← Pioneer Math League home · Bonus problems
Bonus problem solutions (2009–2010)
Here are the bonus problems, with solutions, that I have given so far this year in math league. You can submit answers up to the time when I post the solution on this page. You can discuss the problem with whoever you want, but answers must be submitted in your own handwriting. It is not enough to submit a number as your answer. Some explanation of how you solved it must be provided as well.
Speeding to Aunties House (10/9/09)
Joe 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?
Since it does not matter, assume it is 100k to the aunt’s house.
It took him 50 k / 50 k/hr = 1 hour to go halfway.
If his average speed for the whole distance is to be 80 k/hr then it will take him 100 k / 80 k/hr = 1 1/4 hours total.
That leaves him only 15 minutes to drive the second 50 k.
In order to go 50 k in 15 min, he needs to speed at 50 k / 0.25 hr = 200 k/hr.
Great Discount on Coat (10/16/09)
A coat is discounted 20%, then another 20% then another 20%. If it had been originally discounted to its final price, what percent would the discount have been?
Solution A
To make things simple assume the coat costs $100.
After taking 20% off we have $80, after another 20% we have $64, after another 20%, $51.2.
This makes the discount (100 − 51.2)/100 = 0.488 or 48.8%.
Solution B
What remains will be .8 × .8 × .8 = 0.512. So the amount discounted is 48.8 percent.
(4 correct solutions were submitted)
Dried Apples (10/23/2009)
You have 10 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?
Solution:
Initially you have this proportion: f / 10 = 2 / 100, where f is the weight of everything in the apple except the water (the fiber).
After 2 days some of the water has evaporated, and now the proportion is f / x = 4 / 100, where x is the total weight of the apples after 2 days. f has not changed.
Since the numerator on the right has doubled and we know that f has not changed, the only way for this proportion to stay balanced is if x is half of 10.
Therefore x = 5 kg (the weight after 2 days).
Nuclear Explosion (10/30/2009)
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?
Solution:
In 2 days the radiation will be 1000 roentgens.
In 4 days the radiation will be 500 roentgens.
In 6 days the radiation will be 250 roentgens.
In 8 days the radiation will be 125 roentgens.
In 10 days the radiation will be 62.5 roentgens.
In 12 days the radiation will be 31.25 roentgens.
In 14 days the radiation will be 15.625 roentgens.
In 16 days the radiation will be 7.8125 roentgens.
It will be safe to go out after 16 days. On day 16 there will still be more than 9 roentgens.
Making Moonshine (11/13/2009)
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?
Jungle Animals in a Cage (12/11/2009)
There is a cage at the zoo that holds different jungle animals. In the cage are 11 heads and 20 feet. There are twice as many jungle creatures with 4 feet as those with 2 feet. How many of each type are in the cage?
Let T = the number of animals with 2 feet, F = the number of animals with 4 feet and Z = the number of animals with no feet.
Tabular solution: make a table trying various guesses
T F Z Heads Feet ------------------------------------------------ 1 2 7 11 10 (too few feet) 3 6 2 11 30 (too many feet) 2 4 5 11 20 (just right)
Algebraic solution:
T + F + Z = 11 and 2T + 4F = 20 and 2T = F (because there are twice as many 4-footed creatures as 2-footed).
We have 3 equations and 3 unknowns.
2T + 4(2T) = 20, so 10T = 20 which implies T = 2, so F = 4.
2 + 4 + Z = 11 so Z = 5.
Fishing Pole (1/8/2010)
A child who bought a 5 foot fishing pole wants to get on a bus. The bus driver says “sorry you cannot bring on anything longer than 4 feet”. The boy goes back in the store and gets something else that allows him to bring the fishing rod on the bus. What did he get? Note: he does not bend or dismantle the 5 foot rod.
Men Eating Cakes (1/15/2010)
If a man and a half can eat a cake and a half in a minute and a half, how many men would it take to eat 60 cakes in 30 minutes?
Make a table:
Men Cakes Time 3/2 3/2 3/2 minutes 3 3 3/2 3 6 3 3 60 30
So 3 men could eat 60 cakes in 30 minutes.
Burning Fuses
A man has two fuses that burn for exactly 30 seconds each, but not necessarily at continuous rates of speed. How can he burn the fuses to measure exactly 45 seconds?
More Boys or Girls?
In a certain country a law was passed that a married couple can have as many girls as they want, but have to stop having children after they have one boy. After 10 years will there be more boys than girls or more girls than boys? Explain.
Solution 1: The answer is that the proportion of boys to girls will not change because each pregnancy is an independent event. It seems counter-intuitive at first, but if you just consider the total number of pregnancies that occur in a given interval, and the fact that each is 50% likely to be a boy, then I think you can see that the proportion should not change regardless of government legislation. We are not considering things like early termination if the parents realize that they have a gender they do not want, etc.
Solution 2 (from MindTrap): There will be no effect on the ratio of girls to boys. Assuming that in every birth there is a 50-50 chance of having a boy or girl, if 100 women have children, the odds are that there will be 50 boys and 50 girls. Therefore, according to the law, only the half who had girls could continue to bear children. Out of that half, there will be an equal number of boys and girls produced. In the third round, the same odds will be repeated. At no time does the expected boy/girl ratio deviate from 50/50.
Solution 3: Here is another explanation.
Imagine flipping a coin represents whether it is a boy or girl (H=boy; T=girl).
Flip coins for each family in the country. So for family A, you might have TTH; then for family B: H; for family C: H; for family D: TH; for family E: TTTH; for family F: H.
In other words, the problem has a 1-1 mapping to a string of N random coin flips: TTHHHTHTTTHH — which we know is going to turn out to be about 50% of each in the long run.
Sharing Birthdays
If there are 10 people in a room, what is the probability that two of them share the same birthday?
Painting a Fence
Jill can paint a fence in 2 hours. To paint the same fence, Bob would take 3 hours. How long will it take them if they paint at the same time?