Piggy Bank Day (Nov. 2) Meeting

Piggy Bank Day (Nov. 2) Meeting
(Multiple Topics)
Topic
There are a variety of math topics covered in the problems used for this meeting, including
counting, proportional reasoning, systems of equations, logic and making organized lists. We
have attempted to put the problems in order of difficulty.
Materials Needed
♦ Copies of the Piggy Bank Day problem set (Problems and answers can be viewed here, but
a more student-friendly version in larger font is available for download from
www.mathcounts.org on the MCP Members Only page of the Club Program section.)
♦ Calculators
♦ Coins (real or paper) – optional
♦ Candy or trinkets to sell or auction off – optional
Meeting Plan
Most of the problems used during this meeting do not require any advanced math, but there
are problems that focus on counting, proportional reasoning, systems of equations, logic and
making organized lists. The problems are in order of difficulty, so it might be a good lesson to
have students work on individually for a small portion of the meeting and then come together to
discuss the final questions.
One idea for the lesson is to use paper coins (a sheet you can copy is in the MCP Members
Only page of the Club Program section of www.mathcounts.org) and give a value to each of the
problems. Below, we’ve assigned some values you can use. Students getting a problem correct
will earn that amount of money in paper coins. At the end of the meeting, you can then “sell” or
auction off candy or some other trinkets students might enjoy. (If you have enough real coins to
give out, that’s even better. It’s hoped you’ll get them all back through the sale or auction at the
end of the meeting.)
(1 cent) 1. Using pennies, nickels, dimes and quarters, what is the least number of coins needed to make 68 cents in
change? 2003–2004 School Handbook, Warm-Up 14-1
(2 cents) 2. A piggy bank contains only nickels, dimes and quarters. There is at least one of each
type of coin in the bank. If the total value of the coins in the piggy bank equals 60 cents, how many
quarters are in the bank? 2006–2007 School Handbook, Warm-Up 2-9
(3 cents) 3. Sasha has $3.20 in U.S. coins. She has the same number of quarters and nickels.
What is the greatest number of quarters she could have? 2006 School Competition, Sprint Round #3
(3 cents) 4. Krista put 1 cent into her new bank on a Sunday morning. On Monday she put 2 cents into her bank. On
Tuesday she put 4 cents into her bank, and she continued to double the amount of money she put into her bank each
day for two weeks. On what day of the week did the total amount of money in her bank first exceed $2? 2006 Chapter
Competition, Sprint Round #5
(5 cents) 5. Anjali has 230 quarters, 300 dimes and 165 nickels. Niki has 210 quarters, 316 dimes
and 173 nickels. How many more cents than Niki does Anjali have? 2002 School Competition, Sprint
Round #4
2008–2009 MATHCOUNTS Club Resource Guide
Club Resource Guide.pdf 29
29
8/18/08 11:24:15 AM
(5 cents) 6. Karla has only dimes and quarters in her piggy bank. The value of her dimes is exactly five times the
value of her quarters. What is the least number of coins she could have in her piggy bank if there is at least one
quarter? 2007 School Competition, Countdown Round #53
(10 cents) 7. In how many different ways can 30 cents be made from any combination of quarters, dimes, nickels and/
or pennies? 2000–2001 School Handbook, Warm-Up 1-4
(10 cents) 8. Susan has exactly $1.50 in change. She has 30 total coins (pennies, nickels, dimes and/or quarters).
Half of the coins are nickels. There are two-thirds as many pennies as there are nickels. How many dimes does
Susan have? 2004–2005 School Handbook, Workout 2-6
(10 cents) 9. A coin machine keeps 8.9 cents per $1 of coins inserted. This ratio also is maintained when only a
fraction of a dollar is inserted. If Ashley inserts $50.50 worth of change into the machine, how much money will the
machine give her back? 2004–2005 School Handbook, Workout 3-7
(10 cents) 10. My piggy bank has only nickels, dimes and dollar bills. The ratio of nickels to dimes is 2:3, and the ratio
of dimes to dollar bills is 10:1. What is the ratio of coins to dollar bills? Express your answer in the form a:b, where a
and b are positive integers with no common factors greater than 1. 2007–2008 School Handbook, Warm-Up 11-10
(10 cents) 11. There are equal numbers of pennies, nickels, dimes and quarters in a piggy bank.
Two coins are pulled out, one at a time, and each coin is replaced before the next is drawn. What is
the probability that the sum of the values of the two coins will be less than 15 cents? Express your
answer as a common fraction. 2004–2005 School Handbook, Warm-Up 14-6 (modified)
(15 cents) 12. John’s piggy bank had $1.20 when he went to bed on Monday. On Tuesday morning he put three coins
in the bank. He put in three more coins in the afternoon and then three more in the evening. That night there was a
total of $2.20 in his bank. If no coin is worth more than 25 cents, what is the greatest amount, in cents, John could
have put in the bank on Tuesday evening? 2005–2006 School Handbook, Workout 4-2
(15 cents) 13. Jennifer, Mike and Carol each have a bunch of quarters. Jennifer and Mike have 26 quarters together.
Jennifer and Carol have 20 quarters together. Mike and Carol have 22 quarters together. How many cents does Mike
have? 2006–2007 School Handbook, Warm-Up 10-3
Answers: 7 coins; 1 quarter; 10 quarters; Sunday; 300 cents; 27 coins; 18 ways; 4 dimes; $46.01; 5:3; 3/8; 60
cents; 350 cents
A coin sheet similar to this one is available online with the Club Resources.
Possible Next Steps
Now ask the students to come up with some of their own coin-based problems. You can compile
these and use them at future meetings or in your classroom teaching. We’d also enjoy seeing
what you and your students write.
If you do this activity before the actual week of Piggy Bank Day, we may be able to use
some of your ideas for the Problem of the Week on mathcounts.org. Send the real gems
to info@mathcounts.org with the subject line “MATHCOUNTS Club Program.”
30
Club Resource Guide.pdf 30
2008–2009 MATHCOUNTS Club Resource Guide
8/18/08 11:24:15 AM
$
Piggy Bank Day Meeting
Problem Set
$
The difficulty level of each problem is indicated by the value that is written before it.
(1 cent) 1. __________ Using pennies, nickels, dimes and quarters, what is the least number of
coins needed to make 68 cents in change? 2003–2004 School Handbook, Warm-Up 14-1
(2 cents) 2. __________ A piggy bank contains only nickels, dimes and quarters.
There is at least one of each type of coin in the bank. If the total value of the
coins in the piggy bank equals 60 cents, how many quarters are in the bank?
2006–2007 School Handbook, Warm-Up 2-9
(3 cents) 3. __________ Sasha has $3.20 in U.S. coins. She has the same number of quarters
and nickels. What is the greatest number of quarters she could have? 2006 School Competition,
Sprint Round #3
(3 cents) 4. __________ Krista put 1 cent into her new bank on a Sunday morning. On Monday
she put 2 cents into her bank. On Tuesday she put 4 cents into her bank, and she continued to
double the amount of money she put into her bank each day for two weeks. On what day of the
week did the total amount of money in her bank first exceed $2? 2006 Chapter Competition, Sprint
Round #5
(5 cents) 5. __________ Anjali has 230 quarters, 300 dimes and 165 nickels. Niki has
210 quarters, 316 dimes and 173 nickels. How many more cents than Niki does Anjali
have? 2002 School Competition, Sprint Round #4
(5 cents) 6. __________ Karla has only dimes and quarters in her piggy bank. The value of
her dimes is exactly five times the value of her quarters. What is the least number of coins she
could have in her piggy bank if there is at least one quarter? 2007 School Competition, Countdown
Round #53
(10 cents) 7. __________ In how many different ways can 30 cents be made from any
combination of quarters, dimes, nickels and/or pennies? 2000–2001 School Handbook, Warm-Up 1-4
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
(10 cents) 8. __________ Susan has exactly $1.50 in change. She has 30 total coins (pennies,
nickels, dimes and/or quarters). Half of the coins are nickels. There are two-thirds as many
pennies as there are nickels. How many dimes does Susan have? 2004–2005 School Handbook,
Workout 2-6
(10 cents) 9. __________ A coin machine keeps 8.9 cents per $1 of coins inserted. This ratio
also is maintained when only a fraction of a dollar is inserted. If Ashley inserts $50.50 worth of
change into the machine, how much money will the machine give her back? 2004–2005 School
Handbook, Workout 3-7
(10 cents) 10. __________ My piggy bank has only nickels, dimes and dollar bills. The ratio of
nickels to dimes is 2:3, and the ratio of dimes to dollar bills is 10:1. What is the ratio of coins to
dollar bills? Express your answer in the form a:b, where a and b are positive integers with no
common factors greater than 1. 2007–2008 School Handbook, Warm-Up 11-10
(10 cents) 11. __________ There are equal numbers of pennies, nickels,
dimes and quarters in a piggy bank. Two coins are pulled out, one at a time,
and each coin is replaced before the next is drawn. What is the probability
that the sum of the values of the two coins will be less than 15 cents?
Express your answer as a common fraction. 2004–2005 School Handbook, Warm-Up 14-6 (modified)
(15 cents) 12. __________ John’s piggy bank had $1.20 when he went to bed on Monday. On
Tuesday morning he put three coins in the bank. He put in three more coins in the afternoon
and then three more in the evening. That night there was a total of $2.20 in his bank. If no coin
is worth more than 25 cents, what is the greatest amount, in cents, John could have put in the
bank on Tuesday evening? 2005–2006 School Handbook, Workout 4-2
(15 cents) 13. __________ Jennifer, Mike and Carol each have a bunch of quarters. Jennifer
and Mike have 26 quarters together. Jennifer and Carol have 20 quarters together. Mike and
Carol have 22 quarters together. How many cents does Mike have? 2006–2007 School Handbook,
Warm-Up 10-3
**Answers to these problems are on page 30 of the 2008-2009 Club Resource Guide.**
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
Piggy Bank Day Solutions (2008-2009 MCP Club Resource Guide)
Problem 1. To use the fewest number of coins, we should try to use as many of the quarters,
then dimes, then nickels as possible. 68 cents in change can be made with 2 quarters, 1 dime,
1 nickel and 3 pennies, which is 7 coins.
Problem 2. If there is at least one quarter, one dime and one nickel, those coins come to
40 cents. Therefore, there are only another 20 cents to account for. Those 20 cents can’t be
made with any quarters, so there is only 1 quarter in the piggy bank.
Problem 3. Every quarter must be matched with a nickel, so they come in bundles of 30 cents.
To have the most quarters possible, we must see how many times 30 cents goes into $3.20.
Dividing, we see that the answer is 10 times. The other 20 cents must be dimes. Therefore, the
greatest possible number of quarters is 10.
Problem 4. Krista put 1 cent into her new bank on Sunday. On Monday she put 2 cents in and
on Tuesday she put 4 cents in. Each day she doubles the amount that she put in the day before.
We must find the day of the week that the total amount of money in her bank first exceeded $2.
Let’s look at the first couple of days…
DAY AMOUNT ADDED
TOTAL
1
1
1
2
2
3
3
4
7
4
8
15
Day 1 has a total of 21 – 1 = 2 – 1 = 1 cent.
Day 2 has a total of 22 – 1 = 4 – 1 = 3 cents.
Day 3 has a total of 23 – 1 = 8 – 1 = 7 cents.
Day 4 has a total of 24 – 1 = 16 – 1 = 15 cents.
There is a pattern here. (It’s a good thing to recognize powers of 2 and one less than each
power of 2.) Now we only need to find the first power of 2 that is greater than 200 to find the
number of days. Starting with the power 1, we have 2, 4, 8, 16, 32, 64, 128, 256. So
28 – 1 = 256 – 1 = 255 and it takes 8 days, starting with Sunday, for us to get over 200 cents.
Since Sunday is the first day, it must also be the eighth day.
Problem 5. Anjali has 230(25) + 300(10) + 165(5) = 5750 + 3000 + 825 = 9575 cents = $95.75.
Niki has 210(25) + 316(10) + 173(5) = 5250 + 3160 + 865 = 9275 cents = $92.75. This is
$95.75 – $92.75 = $3.00 or 300 cents.
Problem 6. If Karla has 1 quarter, its value is 25 cents. The value of her dimes is five times the
value of her quarters. This would make her dimes’ value 25 × 5 = 125 cents, which is
impossible. Therefore, Karla does not have just 1 quarter. If she has 2 quarters, their value is
50 cents. The value of her dimes would be 5 × 50 = 250 cents, which is 25 dimes. This is a total
of 2 + 25 = 27 coins.
Problem 7. Making an organized list may be the best approach for this problem. There are
2 ways to make 30 cents with one quarter (no dimes could be used). With no quarters, we can
look at using as many dimes as possible. There is 1 way with three dimes; there are 3 ways with
two dimes; there are 5 ways with one dime; and there are 7 ways with no dimes (and still no
quarters). That’s a total of 18 ways.
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Solution Set
Problem 8. If half of Susan’s 30 coins are nickels, then she has 15 nickels, worth 15 × 5 =
75 cents. Two-thirds of 15 is 10, so Susan has 10 pennies, worth 10 cents. That’s 25 coins
worth 85 cents so far. The other 5 coins must amount to the other 65 cents. One quarter and
4 dimes is the only combination of quarters and dimes that works. Susan has 4 dimes.
Problem 9. The coin machine is going to keep $50.50 × 0.089 = $4.49. This is the fee for
counting the coins, and many people would rather give up that amount of money than take the
time to roll $50.50 worth of coins. Ashley will get $50.50 – $4.49 = $46.01 back.
Problem 10. We need the least common multiple for the dimes in the two comparisons. We can
rewrite the nickels to dimes comparison as 20 to 30 and the dimes to dollar bills comparison as
30 to 3. Then there are 20 + 30 = 50 coins for every 3 dollars, so the coins to dollar bills ratio is
50:3. (NOTE: The answer is printed incorrectly as 5:3 in the hard copy of the Club Resource
Guide.)
Problem 11. In order for two coins to have a value of less than 15 cents, the two coins must be
selected in one of the following ways: PP, NN, PN, NP, DP or PD with P = penny, N = nickel
and D = dime. Since there are an equal number of each type of coin, and the first selection is
replaced, the probability of selecting PN is (1/4)(1/4) = 1/16. This is the same for each of the six
options, so the answer is 6(1/16) = 3/8. Also note that there are a total of 4 × 4 = 16 two-coin
permutations, 6 of which would be successful. This also gets us to 3/8.
Problem 12. John added a total of $2.20 – $1.20 = $1.00 to his piggy bank over the course of
the day. We would like to see if he could have added 75 cents to the bank in the evening since
three quarters is the greatest amount possible in three coins when no coin is worth more than
25 cents. If he did add 75 cents in the evening then the six coins he added in the morning and
afternoon must add up to 25 cents. This is not possible. The next greatest amount possible in
three coins is 60 cents, two quarters and one dime. In this case, John would have to get 40
cents with the other six coins. This is possible with two dimes and four nickels, so 60 cents is
our answer.
Problem 13. If we add 26 + 20 + 22 = 68, we have counted everyone’s quarters twice.
Altogether, Jennifer, Mike and Carol must have 68 ÷ 2 = 34 quarters. Since Jennifer and Carol
have 20 quarters together, Mike must have 34 – 20 = 14 quarters. Fourteen quarters is worth
350 cents.
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Solution Set