Valentine’s Day (Feb. 14) Meeting
(Multiple Topics)
Topic
There are a variety of math topics covered in the problems used for this meeting.
Materials Needed
♦ Copies of the Valentine’s 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
♦ Valentine’s Day treat for your students – optional
If creating Valentine’s Day thank you cards, your students may wish to use the following:
♦ Graph paper
♦ Paper
♦ Red markers
♦ Rulers
♦ Scissors
Meeting Plan
This meeting idea is for use around the time of Valentine’s Day. Students can work together in
groups on the Valentine’s Day problem set provided. Please feel encouraged to add problems,
delete problems or change any of the problems to accommodate your students’ abilities.
Problem #8 was used in last year’s Club Resource Guide, but the other problems are new. We
kept #8 and the Possible Next Step due to the positive feedback we received about the activity
last year. If you would like more Valentine’s Day-related problems, you can find last year’s
complete set at www.mathcounts.org on the MCP Members Only page of the Club Program
section.
1. According to “A History of Valentine’s Day Cards in America” by T.M. Wilson, in 1847 Esther Howland was the first
person to mass-produce Valentine’s Day cards. She made them out of lace, paint and expensive paper, and each one
was individually written by a skilled calligrapher. The average card sold for $7.50 while others cost as much as $50.
If 10 cents in 1847 would be equivalent to $1.85 today, how much would the average card and most expensive card
have cost today? 2/19/2001 Problem of the Week
2. Kelly decided to celebrate Valentine’s Day for an entire month. She started giving her Valentine 1 candy heart on
Jan. 14, 2 candy hearts on Jan. 15 , 4 candy hearts on Jan. 16 , and continued doubling the amount of hearts each
day through Feb. 14. If 200 candy hearts come in a bag, how many bags of candy hearts would Kelly need just for
Feb. 14? 2/19/2001 Problem of the Week
3. For Valentine’s Day Kevin wanted to send Mary Beth 11 balloons since that was her favorite number. In the store,
plain-colored balloons cost $0.75 each, multi-colored balloons cost $1.30 each, and extra-large balloons cost $1.50
each. How many different combinations of 11 balloons can Kevin buy if he has only $12.00? 2/19/2001 Problem of
the Week
4. Kia decided that she wants to give each of her friends a small pouch of candy hearts. She’ll use fabric and ribbon
to make and tie the pouch, and each one will contain four candy hearts. The hearts come in six colors: white, orange,
pink, green, purple and yellow. If each pouch contains exactly four hearts, such that no two hearts are the same color,
how many possible combinations of hearts could be in a pouch? 2/14/2005 Problem of the Week
2008–2009 MATHCOUNTS Club Resource Guide
Club Resource Guide.pdf 43
43
8/18/08 11:24:16 AM
5. Mrs. Stuver’s art class has used geometric shapes to design a valentine in the shape of a heart. They have placed
two adjacent semicircles along one side of an equilateral triangle so that the diameters of the semicircles and one
side of the triangle are concurrent. The diameter of each semicircle is exactly one-half the length of the side of the
triangle. The length of each side of the triangle is 4 inches. What is the area of the valentine in square inches?
Express your answer as a decimal to the nearest tenth. 2/12/2007 Problem of the Week
6. Mrs. Stuver’s class then decorated the perimeter of each heart-shaped valentine with lace. What is the length, in
inches, of the perimeter of the valentine? Express your answer as a decimal to the nearest tenth. 2/12/2007 Problem
of the Week
7. Two pink valentines and two green valentines are delivered at random to two girls and two boys so that each girl
and each boy receives exactly one valentine. What is the probability that each girl receives a pink valentine and each
boy receives a green valentine? Express your answer as a common fraction. 2/12/2007 Problem of the Week
8. On some graph paper, graph the following segments:
y = x for 0 ≤ x ≤ 2
y = 2x – 2 for 2 ≤ x ≤ 3
x = 3 for 4 ≤ y ≤ 6
y = –x + 9 for 2 ≤ x ≤ 3
y = 7 for 1 ≤ x ≤ 2
y = x + 6 for 0 ≤ x ≤ 1
Now reflect each segment over the y-axis. What popular shape have you drawn?
2/09/2004 Problem of the Week
Answers: $138.75 and $925; 10,737,419 bags (since rounding to 10,373,418 would not be enough);
24 combinations; 15 combinations; 10.1 square inches; 14.3 inches; 1/6; heart
**Complete solutions to the Problems of the Week are available in the Problem of the Week Archive section
of www.mathcounts.org.**
Once students finish working on these problems, they should be encouraged to present their
solutions to the group. Additionally, students can be asked to come up with some of their own
Valentine’s Day-related problems for either your future use with clubs or that can be sent to the
elementary school teachers in your district to be used by their students.
Possible Next Step
Your students also may like to create a math Valentine similar to the one
shown here (using #8 of the problem set) to thank special sponsors of
the math club or key supporters of the math club or the teacher who
lets you hold meetings in his room! A special thank you signed by
your club members will go a long way in keeping your supporters
excited about the program.
44
Club Resource Guide.pdf 44
2008–2009 MATHCOUNTS Club Resource Guide
8/18/08 11:24:16 AM
Valentine’s Day Meeting
Problem Set
,
1. _________________
According to “A History of Valentine’s Day Cards in America” by T.M.
Wilson, in 1847 Esther Howland was the first person to mass-produce Valentine’s Day cards.
She made them out of lace, paint and expensive paper, and each one was individually written
by a skilled calligrapher. The average card sold for $7.50 while others cost as much as $50. If
10 cents in 1847 would be equivalent to $1.85 today, how much would the average card and
most expensive card have cost today?
2. __________ Kelly decided to celebrate Valentine’s Day for an entire month. She started
giving her Valentine 1 candy heart on Jan. 14, 2 candy hearts on Jan. 15, 4 candy hearts on
Jan. 16, and continued doubling the amount of hearts each day through Feb. 14. If 200 candy
hearts come in a bag, how many bags of candy hearts would Kelly need just for Feb. 14?
3. __________ For Valentine’s Day Kevin wanted to send Mary Beth 11 balloons
since that was her favorite number. In the store, plain-colored balloons cost $0.75
each, multi-colored balloons cost $1.30 each, and extra-large balloons cost $1.50
each. How many different combinations of 11 balloons can Kevin buy if he has
only $12.00?
4. __________ Kia decided that she wants to give each of her friends a small pouch of candy
hearts. She’ll use fabric and ribbon to make and tie the pouch, and each one will contain four
candy hearts. The hearts come in six colors: white, orange, pink, green,
purple and yellow. If each pouch contains exactly four hearts, such that no
BE LEO
two hearts are the same color, how many possible combinations of hearts
V
MIN YA E
could be in a pouch?
5. __________ Mrs. Stuver’s art class has used geometric shapes to design a valentine in the
shape of a heart. They have placed two adjacent semicircles along one side of an equilateral
triangle so that the diameters of the semicircles and one side of the triangle are concurrent. The
diameter of each semicircle is exactly one-half the length of the side of the triangle. The length
of each side of the triangle is 4 inches. What is the area of the valentine, in square inches?
Express your answer as a decimal to the nearest tenth.
6. __________ Mrs. Stuver’s class then decorated the perimeter of each heart-shaped valentine
with lace. What is the length, in inches, of the perimeter of the valentine? Express your answer
as a decimal to the nearest tenth.
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
7. __________ Two pink valentines and two green valentines are delivered at random to two
girls and two boys so that each girl and each boy receives exactly one valentine. What is the
probability that each girl receives a pink valentine and each boy receives a green valentine?
Express your answer as a common fraction.
8. __________ On some graph paper or below, graph the following segments:
y = x for 0 ≤ x ≤ 2
y = 2x – 2 for 2 ≤ x ≤ 3
x = 3 for 4 ≤ y ≤ 6
y = –x + 9 for 2 ≤ x ≤ 3
y = 7 for 1 ≤ x ≤ 2
y = x + 6 for 0 ≤ x ≤ 1
Now reflect each segment over the y-axis. What popular shape have you drawn?
**Answers to these problems are on page 44 of the 2008-2009 Club Resource Guide.**
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
Valentine’s Day Meeting
Problem Set SOLUTIONS
,
1. _________________
According to “A History of Valentine’s Day Cards in America” by T.M.
Wilson, in 1847 Esther Howland was the first person to mass-produce Valentine’s Day cards.
She made them out of lace, paint and expensive paper, and each one was individually written
by a skilled calligrapher. The average card sold for $7.50 while others cost as much as $50. If
10 cents in 1847 would be equivalent to $1.85 today, how much would the average card and
most expensive card have cost today?
Since we are given a ratio of 10 cents to $1.85, we can set up 2 more ratios to find
what $7.50 and $50 would convert to. Just remember to convert 10 cents to $.10 before
beginning. An extended proportion would say that .10/1.85 = 7.50/x = 50/y. By crossmultiplying and solving for x and y, we would see people were spending what would be
equivalent to $138.75 and $925 for us today!
2. __________ Kelly decided to celebrate Valentine’s Day for an entire month. She started
giving her Valentine 1 candy heart on Jan. 14, 2 candy hearts on Jan. 15, 4 candy hearts on
Jan. 16, and continued doubling the amount of hearts each day through Feb. 14. If 200 candy
hearts come in a bag, how many bags of candy hearts would Kelly need just for Feb. 14?
This idea may have sounded like a good one at first, but Kelly’s probably regretting it
now! This is an exponential growth problem that shows how quickly an amount can grow
when repeatedly doubled. The first day she gave 1 candy. The second day she gave 1 ×
2 candies. The third day she gave 1 × 2 × 2 candies. She will keep multiplying by 2 until
she gets to the 32nd day. Therefore, the amount of candy she’ll need just for Feb. 14 is 1
× 2 31. This is 2,147,483,648 pieces of candy. Dividing this by 200 for each bag of candy
means she’ll need 10,737,419 bags just to cover Valentine’s Day!
3. __________ For Valentine’s Day Kevin wanted to send Mary Beth 11
balloons since that was her favorite number. In the store, plain-colored
balloons cost $0.75 each, multi-colored balloons cost $1.30 each, and extralarge balloons cost $1.50 each. How many different combinations of 11
balloons can Kevin buy if he has only $12.00?
Making an orderly chart may be the best way to approach the problem. Starting with
buying as many of the extra-large balloons as possible, then methodically subtracting
an extra-large balloon, and so on. Though he can afford 8 extra-large balloons, he then
could not afford 3 more to make the 11 balloons needed. The most extra-large balloons
he can afford then is 5 ($7.50) leaving him just enough to buy 6 plain-colored balloons
($4.50). Then find possibilities with 4 extra-large balloons. Notice exchanging a multicolored balloon for a plain-colored balloon raises the cost $.55. This may help when
determining possibilities and finding patterns. Eventually you will find 24 possible
combinations!
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
4. __________ Kia decided that she wants to give each of her friends a small pouch of candy
hearts. She’ll use fabric and ribbon to make and tie the pouch, and each
one will contain four candy hearts. The hearts come in six colors: white,
BE LEO
orange, pink, green, purple and yellow. If each pouch contains exactly four
V
MIN YA E
hearts, such that no two hearts are the same color, how many possible
combinations of hearts could be in a pouch?
This is a problem that is really easier if we answer a different question. Rather than
counting the number of different groups of four hearts to put into the pouch, it’s easier to
count the number of combinations of two hearts to leave out of each pouch. Each of the
six colors could be matched with one of the five remaining colors, to make a total of 6 ´ 5
= 30 ways to chose the two colors to leave out. However, this way counts yellow/pink as
different from pink/yellow. So every pair of two colors is represented twice. Therefore,
there are only 15 combinations of two colors since the order doesn’t matter. Each of
these combinations of hearts to leave out has a corresponding combination of hears to
put in, so there are also 15 different four-heart groupings that could be used.
5. __________ Mrs. Stuver’s art class has used geometric shapes to design a valentine in the
shape of a heart. They have placed two adjacent semicircles along one side of an equilateral
triangle so that the diameters of the semicircles and one side of the triangle are concurrent. The
diameter of each semicircle is exactly one-half the length of the side of the triangle. The length
of each side of the triangle is 4 inches. What is the area of the valentine, in square inches?
Express your answer as a decimal to the nearest tenth.
The area of the heart-shaped valentine is the sum of the areas of the equilateral triangle
and the two semicircles. The altitude of an equilateral triangle is perpendicular to the
base of the triangle and bisects the base. The Pythagorean Theorem can be used to find
the altitude of the equilateral triangle: a2 + b2 = c2 or 22 + b2 = 42. Solving for b, the
altitude is 2√(3). The area of the triangle is (base × height) ÷ 2 = (4 × 2√(3)) ÷ 2. The area
of the two semicircles is the same as the area of a circle with diameter = 2 or radius = 1:
(π × radius2) = π ×12 = π. (4 × 2√(3)) ÷ 2 + π = 10.1 square inches.
6. __________ Mrs. Stuver’s class then decorated the perimeter of each heart-shaped valentine
with lace. What is the length, in inches, of the perimeter of the valentine? Express your answer
as a decimal to the nearest tenth.
The length of the perimeter of the valentine is two times the side length of the triangle
plus the length of the circumference of the two semicircles:
(2 × 4) + (2 × π) = 14.3 inches.
7. __________ Two pink valentines and two green valentines are delivered at random to two
girls and two boys so that each girl and each boy receives exactly one valentine. What is the
probability that each girl receives a pink valentine and each boy receives a green valentine?
Express your answer as a common fraction.
P(girl, pink) = 2/4, P(girl2, pink) = 1/3, P(boy1, green) = 2/2, P(boy2, green) = 1/1; 2/4 × 1/3
× 2/2 × 1/1 = 1/6. The result is 1/6 no matter what order the distribution is done. Therefore
the probability is 1/6 that each girl receives a pink valentine and each boy receives a
green valentine.
Another way to think of the solution is to consider the six distributions of the pink and
green valentines to the two girls and two boys. Only one of the six distributions shows
each girl receiving a pink valentine and each boy receiving a green valentine.
Girl 1
Pink
Pink
Pink
Green
Green
Green
Girl 2
Pink
Green
Green
Pink
Pink
Green
Boy 1
Green
Pink
Green
Pink
Green
Pink
Boy 2
Green
Green
Pink
Green
Pink
Pink
8. __________ On some graph paper or below, graph the following segments:
y = x for 0 ≤ x ≤ 2
y = 2x – 2 for 2 ≤ x ≤ 3
x = 3 for 4 ≤ y ≤ 6
y = –x + 9 for 2 ≤ x ≤ 3
y = 7 for 1 ≤ x ≤ 2
y = x + 6 for 0 ≤ x ≤ 1
Now reflect each segment over the y-axis. What popular shape
have you drawn?
The first segment connects the points (0, 0) and (2, 2). The second segment connects the
points (2, 2) and (3, 4). The third segment connects the points (3, 4) and (3, 6). The fourth
segment connects the points (3, 6) and (2, 7). The fifth segment connects the points (2,
7) and (1, 7). Finally, the sixth segment connects the points (1, 7) and (0, 6). This should
appear as half a heart. Once the reflection is done, you should have the shape of a heart
with the y-axis running down the center.
Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set