11 = 36. Tom gives away 1/3 of these pens: 36/3 = 12 pens given away. So, he has 78-12 = 66 pens left. 💡 Tip: Identify the specific boxes and pens involved in the action.
5. 5. A — The probability that the first player wins the game on their first turn is indeed 1/6, as that's the probability of rolling a 6. However, if they don't win on the first turn, the game essentially restarts, and they have the same chance of winning as they did initially, but now it's the probability of not winning on the first round and then winning. Since there are 5 players, the probability that the first player wins on any given round after the first is (5/6)*(1/6), because the first player must not win on their first turn (4/5 chance for each of the other players not to win, combined as (4/5)^4 for all other players), and then win on their subsequent turn. However, simplifying the calculation to its essence, the first player's chance of winning on the first turn is 1/6. The chance of the game continuing beyond the first round involves complex considerations of the other players' turns, but essentially, the first player's chance of winning overall remains 1/6 because the structure of the game does not change their odds of winning on their turn. 💡 Tip: The key insight here is recognizing that the probability of the first player winning does not change based on when they win, as long as the structure of the game remains the same. This requires understanding the nature of probability in repeating events.
6. 6. B — The volume of the larger cube is 216 cm^3. Since volume = side^3, the side length of the larger cube is the cube root of 216, which is 6 cm (because 6^3 = 216). The maximum number of smaller cubes (1 cm on each side) that can fit into this larger cube is found by dividing the volume of the larger cube by the volume of a smaller cube: 216 cm^3 / 1 cm^3 = 216. Thus, 216 smaller cubes can fit into the larger cube. 💡 Tip: Understand that the maximum number of smaller cubes is directly related to the volume of the larger cube divided by the volume of a single smaller cube.
7. 7. B — Let x be the number of whole wheat loaves and y be the number of white loaves. We have x + y = 250 and 0.20x + 0.15y = 40. We can solve this system of equations for x. First, multiply the first equation by 0.15 to get 0.15x + 0.15y = 37.5. Then subtract this from the second equation to find 0.05x = 2.5, so x = 50. However, this initial step was incorrect; the right approach involves solving these equations correctly: from 0.20x + 0.15y = 40, we express y as y = (40 - 0.20x) / 0.15. Substituting y in x + y = 250 gives x + (40 - 0.20x) / 0.15 = 250. Simplifying, we get x + (400 - 4x) / 3 = 250, leading to 3x + 400 - 4x = 750, which simplifies to -x = 350, and x = -350 is not valid. Let's correct the algebraic manipulation: 0.20x + 0.15(250 - x) = 40, simplifying to 0.20x + 37.5 - 0.15x = 40, which leads to 0.05x = 2.5, and x = 50 is also not the correct step here. Correcting the calculation directly from the equation: 0.20x + 0.15y = 40 and x + y = 250, we should directly solve for x without confusion. The confusion arose from incorrect manipulation. The right way involves setting up the equation correctly and solving for the variables without the incorrect intermediate steps. The actual step should involve using substitution or elimination correctly, acknowledging the mistake in algebraic manipulation. 💡 Tip: To solve systems of linear equations, use either substitution or elimination method, and ensure correct algebraic manipulations.
8. 8. B — For the 2 games won, they score 2 * 2 = 4 points. For the 3 tied games, they score 3 * 1 = 3 points. For the 1 game lost, they score 0 points. Adding these together: 4 + 3 + 0 = 7 points. However, considering the average scoring and the specific outcomes, the correct approach directly calculates the points from wins and ties without needing to adjust for losses, since losses contribute 0 points. Thus, a closer examination shows the straightforward calculation is correct but let's ensure clarity: 2 wins * 2 points = 4 points, 3 ties * 1 point = 3 points, totaling 7 points. This matches the initial calculation, confirming the total score is indeed 7 points from wins and ties, with no contribution from losses. 💡 Tip: Calculate points from each type of game outcome separately and sum them.
9. 9. C — First, we find the combined rate of pipes A and B. Pipe A fills 1/4 of the tank per hour, and pipe B fills 1/6 of the tank per hour. Their combined rate is 1/4 + 1/6 = (3/12) + (2/12) = 5/12 of the tank per hour. After 2 hours, they fill 2 * (5/12) = 10/12 = 5/6 of the tank. Then, only pipe B remains, filling 1/6 of the tank per hour. To fill the remaining 1/6 of the tank, pipe B takes 1 hour. So, the total time to fill the tank is 2 hours (with both pipes) + 1 hour (with pipe B alone) = 3 hours, but since we're considering the time from the start until the tank is full, and given that the initial calculation aimed to demonstrate the process but contained a miscalculation in determining the exact time needed after the first 2 hours, let's correct the approach: After 2 hours of both pipes working together, 5/6 of the tank is filled, leaving 1/6 of the tank to be filled by pipe B. Since pipe B fills 1/6 of the tank per hour, it indeed takes 1 hour to fill the remaining 1/6 of the tank after pipe A is closed. Thus, the correct total time is the initial 2 hours plus the additional time for pipe B to fill the rest, which we've established is 1 hour for the remaining fraction but requires a correct calculation for the combined initial effort and the subsequent solo filling by pipe B. 💡 Tip: The problem requires calculating the combined rate of the two pipes and then determining how much of the tank is filled after 2 hours. After that, it's a matter of finding out how long it takes the remaining pipe to fill the rest of the tank.
10. 10. B — When a square is rotated 45 degrees, the overlapping region forms a smaller square. The diagonal of this smaller square is equal to the side length of the original square (because the rotation creates an isosceles right triangle when diagonals are drawn). The diagonal of the overlapping square is 10 units. The area of a square from its diagonal can be found using the formula Area = (diagonal^2)/2. So, the area = (10^2)/2 = 100/2 = 50. However, this thought process actually describes the area of the overlapping region being related to the side of the square formed by the overlap differently. The correct insight is recognizing the overlap forms a square where the side length can be determined by geometry principles. The correct calculation directly relates to understanding the geometric properties of the rotation and overlap, which simplifies to using the side length of the original square to find the area of overlap directly through geometric principles, specifically considering the rotation and resulting geometric figures involved. 💡 Tip: Visualize the rotation and the resulting geometric figures to understand how the area of the overlapping region is determined.
11. 11. A — The rate of filling for the first pipe is 1/4 tank per hour, and for the second pipe, it's 1/6 tank per hour. When both pipes are open, their combined rate is 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tank per hour. To find the time to fill the tank, take the reciprocal of this rate: 12/5 hours, which equals 2.4 hours. 💡 Tip: Calculate the combined rate of work for multiple agents working together.
12. 12. B — First, find the combined rate at which both pipes fill the tank. Pipe A fills 1/4 of the tank per hour, and pipe B fills 1/6 of the tank per hour. Together, they fill (1/4 + 1/6) of the tank per hour. The common denominator is 12, so (3/12 + 2/12) = 5/12 of the tank per hour. In 2 hours, with both pipes open, they fill 2 * (5/12) = 10/12 = 5/6 of the tank. After 2 hours, 1/6 of the tank remains to be filled. Pipe B, which is left open, fills 1/6 of the tank per hour. So, it will take 1 hour for pipe B to fill the remaining 1/6 of the tank. Therefore, the total time to fill the tank is 2 hours (with both pipes) + 1 hour (with pipe B alone) = 3 hours, but considering the initial fill rate and the time pipe A is open, the calculation of the combined fill rate and the subsequent fill by pipe B alone after A is closed needs careful handling of the fractions and the time, leading to the realization that after 2 hours, the remaining fraction is indeed filled by pipe B, which actually takes more time to fill the remaining part than initially calculated, given its slower rate. Thus, the step involving the calculation of time for the remaining fraction to be filled by pipe B should consider its rate properly, and the accurate calculation reflects the time it takes for pipe B to fill the tank after A is closed, aligning with the provided options and correcting the oversight in initial calculation steps. 💡 Tip: Calculate the combined rate of filling and then adjust for the change in filling rate when one pipe is closed.
13. 13. B — The sequence starts with 1, 2. The third number is the sum of the first two plus 1, so it's 1 + 2 + 1 = 4. The fourth number is the sum of the second and third numbers plus 1, so it's 2 + 4 + 1 = 7. The fifth number is the sum of the third and fourth numbers plus 1, so it's 4 + 7 + 1 = 12. Therefore, the 5th number in the sequence is 12. 💡 Tip: This problem requires understanding the pattern of the sequence and applying it to find the subsequent numbers. It's essential to carefully follow the pattern and calculate each number step by step.
14. 14. B — For the cylinder to be inscribed in the cone, the diameter of the cylinder (which is 2 * radius = 8 cm) must be equal to the radius of the cone's base at the level where the cylinder's top or bottom touches the cone. This relationship comes from similar triangles formed by the radii and heights of the cone and cylinder. Since the height and radius of the cone and cylinder are proportional, we can set up a proportion based on similar triangles to find the radius of the cone's base. However, given the cylinder fits exactly inside the cone and touches at the top and bottom, and considering similar triangles, the radius of the cone at its base is actually determined to be the same as the radius of the cylinder because of how the shapes relate geometrically when inscribed. Thus, the radius of the cone is equal to the radius of the cylinder, which is 4 cm. 💡 Tip: Use the concept of similar triangles and the geometric relationship between the inscribed cylinder and the cone to solve the problem.
15. 15. B — First, subtract the friends who won't eat any candy: 8 - 2 = 6 friends. Then, divide the total number of candies by the number of friends who will eat: 48 / 6 = 8. 💡 Tip: Adjust the number of recipients based on changes in the group before dividing the total amount.
16. 16. A — Let's denote the cost of the ball as x. The bat costs $1.00 more than the ball, so the bat costs x + $1.00. Together, the bat and the ball cost $1.10, so we can write the equation: x + (x + $1.00) = $1.10. Simplifying, 2x + $1.00 = $1.10. Subtract $1.00 from both sides: 2x = $0.10. Divide both sides by 2: x = $0.05. Therefore, the ball costs $0.05. 💡 Tip: Set up an equation based on the given information and solve for the unknown.
17. 17. D — Using the Pythagorean theorem, a^2 + b^2 = c^2, where c is the length of the hypotenuse (10 inches), and one of the legs (let's say a) is 6 inches. We need to find the length of the other leg (b). So, 6^2 + b^2 = 10^2. This simplifies to 36 + b^2 = 100. Subtracting 36 from both sides gives b^2 = 64. Taking the square root of both sides gives b = sqrt(64) = 8 inches. However, considering the exact calculation with the Pythagorean theorem: b^2 = 100 - 36 = 64, and thus b = sqrt(64) = 8 inches. But to be precise, the calculation should reflect the exact values without rounding during intermediate steps, confirming the result is indeed an exact value when calculated correctly with the theorem. 💡 Tip: The Pythagorean theorem is key to solving this problem. It's crucial to apply the theorem correctly and calculate the length of the unknown leg accurately.
18. 18. B — Given the sides are in the ratio 3:4:5, and the shortest side (3 parts) is 9 cm, we can find the length of one part by dividing the shortest side by 3. So, 9 cm / 3 = 3 cm per part. The longest side is 5 parts, so 5 parts * 3 cm/part = 15 cm. 💡 Tip: Use the ratio of the sides to find the length of one part, then calculate the length of the longest side.
19. 19. C — Subtract the base fee from the total cost to find the cost due to mileage: $50 - $20 = $30. Then, divide this by the cost per mile to find the number of miles driven: $30 / $0.15 = 200 miles. 💡 Tip: Isolate the variable of interest by subtracting constants and then dividing by the rate.
20. 20. B — To find the total number of different outfits, multiply the number of shirts by the number of pants: 8 shirts * 5 pants = 40 outfits. However, considering the basic principle of combinations in this context, the correct calculation directly multiplies the options for shirts by the options for pants, yielding the total number of unique combinations without needing further adjustment, thus confirming the calculation of 40 outfits as a starting point but recognizing the error in not directly affirming this product as the solution from the outset. 💡 Tip: Use the multiplication principle for counting.