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Solution |
1) Lucy is riding an elevator up a very tall building. She has just reached 1/3 of the way to the top.
She goes 6 more floors up and is now 2/5 of the way to the top. How many floors does the building have?
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- Compute what fraction of the building's height Lucy goes from 1⁄3 of its height to 2⁄5 of its height:
2⁄5 - 1⁄3 = 6⁄15 - 5⁄15 = 1⁄15 of its height
- 1⁄15 of its height = 6 floors, so the height of the building is 6x15 = 90 floors
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2) In the following sequence, how many
squares are needed to construct the nth figure?
(Express your answer in terms of n)
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- Each sequence has 2 squares on top
- Notice the last row contains n+2 squares
- Also notice that that there are n of these last rows. For example for n=3 (the last figure) the bottom row contains 5 squares (which is n+2) and there are n copies of this last row above it
- So, the other squares (other than the 2 on top) are (n+2)n squares
- So, the nth figure contains n(n+2) + 2 squares, or
n2 +2n +2
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3) The sum of three integers is 107. The first integer is 1 more than 1/4 of the third. The second
integer is 1 less than twice the first. What is the mean of the three integers? Round your answer to
the nearest tenth.
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The mean (average) of a set of 3 integers is merely their sum divided by 3 =
107/3 = 35.666, rounded to the nearest 10th = 35.7
They were just trying to mislead you with all that logic about what each element of the sequence was! You don't need it!
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4) Sam takes two steps forward and one back, three steps forward and two back, and four steps
forward and three back. If Sam repeats this pattern 4 more times (forward and back), how many
total steps will Sam have taken?
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- Counting the number of steps (forward and back) for each time, we get an arithmetic sequence:
3 5 7 . .
- Extending this 4 more steps:
3 5 7 9 11 13 15
- So the last element of this sequence is 15. The equation for the sum of a sequence is:
Sum = (A1 + An) n / 2
- Substituting our values:
Sum = (3 + 15) x 7 / 2 = 63
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5) How many integers satisfy the following equation?
(x + 2) (x - 5) ≤ 0
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- Note that when x gets above 5 both expressions go positive and the total equation is not true.
- Note also that when x goes below -2 that the first expression (x + 2) goes negative and along with the second negative expression, the whole expression becomes positive.
- So the range of x for which the expression is valid is from -2 to +5 which is 8 integer values
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