Puzzle of the week
Past week's problems and solutions
Nov 29, 2020
Problem: An integer from 1 - 100 is randomly picked. What is the probability that the integer is divisible by 2, 3, or 5?
Solution: This problem falls under the category of Probability. The formula for probability is desired outcomes divided by total outcomes. Now let's find the desired outcomes. There are 50 multiples of 2, 33 multiples of 3, and 20 multiples of 5. But we are over counting the multiples of 10, 15, 6. There are 10 multiples of 10, 6 multiples of 15, and 16 multiples of 6. We have to subtract these from the total. Finally, when we are subtracting, we are over counting the multiples of 30. There are 3 multiples of 30 which w need to add to the total. So our desired outcomes is 50 + 33 + 20 - 16 - 10 - 6 + 3 = 74. The total outcomes is 100. So desired/total outcomes is 74/100 or 37/50.
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Nov 15, 2020
Problem: 2 angles of a convex hexagon are congruent. Each of the other angles has a degree measure that is 4 times the measure of the first 2 angles. Find the degree measure of the larger angles.
Solution: This problem falls under the category of Geometry. The sum of the angles in a hexagon is 180(n-2) which is 720 (n is the number of sides). We can call the 2 congruent angles x which makes each of the other 4 angles 4x. Since 4 of the angles are 4x that makes the sum of those angles 16x and the 2 congruent angle sum is 2x. 2x + 16x = 18x so 18x = 720. That means that x is 40 but the larger angles is 4 times of this so 160 is the answer.
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Nov 8, 2020
Problem: Jack drove 30 mph to work. As soon as he got to work, he remembered that he forgot to feed his dogs. So, he sped back home, driving 45 mph. What was his average speed during his round-trip to work and back home?
Solution: This problem falls under the category of Speed Distance Time. Let's say that the distance he drove is x miles. We know that distance/speed = time. That means that he spent x/30 hours for his first trip. He is driving the same x miles for his second trip so he is spending x/45 hours. He drove 2x miles altogether and when we divide by the total time, we will be able to find the average speed. x/30 + x/45 = 5x/90 = x/18. 2x/(x/18) = 36 so his average speed is 36 mph. You could have also solved this by doing harmonic mean which is the reciprocal of the average of the reciprocals of the 2 speeds or in other word 2ab/(a+b). 2*45*30/ 75 also gives you 36 mph.
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Nov 1, 2020
Problem: sqrt(a+b/c) = a * sqrt(b/c). Find c in terms of a and b.
Solution: This problem falls under the category of Algebra. We start by squaring both sides of the equation to get a+b/c = a^2 * b/c. Multiplying 'c' on both sides you get ac + b = a^2*b. Subtracting 'b' on both sides and dividing both sides by 'a' you get c = (a^2*b-b)/a. You can further split the fraction into c = ab - b/a.
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Oct 25, 2020
Problem: Bob can paint a wall in 6 days and Charlie can paint the same wall in 14 days. They start working together. After 4 days Charlie leaves and Bob finishes the job alone. How many days does it take Bob to finish painting the wall?
Solution: This problem falls under the category of Rates & Proportions. Bob can paint 1/6 of the wall in 1 day and Charlie can paint 1/14 of the wall in 1 day. Adding these 2 fractions you get 5/21 and multiplying by 4 (because they worked together for 4 days) you get 20/21. So Bob still has to finish 1/21 of the wall. To find the number of days it will take Bob to finish the wall, you just divide 1/21 and 1/6. So our final answer is 2/7.
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Oct 18, 2020
Problem: For Halloween, Mrs. Smith bought 100 bags of candies. Each bag had 6 pieces of M&M candies. If Bob and his friends can eat 15 M&Ms per minute how many hours will it take them to finish the candies? Express your answer as a fraction.
Solution: This problem falls under the category of Rates & Proportions. When you multiply 100 and 6, you get 600 so there are 600 total M&M candies. Since Bob and his friends can eat 15 candies per minute, you divide 600 by 15 to find the total number of minutes it takes to finish the candies, which is 40. We divide 60 to convert to hours and find 2/3 hours.
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Oct 11, 2020
Problem: For what value of x does x @ 6 = 3, if a @ b = (3a + 2b) / (2b - a)
Solution: This problem falls under the category of Operations. If you substitute x for a, 6 for b and equate this to 3 you get, (3x + 12) / (12 - x) = 3. Now this becomes a one-variable linear equation. Multiplying (12-x) on both sides, you get 3x + 12 = 36 - 3x. Solving for x, you get x = 4.