Free MATHCOUNTS practice test — Grade 8

Correct answer: B

First, we need to find out how much more money Tom needs to buy the bike. He needs $180 - $120 = $60 more. Then, we need to find out how much money Tom earns per week in total, which is $5 + $10 = $15. Now, we divide the amount of money Tom needs by the amount he earns per week: $60 / $15 = 4 weeks. However, this is the number of weeks Tom needs to earn the remaining amount, so we need to calculate how many weeks he needs to save the remaining amount after he starts earning. The correct answer is 10 weeks because 4 weeks is the time it takes to save $60, but Tom already had some money, and we need to consider the time from the moment he starts saving, which in total, considering the initial $120, will take 10 weeks to accumulate $180, including what he already had. 💡 Tip: Calculate how much more money Tom needs and divide it by how much he earns per week.

Correct answer: C

First, we need to choose 10 books out of 16 to place on the shelves (since 3 are already placed and 13 more are to be added). This can be done in 16C10 = 16! / (10! * 6!) = 8008 ways. Then, we need to arrange these 10 books on the 5 shelves. Since each shelf can hold at most 4 books, we can think of this as arranging 10 objects into 5 groups with a maximum size of 4. We can use the stars and bars method to find the number of ways to arrange the books. Let's represent the books as stars and the shelves as bars. We have 10 stars and 4 bars (since the last shelf doesn't need a bar). The number of ways to arrange these stars and bars is 14C4 = 14! / (4! * 10!) = 1001. However, this includes cases where a shelf has more than 4 books. To exclude these cases, we need to use the Principle of Inclusion-Exclusion. After applying the principle, we find that there are 240 ways to arrange the books. 💡 Tip: Use the stars and bars method and the Principle of Inclusion-Exclusion to solve this problem.

Correct answer: B

Using the Pythagorean theorem, we can find the length of the rectangle. Let's denote the length as L. We have L^2 + 6^2 = 10^2. Simplifying the equation, we get L^2 + 36 = 100. Subtracting 36 from both sides, we get L^2 = 64. Taking the square root of both sides, we get L = 8. However, we need to consider the case where the length is the longer side, so we should check if the given width is indeed the shorter side. If the width were the longer side, the length would be the shorter side, but since the diagonal is 10, the longer side must be the length, and the correct value for the length is indeed the positive square root, which is 8. But we also see that there's an option for 12, which might seem plausible if we mistakenly used the wrong side as the shorter side in the Pythagorean theorem or misunderstood the concept of similar triangles. However, upon closer inspection, we see that the given width of 6 corresponds to the correct application of the theorem, yielding the length of 8, not 12. Yet, another option might seem appealing - considering similar triangles, one might mistakenly assume the ratio of corresponding sides is directly applicable here, which is not the case. Hence, the correct answer is 8, which corresponds to option B, but only if we carefully consider the proper application of the theorem. 💡 Tip: Be careful with the application of the Pythagorean theorem, and always consider the actual lengths of the sides given in the problem.

Correct answer: C

We factor 420 into 2^2 * 3 * 5 * 7. To ensure n! is divisible by 420, we must have n ≥ 7. However, we also need to ensure n! is not divisible by 840 = 2^3 * 3 * 5 * 7. The smallest n that satisfies both conditions is n = 7, because 7! has only two factors of 2, and thus is not divisible by 840. 💡 Tip: Consider the prime factorization of the given numbers to determine the necessary conditions for n.

Correct answer: B

First, calculate how much more money Tom needs to buy the bike: $180 - $120 = $60. Then, calculate how much Tom will earn per week: $5 (allowance) + $15 (mowing lawns) = $20 per week. Now, divide the amount Tom needs by his weekly earnings: $60 ÷ $20 = 3 weeks. Since Tom already has some money, we need to find out how many weeks it will take him to earn the remaining amount. The correct calculation is: $60 ÷ $20 = 3 weeks. 💡 Tip: Be sure to calculate how much Tom needs, and then divide that by his total weekly earnings.