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Daily Quiz Solutions #1 and #2

Here is how to work the quizzes from this weekend. Unfortunately, this means if you haven’t posted your answers for these questions yet, you don’t get points for them. But even so, work the problems anyway – they’re great practice!

Got questions? Post them in the comments!

1. We want to find the number of visits at which the amount paid by a subscriber equals the amount paid by the non-subscriber. If the subscriber’s cost is given by 99 + 5x and the non-subscriber’s by 18x, solving the equation 99 + 5x = 18x for x yields the answer of 8 visits. Remember to round up, because rounding down (7 visits) would still cost me more.

2. We’ll work this problem using the blanks method. There are two blanks, because two books are being chosen. The first blank contains 70/150, and the second blank contains 80/149. But wait – we also have to multiply by 2 because we could have chosen a nonfiction book first instead of a fiction book. Evaluating (70/150) x (80/149) x 2 yields the answer, 224/447. Yuk.

3. Here we’re solving a system of equations in three variables. The equations are (1) bac – 8, (2) abbc – 14, and (3) b = 3a. We can subtract a from both sides in (1) and subtract b from both sides in (2). Then we substitute for b in (1) from (3) to get (4) 3ac – 8. Then, we subtract the modified (2) and (4), eliminating the c term, to get 2a = 6 and a = 3. From (3), b = 9, and from (1) c = 17. The sum abc29.

4. To convert 17268 to base 10, we determine what place value each digit represents. Since the place values are powers of 8, the value in base 10 is 1 x 83 + 7 x 82 + 2 x 81 + 6 x 80 = 982. Performing the same steps on 324 using base 6, we get 124. 982 + 124 = 1 106.

5. The only way to imagine this trapezoid (call it ABCD) is like this: AB and BC are 300 meters, angles B and C are right angles, and the diagonal DB is 500 meters. If it doesn’t make sense, I recommend making a drawing. We are looking for diagonal AC, and we quickly realize that since both sides of the triangle (AB and BC) measure 300 meters, the hypotenuse must be 300\sqrt{2}, or 424.3 meters.

6. This problem can be solved by remembering that the volume of a cone is proportional to the square of the radius and to the height. Since the tall cone is 3 times as tall as the regular, it has 3 times the volume. Since the wide one has twice the diameter, it therefore has twice the radius and four times the volume. So the wide cone holds more ice cream than the tall, at a factor of 4/3.

7. This problem is similar to #2. First, let’s figure out how many ways we could flip the coin: 2 x 2 x 2 = 8. Then, let’s draw three blanks, ____ ____ ____. If heads A and B each have to go into one of those blanks, there are 3 blanks for heads A and 2 left for heads B. 3 x 2 = 6 ways, but we have to divide by 2 to eliminate the ways where heads A and B are switched. The answer is 3/8.

8. We have ten consecutive numbers, n, n + 1, n + 2, and so on up to n + 9. Their sum is 395, and when we write this as an equation and collect like terms, we get 10n + 45 = 395 and n = 35. This is the lowest of the consecutive numbers. To quickly add up the 5 odd numbers {35, 37, 39, 41, 43}, remember that their average is 39 and multiply by 5 numbers to get 195.

9. First, figure out how much lawn each person could mow in an hour: 1/2 lawn and 2/3 lawn. The amount of lawn both would mow together is 1/2 + 2/3 = 7/6. Setting a proportion 7/6 = 10.5/t, we find that it will take them 9 hours to mow the lawn.

10. The quadratic equation that includes the given values is -16t2 + 40t + 24 = 0. We can divide the equation by -8 to make the numbers easier: 2t2 – 5t – 3 = 0. Factoring by grouping, we get (2t + 1)(t – 3) = 0. The positive solution is 3 seconds.

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