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4rd Test - Team test for Bonus Problems
1) Justin is traveling to his Aunt's house in another state. For the first half of the distance he averaged 50 kph (kilometers per hour). If he wants his average speed for the trip to be 80 kph, how fast (on average) does he have to drive for the second half of the distance?
Suppose the distance is 100k
It takes 1 hour to drive 50k (the first half).
If he wants to average 80 kph, then
100k / (1 + x) hr = 80 k/hr
100 = 80 + 80x
x = 1/4 hr
This means he has only 1/4 hour to drive 50 k so
50 k / (1/4 hr) = 200 k/hr
2) A coat is discounted 10%, then another 10% then another 10%. If it had been originally discounted to its final price, what percent would the discount have been?
approach one:
100 - 10 = 90
- 9 = 81
- 8.1 = 72.9 so the discount is 27.1%
approach two:
100 * 9/10 * 9/10 * 9/10 = 100 * 729 / 1000 = 729/10 = 72.9, or a 27.1% discount.
3) You have 20 kilograms of apples. You are told they are 98% water.
After leaving them outdoors for 2 days, you are told they are now 96% water.
How much do the dehydrated apples weigh?
We know w + f = 20k initially (f = apple fiber; the part that is not water, and w is the water weight)
f / 20 = 2/100 => f=2/5 k
after 2 days we have
2/5 / W = 4/ 100 (where W is the total weight after 2 days in kilograms)
W = 100/4 * 2/5 = 10k
4) After a nuclear explosion there is a deadly amount of radiation in the air. The amount of radiation decreases by 1/2 every 2 days. If initially the radiation is measured at 2000 roentgens, how long must a survivor wait before they can safely leave the bomb shelter if a safe level of radiation is considered to be anything less than 9 roentgens?
Keep dividing by 2 until you get a number less than 9. Then it is safe
at time 0 we have 2000
after 2 days we have 1000
after 4 days we have 500
after 6 days we have 250
after 8 days we have 125
after 10 days we have 62.5
after 12 days we have 31.25
after 14 days we have 15.625
after 16 days we have 7.8125 r (less than 9)
Safe after 16 days of waiting.
5) Joe makes moonshine from barley. Joe keeps 10% of the moonshine produced as payment for his work. If Skip gets 1 liter of moonshine for the barley he brought Joe, how much moonshine did his barley make?
Let x be the amount of barley that was made. Joe keeps one tenth of that, so
x - x/10 = 1
so
9x/10 = 1
x = 10/9 liters (or 1 1/9 liters)
Suppose the distance is 100k
It takes 1 hour to drive 50k (the first half).
If he wants to average 80 kph, then
100k / (1 + x) hr = 80 k/hr
100 = 80 + 80x
x = 1/4 hr
This means he has only 1/4 hour to drive 50 k so
50 k / (1/4 hr) = 200 k/hr
2) A coat is discounted 10%, then another 10% then another 10%. If it had been originally discounted to its final price, what percent would the discount have been?
approach one:
100 - 10 = 90
- 9 = 81
- 8.1 = 72.9 so the discount is 27.1%
approach two:
100 * 9/10 * 9/10 * 9/10 = 100 * 729 / 1000 = 729/10 = 72.9, or a 27.1% discount.
3) You have 20 kilograms of apples. You are told they are 98% water.
After leaving them outdoors for 2 days, you are told they are now 96% water.
How much do the dehydrated apples weigh?
We know w + f = 20k initially (f = apple fiber; the part that is not water, and w is the water weight)
f / 20 = 2/100 => f=2/5 k
after 2 days we have
2/5 / W = 4/ 100 (where W is the total weight after 2 days in kilograms)
W = 100/4 * 2/5 = 10k
4) After a nuclear explosion there is a deadly amount of radiation in the air. The amount of radiation decreases by 1/2 every 2 days. If initially the radiation is measured at 2000 roentgens, how long must a survivor wait before they can safely leave the bomb shelter if a safe level of radiation is considered to be anything less than 9 roentgens?
Keep dividing by 2 until you get a number less than 9. Then it is safe
at time 0 we have 2000
after 2 days we have 1000
after 4 days we have 500
after 6 days we have 250
after 8 days we have 125
after 10 days we have 62.5
after 12 days we have 31.25
after 14 days we have 15.625
after 16 days we have 7.8125 r (less than 9)
Safe after 16 days of waiting.
5) Joe makes moonshine from barley. Joe keeps 10% of the moonshine produced as payment for his work. If Skip gets 1 liter of moonshine for the barley he brought Joe, how much moonshine did his barley make?
Let x be the amount of barley that was made. Joe keeps one tenth of that, so
x - x/10 = 1
so
9x/10 = 1
x = 10/9 liters (or 1 1/9 liters)