Portions of Portions Meeting
(Fractions of Fractions &
Percents of Percents)
Topic
The problems for this meeting cover a variety of ways students can work with portions of
portions.
Materials Needed
♦ Copies of the Portions of Portions problem set (Problems and answers can be viewed below.
Complete solutions and a more student-friendly version of the problems—with pictures and
larger font—are available for download from www.mathcounts.org on the MCP Members
Only page of the Club Program section.)
♦ Scrap Paper to draw representations of the problems
Meeting Plan
Many of the problems can be solved nicely using diagrams to represent the “whole” amounts
and then systematically breaking the “wholes” into portions according to the problems.
(Examples of these representations are provided below for #1 and #2.) Ask students to
represent at least five of the problems (including #1 and #2) with diagrams. You may wish to
do #1 and/or #2 as a group to demonstrate what you are asking the students to do. Then have
students share their representations with the rest of the club members. Allowing students to
work in pairs or groups is suggested.
1. Kamera and a friend order one pizza that is half-pepperoni and half-mushroom.
Kamera eats 1/3 of the pepperoni portion and 1/4 of the mushroom portion. What fraction
of the pizza did Kamera eat? Express your answer as a common fraction. 2002-2003
School Handbook Warm-Up 4-9
stayed
left
left
left
left left
9
P
M
P
M
= 1/6
P
M
= 1/8
1/6 + 1/8 = 4/24 + 3/24 = 7/24
Whole Audience
left
Whole Pizza
2. Boar Ring gave a speech to an assembly of colleagues. After 5 minutes, half of the
audience left; 10 minutes later a third of the remaining audience left. After 20 more
minutes, half of the remaining audience left, leaving only 3 people in the audience. How
many people were in the audience at the beginning of Mr. Ring’s speech? 2007 School
Competition Target Round #8
3
3 3 3 = 18
3. An oil tanker containing 108,000 gallons of oil releases one-third of its remaining volume every two hours. How
many gallons have been released after the first six hours? 2002-2003 School Handbook Warm-Up 3-6
2009–2010 MATHCOUNTS Club Resource Guide
49
4. What is the single discount that is equivalent to a discount of 10% off followed by a discount of 20% off the
discounted price? 2006-2007 School Handbook Warm-Up 7-9 modified
5. Connie went shopping and spent one-half of her money at Allen’s Grocery Store. She then spent one-third of what
she had left at Bryan’s Bar-B-Q Cafe. After that, she then had $6 left. How much money did she spend at Allen’s
Grocery Store? 2006 School Competition Sprint Round #23
6. A delivery truck made three stops after leaving the warehouse. At the first stop, the driver unloaded 1/4 of his
crates. At the second stop, he unloaded 2/3 of the crates remaining on his truck. At the last stop, he unloaded 80%
of the rest of his crates at that point, and then he returned to the warehouse with 20 crates. How many crates did he
unload at the first stop? 2006-2007 School Handbook Warm-Up 14-1
7. In algebra class, half of the students are boys. One-third of the students are wearing glasses. Half of the boys are
wearing glasses. What fraction of the girls is wearing glasses? Express your answer as a common fraction. 20062007 School Handbook Warm-Up 8-4
8. In Mathopolis, an adult is a person 21 years of age or older and a child is a person under 21 years of age. Exactly
50% of the adults in Mathopolis are female, and exactly 50% of the female adults have exactly one biological child.
Nobody else has a child, and there are no other children. What percent of the people of Mathopolis are children?
2007 School Competition Sprint Round #10
9. Every other time that Cheri Linn saw a friend today, she gave away half of her flowers. At the end of the day,
she had five flowers left. How many flowers did she start the day with if she saw eight friends? 2005-2006 School
Handbook Warm-Up 9-4
10. I have two one-quart jars; the first is filled with water, and the second is empty. I pour half of the water in the first
jar into the second, then a third of the water in the second jar into the first, then a fourth of the water in the first jar into
the second, then a fifth of the water in the second jar into the first, and so on. How much water, in quarts, is in the first
jar after the 10th pour? Express your answer as a common fraction. 2007 School Competition Team Round #9
Answers: 7/24; 18 people; 76,000 gallons; 28%; $9; 100 crates; 1/6; 20%; 80 flowers; 6/11 quarts
Possible Next Steps
Ask students to write three Portions of Portions problems of their own. Each one should have
at least three “breakdown steps” where a portion is asked for (see example below). Note that a
“breakdown step” might actually be an increase... such as “1/3 more” or “an increase of 25%.”
To add difficulty, tell students (1) the answer to the first problem must be 3, (2) the initial starting
value of the second problem must be 27 and (3) the third problem must have an initial starting
value of 48 and an answer of 7.
Sample answer for third student-generated problem:
Note: The “length” of a bracelet or necklace is the number of beads used to create it.
Fisher had 48 identical beads with which to make a bracelet for himself and for each of his two
older brothers (Grady and Noah). He decided to make three bracelets with lengths in the ratio
of 2:3:3. He kept the smaller bracelet and gave his older brothers the longer bracelets (breakdown
step #1). When Fisher tried on his bracelet, he realized that he would prefer a necklace, so he
went to the bead store and bought enough beads to increase the length of his bracelet by 75%
(breakdown step #2). Finally happy with his creation, he went to visit his three cousins (Sophia, Ella
and Ava). When he saw how much they liked his necklace, he decided to disassemble it and
create a bracelet for each of his cousins that was one-third the length of his necklace (breakdown
step #3). How many beads did Sophia’s bracelet have? Answer: 7 beads
50 2009–2010 MATHCOUNTS Club Resource Guide
Portions of Portions
1. ___________ Kamera and a friend order one pizza that is halfpepperoni and half-mushroom. Kamera eats 1/3 of the pepperoni
portion and 1/4 of the mushroom portion. What fraction of the pizza
did Kamera eat? Express your answer as a common fraction.
pple Boar Ring gave a speech to an assembly of
2. ___________
colleagues. After 5 minutes, half of the audience left; 10 minutes
later a third of the remaining audience left. After 20 more minutes,
half of the remaining audience left, leaving only 3 people in the
audience. How many people were in the audience at the beginning
of Mr. Ring’s speech?
Whole Pizza
P
M
Whole Audience
left
stayed
gal An oil tanker containing 108,000 gallons of oil releases one-third of its
3. ___________
remaining volume every two hours. How many gallons have been released after the first
six hours?
% What is the single discount that is equivalent to a
4. ___________
discount of 10% off followed by a discount of 20% off the discounted
price?
F
F
O
%
10
Addit
io
20% nal
Off
5. $___________ Connie went shopping and spent one-half of her money at Allen’s
Grocery Store. She then spent one-third of what she had left at Bryan’s Bar-B-Q Cafe.
After that, she then had $6 left. How much money did she spend at Allen’s Grocery
Store?
Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Problem Set
crates A delivery truck made three stops after leaving the
6. ___________
warehouse. At the first stop, the driver unloaded 1/4 of his crates.
At the second stop, he unloaded 2/3 of the crates remaining on his
truck. At the last stop, he unloaded 80% of the rest of his crates at
that point, and then he returned to the warehouse with 20 crates.
How many crates did he unload at the first stop?
E
AT RY
R
C IVE
L
DE
7. ___________ In algebra class, half of the students are
boys. One-third of the students are wearing glasses. Half
of the boys are wearing glasses. What fraction of the girls
is wearing glasses? Express your answer as a common
fraction.
% In Mathopolis, an adult is a person 21 years of age or older and a child
8. ___________
is a person under 21 years of age. Exactly 50% of the adults in Mathopolis are female,
and exactly 50% of the female adults have exactly one biological child. Nobody else has
a child, and there are no other children. What percent of the people of Mathopolis are
children?
flowers Every other time that Cheri Linn saw a friend today, she gave
9. ___________
away half of her flowers. At the end of the day, she had five flowers left. How
many flowers did she start the day with if she saw eight friends?
quarts I have two one-quart jars; the first is filled with water, and the second
10. ___________
is empty. I pour half of the water in the first jar into the second, then a third of the water
in the second jar into the first, then a fourth of the water in the first jar into the second,
then a fifth of the water in the second jar into the first, and so on. How much water, in
quarts, is in the first jar after the 10th pour? Express your answer as a common fraction.
Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Problem Set
Portions of Portions Meeting Solutions
(2009-2010 MCP Club Resource Guide)
Problem 1. Answer: 7/24. The representation is on pg. 49 of the 2009-2010 Club Resource
Guide.
Problem 2. Answer: 18.The representation is on pg. 49 of the 2009-2010 Club Resource Guide.
Problem 3.
36,000 + 24,000 + 16,000 = 76,000 gallons released
Problem 4.
100% – 72% = 28%
Problem 5.
$9 was spent at Allen’s
Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Solution Set
Problem 6.
100 crates were unloaded at the first stop
Problem 7. Let the area of a rectangle represent the boys and girls in the algebra class. The
class is divided into half boys and half girls. Now half of the boys are wearing glasses, which
means half of half of the class, or (1/2)(1/2) = 1/4 of
the class is “boys wearing glasses.” We need a third
of the class to be wearing glasses, so 1/3 – 1/4 =
4/12 – 3/12 = 1/12 of the class must be “girls
wearing glasses.” If we divide our class into
twelfths, we see that we get a third (or 4/12) of the
class wearing glasses when we have 3/12 = 1/4 of
the class being the “boys wearing glasses” and 1/12
of the class being the “girls wearing glasses.” From
the figure we can see that 1/6 of the girls are “girls
wearing glasses.”
Problem 8.
Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Solution Set
Problem 9.
Problem 10.
START:
After pour 1, jar 1 keeps 1/2, jar 2 gets rest
After pour 2, jar 2 keeps 2/3, jar 1 gets rest
After pour 3, jar 1 keeps 3/4, jar 2 gets rest
After pour 4, jar 2 keeps 4/5, jar 1 gets rest
After pour 5, jar 1 keeps 5/6, jar 2 gets rest
After pour 6, jar 2 keeps 6/7, jar 1 gets rest
After pour 7, jar 1 keeps 7/8, jar 2 gets rest
After pour 8, jar 2 keeps 8/9, jar 1 gets rest
After pour 9, jar 1 keeps 9/10, jar 2 gets rest
After pour 10, jar 2 keeps 10/11, jar 1 gets rest
Jar 1: 1
Jar1: (1/2)(1) = 1/2
Jar 1: 1 – 1/3 = 2/3
Jar 1: (2/3)(3/4) = 1/2
Jar 1: 1 – 2/5 = 3/5
Jar 1: (5/6)(3/5) = 1/2
Jar 1: 1 – 3/7 = 4/7
Jar 1: (7/8)(4/7) = 1/2
Jar 1: 1 – 4/9 = 5/9
Jar 1: (9/10)(5/9) = 1/2
Jar 1: 1/2 – 5/11 = 6/11
Jar 2: 0
Jar 2: 1 – 1/2 = 1/2
Jar 2: (2/3)(1/2) = 1/3
Jar 2: 1 – 1/2 = 1/2
Jar 2: (1/2)(4/5) = 2/5
Jar 2: 1 – 1/2 = 1/2
Jar 2: (6/7)(1/2) = 3/7
Jar 2: 1 – 1/2 = 1/2
Jar 2: (8/9)(1/2) = 4/9
Jar 2: 1 – 1/2 = 1/2
Jar 2: (10/11)(1/2) = 5/11
Notice that if n is odd, then jar 1 and jar 2 each will have 1/2 after pour n.
If n is even, then jar 2 will have (1/2)(n/(n+1)) and jar 1 will have 1 - (1/2)(n/(n+1)) after pour n.
Copyright MATHCOUNTS, Inc. 2009. MATHCOUNTS Club Resource Guide Solution Set