UBC Grade 8–10 Workshop Problems, 2011–2012
1. A kitchen has two electronic timers. One of them, after it is turned on, beeps every five minutes until
it is shut off. The other is similar, but beeps every nine minutes. Use these these timers to time a threeminute egg.
2. Henry has 6 candles that he must arrange in the shape of a pyramid for his art class. Of the 6 candles,
2 are red, 2 are blue, 1 is yellow and 1 is green. How many different ways can he rearrange the
candles in the shape of a pyramid? Only switching two candles of the same colour is not a new
arrangement, and all 6 candles must be used in the pyramid. (Julie’s problem)
3. Two families go camping together. Family A consists of 3 adults and 2 children. Family B has 2
adults and 1 child. They agree to split food costs in proportion to family size, counting each child as
half an adult. If total food costs were $130, how much should family A pay?
4. WirelessMath charges a flat rat charge for the first three minutes and a charge for each additional
minute in long distance calls from Vancouver to Toronto. If a twelve minute call costs $12.24 and a
five minute call costs $5.52, what is the flat rate charge and the cost of each additional minute?
(Zoey’s problem)
5.
The area of one side of the box is 120 , the area of another side is 70 and the area of the top of the
box is 84. What is the volume of the box? (Tracey’s problem)
6. Bobby’s grandmother baked a large cake. She left it on the table to cool. While it was cooling,
Bobby’s older brother Tim came by and took half the cake on the table. Then, Bobby’s sister Jessica
also came by and took ½ of the cake that was on the table. Bobby’s little brother Sasuke, who happens
to be a ninja, took ¼ of the original amount of the cake sometime after it was put on the table. In the
end, when Bobby came to get some cake, there was only 1/8 of the cake left. When did Sasuke take
his share?
7. A man has two daughters, Ada and Zoe. One of them is married and the other is not. Ada always tells
the truth, while Zoe always lies. You are allowed to ask just one question of one of the daughters to
find out which daughter is the married one. The catch is, your question can not contain more than
three words. Ada and Zoe are identical twins so you do not know which daughter is which. What
question do you ask and how do you identify who is married?
8. Miles is thinking of a three digit number. He tells us that none of the digits are zero, that the number
can be divided by 6(i.e. there is no remainder) and that taking the first digit and subtracting the second
is equal to the second and third digits being subtracted. Find the smallest such number, when the first
number > second number>third number
9. Jack and Jill have been accumulating debt since entering university together. Jill got more financial
assistance and after 5 years of University studies, took out 2 times as much money in total in student
loans than Jack. In total, Jill used 3/5 of her student loan. Jack used 1/3 of his loan every year for 5
years. Now Jack has $13,250 of unused student loans. How much does Jill have in unused student
loans?
10. A muffin bakery sells four different flavour muffins: carrot, banana, blueberry and pumpkin cream.
At the end of the day, there were 150 muffins sold. The owner found that the number of banana
muffins sold is 75% of the carrots, the blueberry muffins sold are 2/3 of the bananas and the pumpkin
cream muffins sold are three times of the amount of the blueberry muffins. Find the exact amount sold
of each flavour of muffin.
11. Three watermelons are weighed two at a time in all possible ways. The weights of the watermelons are
13kg, 14 kg and 17kg. How much does the median watermelon weigh?
12. Lando bought a spacecraft and sold it to Luke for half what he paid. Luke didn’t like it after a month and
sold it to Han for 5/6 what he paid. Han managed to make 1000 dollars profit selling it to Jabba. Jabba is a
terrible salesman and sells the spacecraft to Chewbacca for 25% of what he paid. If Chewbacca pays 3000
dollars for it, how much did Lando spend on the spacecraft originally?
13. Of my 18 cousins, 2/3 of them are male and 1/3 of them are female. The average age of my female cousins
is 20 years old. I am 21 years old. The average age of all my cousins and myself is 15 years old. What is the
average age of my male cousins?
14. A 600 pound pumpkin was entered in a contest. When it arrived, it was 99% water. The pumpkin sat for
days in the hot sun, lost some weight (water only) and is now 98.5% water. How much does it weigh?
15. James, Mike and Susan are three friends who love chocolate. At the moment, James has 36 chocolates,
Mike has 21 chocolates, and Susan has 9 chocolates. Every day James eats 2 chocolates. Every other day Mike
eats 3 chocolates and steals 1 from James and Susan. Susan eats a chocolate every third day. But James likes
Susan, so on every day that he eats chocolate, he gives her one. After 8 days, how many chocolates do they
each have?
16. Bernadette loves math but she hates this problem. Can you help her solve it? The sum of four numbers is
79. If you subtract 5 from the second number, divide the last number by 4, add 2 to the first number, and
multiply the third number by 3, then all the results will be equal. What is the difference between the smallest
and the largest of the original four numbers?
17. A lollipop machine that randomly dispenses one lollipop at a time contains 15 cola, 3 mango, 1 secret and
7 apple flavoured lollipops. What is the least number of lollipops that Zoey must buy to guarantee that she
receives 3 lollipops of the same flavour?
18. There is a river and one boat. 3 people and 3 wolves are stuck on one side of the river. You must get
everyone and all three wolves across before nightfall to remain safe. You are one of the 3 people. The boat fits
you and 2 other individuals. If on either side of the river (or on the boat) the wolves outnumber humans, they
will eat the human. How do you bring everyone over? (The wolves cannot get on the boat themselves).
19. Five distinct positive integers add up to 100. The difference between any two of the integers is divisible by
a fixed prime number p. The difference between the smallest and the largest of these numbers is 4p. How many
different sets of integers are there?
20. Kyle walks at 3km/hr and bikes at 8km/hr. He saves 4 ½ minutes by biking instead of walking from his
house to his school. What is the distance, in kilometres, from his house to his school?
21. Jon is buying a toy for $30. While in the checkout line, he sees a sign that reads “buy any 3 action toys and
get 75% off the cheapest one.” Jon then picks up two more toys, one for $70. And the other he can’t remember.
But he knows he has to pay $137.50. How much was the other toy?
22. Frank goes to the pet store to buy two guinea pigs, one male and one female. In one box there are two
white guinea pigs and three brown guinea pigs, all of which are female. In another box, there are four male
guinea pigs, three of which are black and one of which is, surprisingly, blue. What is the probability that the
boy buys the blue guinea pig and a white guinea pig, assuming that he chooses randomly?
Solutions:
6. There are 3 possibilities: Sasuke takes his portion of the cake first, second or third. If first, Bobby sees 1 – ¼
= ¾ ; ¾ x ½ = 3/8 ; 3/8 x ½ = 3/16. If second, Bobby sees 1x ½ = ½; ½ - ¼ = ¼; ¼ x ½ = 1/8. If third, Bobby
sees 1 x ½ = ½ ; ½ x ½ = ¼ ; ¼ - ¼ = 0. So Sasuke took the cake second.
7. The question is “Are you married?” In both cases, regardless of which daughter you ask, a “yes” answer
means that Ada is married and Zoe is not and a “no” answer means that Zoe is married and Ada is not.
8. 432
9. Let x = Jack’s yearly student loan, then Jill’s yearly student loan is 2x and 1/3 x = how much Jack uses of
his student loan. 5x – 5(1/3 x) = 13250 Jack’s student loan left. Thus 10/3 x = 13250 and x = 3975. For Jill
5(2x) - (3/5)(5(2x)) = student loan left or 4x = student loan left = 15900 since x = 3975.
10. Let n = # of carrot muffins sold. Then ¾ n banana muffins are sold, (¾ n)(2/3) = ½ n blueberry muffins are
sole and (3/4 n)(2/3)(3) = 3/2 n pumpkin muffins are sold. So n + ¾ n + ½ n + 3/2 n = 150. Thus n = 40
number of carrot muffins. Hence there are 30 banana muffins, 20 blueberry muffins and 60 pumpkin muffins.
11. 13 = small + medium < 14 = small + large < 17 = medium + large. Thus small = 5, medium = 8, large = 9.
12. Left to reader.
13. Average age of male cousins is 12 years old.
14. It weighs 400 pounds.
15. James has 8 chocolates, Mike has 17 and Susan has 11.
16. The 4 numbers are a = 10, b = 17, c = 4 and d = 48 giving 48 – 4 = 44.
17. The least number she must buy to be sure is 8, since if she bought 7 she could have 2 cola, 2 mango, 1
secret, and 2 apple flavour.
18. Bring one wolf to other side, come back, bring your two friends back, leave one with the wolf and bring
the other one back to avoid being attacked by the remaining wolves. Bring this friend and a new wolf over.
You drop both of them off. Come back and bring the last wolf over. 30.
19. There are 4 different sets of integers. They are x – 2p, x – p, x, x + p, x + 2p for p = 2, 3, 5 and 7.
20. If d is the distance from home to school in km, it takes him d/3 hours to walk to school and d/8 hours to
bike to school. d/3 – d/8 = 3/40 since 4 ½ minutes is 3/40 hours. Thus d = 9/25km.
21. $137.50 - $70. - $30. = $37.50. This means that the other toy is more expensive than $30.00 because
$37.50(4/3) > $30.00. So $30 times ¼ is $7.50. Thus $137.50 - $7.50 - $70.00 = $60.00. Thus the other toy
was $60.00.
22. 1/10 = 2/5 times 1/4.