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Washington State Math Olympiad
Hints and Solutions
2012 Grade 8 Algebra
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1) Tim is notified that he has a $50 library fine in August. A fee of 20% of the current amount owed is added to the balance on the first of each month. If he does nothing to pay off his fine or incur new fines what will the total fine be in January?
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- September to January is 5 months.
- $50(1.29)5 = 50 x 2.488 = $124.42
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2) Consider the sequence of numbers that starts with: 2, 5, a, b, 30. Suppose that in this sequence, after the first three terms (2,5,a), each subsequent term is found by adding the
previous three terms. For example, the fourth term is 2+5+a. Using this fact and noting that the fifth term of the sequence is 30, what is the value of b?
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This is 2 equations and 2 unknowns
The 4th term (b) = 2 + 5 + a
The last term (30) is 5 + a + b
Making the first equation an equation for a:
a = b - 7
Substituting this in the last equation:
5 + (b - 7) + b = 30
5 + 2b -7 = 30
2b = 32, therefore b = 16
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3) Biffy and Scott are hiking off of a mountain from different rest areas. Biffy's descent to the bottom of the trail can be represented by B=500-5t, where B is in yards and t is in
seconds. Scott's descent can be represented by the function S=200-4t, where S is in yards and t is in seconds. If they arrive at the bottom of the trail at the same time, how many
seconds later than Biffy did Scott leave his rest area?
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- Biffy starts at 500 yards elevation and descends at a rate of 5 yards/sec
- Scott starts at 200 yards elevation and descends at a rate of 4 yards/sec
- So, at elevation 0, Biffy has taken 500/5 = 100 seconds to get there.
- At elevation 0, Scott has taken 200/4 = 50 seconds to get there.
- Therefore, if they got to elevation 0 (the bottom of the trail) at the same time, Scott would have had to wait 50 seconds after Biffy started to start his descent.
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