1999–2000 MATHCOUNTS School Handbook
WORKOUTS
Answers to the Workouts include one-letter codes, in parentheses, indicating appropriate problem solving
strategies, as desribed in the Problem Solving section. It should be noted that the strategies indicated
may not be the only applicable strategies. A calculator icon indicates problems which may be more
easily solved with a calculator.
The following codes will be used in the answer keys:
(C)
(F)
(M)
(T)
(G)
(S)
(E)
(P)
Compute or Simplify
Use a Formula
Make a Model or Diagram
Make a Table, Chart or List
Guess, Check and Revise
Consider a Simpler Case
Eliminate
Look for Patterns
The answer key to each Workout appears on the following page. A detailed solution to one of the ten
problems is also provided on the accompanying answer key, and, as appropriate, a mathematical
connection to a problem or an investigation and exploration activity has been noted.
MATHCOUNTS Symbols and Notation
Standard abbreviations have been used for units of measure. Complete words or symbols are also
acceptable. Square units or cube units may be expressed as units2 or units3.
Typesetting of the MATHCOUNTS handbook and competition materials provided by EducAide Software, Vallejo, California.
WORKOUT 1
1.
A 24-exposure disposable camera sells for $7.99. During a
promotion, the company sold 27-exposure cameras for the same
price. How many cents per exposure are saved by purchasing the
promotional camera? Express your answer as a decimal to the
nearest tenth.
1.
2.
A rope is tightly stretched from the top of a 50-foot pole to the
top of a 20-foot pole. The two poles are 16 feet apart. How many
feet are in the length of the rope?
2.
3.
Jamal burned 672 calories during 45 minutes of his
one-hour workout. Assuming he continued to exercise at the same
rate, what is the total number of calories that he burned by the
end of his workout?
3.
4.
In the multiplication shown, each ∗ represents a digit. What is the
sum of all possible products?
4.
2∗
×
∗7
∗∗∗
∗ ∗∗
2 ∗∗1
5.
How many positive integer factors does 2000 have?
5.
6.
Teisha has grown a perfect rose bush. The bush has 24 branches.
Each branch has 12 limbs, each limb has 6 twigs, and each twig
bears 3 roses. How many roses are on the bush?
6.
7.
The length of a rectangle is increased by 10%. By what percent
must the width be decreased for the area to remain the same?
Express your answer as a decimal to the nearest tenth.
7.
8.
In the triangle shown, the value of y is three times the value of x.
What percent of z is y? Express your answer to the nearest whole
number.
8.
9.
What is the 4037th digit following the decimal point in the
1
?
expansion of 17
9.
10. A high school football player runs 40 yards in 4.5 seconds. If he
were to maintain that speed, how many miles would he run in
one hour? Express your answer as a decimal to the nearest tenth.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 1
1.
3.7
(C)
2.
34
(FM)
3.
896
(C)
4.
4232
(TP)
5.
20
(TP)
6.
5184
(C)
7. 9.1
10. 18.2
(FM)
(C)
8.
95
(M)
9.
2
(SP)
SOLUTION
Problem #5
FIND OUT
What do we want to know? The number of positive integer factors of 2000.
CHOOSE A
STRATEGY
It’s possible to check every integer 12000 to find the divisors of 2000, but that would take an
incredible amount of time. Instead, look at the prime factors of 2000 and see if there’s a way to
calculate the number of divisors without actually finding every one of them.
SOLVE IT
The prime factorization of 2000 is easy to find, because it contains only two prime factors,
namely 2 and 5. The prime factorization is 24 × 53 .
That information can be used to determine the total number of positive integer divisors
of 2000. Any positive integer divisor will contain from 0 to 4 powers of 2 and from 0 to 3
powers of 5. (For example, both 22 · 50 = 4 and 23 · 52 = 200 are both divisors of 2000.) Because
there are five possible powers of 2 and four possible powers of 5 from which to choose, the total
number of positive integer divisors of 2000 is 5 × 4 = 20.
LOOK BACK As a check, list all the divisors of 2000. Because the solution above established a strategy for
counting them, making the list is easy:
50
51
52
53
20
1
5
25
125
21
2
10
50
250
22
4
20
100
500
23
8
40
200
1000
24
16
80
400
2000
The chart shows that there are, indeed, 20 divisors of 2000.
MAKING CONNECTIONS. . . to Exercise
Problem #3 & 10
A company that makes in-home fitness equipment claims that a healthy 40-year old male who weighs 150 pounds
will burn 890 calories an hour using their cross-country ski machine. A monitor on the machine indicates
progress. After 45 minutes, it’ll tell a 40-year old, 150-pound man that he’s burned 668 calories. Unfortunately,
after 45 minutes on the machine, it’ll tell a 25-year old, 120-pound woman that she’s burned 668 calories, too.
If only it were that simple. There are many factors which contribute to how many calories are burned during
exercise. Weight, metabolic rate and intensity of the workout are just a few of the factors. Based on the exercise,
though, the number of calories burned increases proportionally with weight. A backpacker, for instance, will burn
3.18 times her weight in pounds every hour, and a mountain biker will burn 3.86 times his weight. Even an
aerobic dancer will burn 2.72 times her weight each hourwhich means that the 20-year old, 120-pound woman
will burn 245 calories in 45 minutes.
To find out more about how many calories are burned during various activities, check out
http://caloriecontrol.org/exercalc.html.
c MATHCOUNTS 19992000
WORKOUT 2
1.
A baseball team won 50% of the first 120 games it played in a
162-game season. What is the minimum number of its remaining
games that the team must win in order to win at least 60% of its
games this season?
1.
2.
A retailer purchases a CD player from a manufacturer for $180.
The retailer sets the price so that he can yield a 35% profit
over the manufacturer’s cost when the CD player sells at a
20% discount. How many dollars are in the price that the retailer
set? Express your answer as a decimal to the nearest hundredth.
2.
3.
The exam grades of a pre-algebra class were arranged in a stem
and leaf plot as illustrated. What is the arithmetic mean of the
median and the mode of the given data?
3.
4
1
5
2
6
7
8
8
7
1
1
2
3
3
3
5
6
8
0
4
4
6
6
6
6
8
9
1
3
5
5
7
8
4.
A human blinks once every 5 seconds. How many times does a
human blink per day?
4.
5.
Hepta House offers a circular 14-inch diameter pizza for $7.99.
The pizza is cut into 17 slices. What is the average number of
cents in the price of one slice?
5.
6.
What is the number of square
inches in the total surface area
of the resulting figure when a
2 00 × 2 00 × 2 00 cube is removed from
a 4 00 × 4 00 × 4 00 cube as shown?
6.
7.
Set U contains all prime numbers less than 30, set A contains all
prime numbers less than 30 that end in 1 or 3, and set B contains
all factors of 397,670. Find the sum of all elements in U that are
not in A ∩ B.
7.
8.
What is 25% of the sum of the first 11 prime numbers?
8.
9.
What is the product of the least common multiple and the greatest
common factor of 22 and 48?
9.
10. A weird number is a number that is the product of two consecutive
prime numbers, such as 7 × 11 = 77. What is the least common
multiple of the four smallest weird numbers? (Problem submitted
by alumnus Matthew Mendicino.)
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 2
1.
38
(CS)
2.
303.75
(FG)
3.
82
(FT)
4.
17,280
(C)
5.
47
(C)
6.
96
(FM)
7.
93
(TEP)
8.
40
(CT)
9.
1056
(F)
10. 2310
(TP)
SOLUTION
Problem #3
FIND OUT
For what are we looking? The mean of the median and mode of a set of data which is
represented in a stem-and-leaf plot.
CHOOSE A
STRATEGY
Because this problem involves three measures (mean, median and mode), we must understand
the difference among them. Then, the median and mode of the data set can be found, and the
mean of those two numbers can be calculated.
SOLVE IT
The median of a set of data is the middle element. (If a set has an even number of elements,
the median is the average of the two middle elements.) The set in this problem has 27 elements,
so identify the 14th element when the data is arranged in order. Because the quiz grades appear
in a stem-and-left plot, identifying the median is easy; simply count until the 14th element is
reached. For this set, the median is the last element in the fourth row, 78.
The mode of a set of data is the most frequently occurring element. (If more than one element
occurs most frequently, the set is said to contain several modes. Multiple modes are listed if
there is more than one; unlike the median, they are not averaged to find one unique mode.)
There are four 6’s in the fifth row of the stem-and-leaf plot, so 86 is the mode of this set of data.
The mean of a set of data is the sum of all elements, divided by the number of elements in
the set. The median and mode are two numbers, so the mean of the median and mode is
78+86
= 82.
2
LOOK BACK Is the answer reasonable? Yes. The elements in the data set range from 41 to 97; it makes
sense that the median and mode fall within that range. Further, the mean of a set should fall
within the range of the set, so it makes sense that the mean of the median and mode falls
between 78 and 86.
MAKING CONNECTIONS. . . to Human Anatomy
Problem #4
It’s true. Humans blink almost 20,000 times a day. When you think about it, it’s truly amazing what the body
does automatically. An average adult’s heart beats about 70 times per minutenearly 100,000 beats per day!
On average, humans breathe about 16 times per minutealthough that number can drop to as low as 6 to
8 breaths per minute when sleeping, and it can increase to as much as 100 breaths per minute when under
extreme stress. Over the course of a lifetime, that translates to 75,000,000 gallons of air breathed.
And all this happens while you’re playing soccer, reading a book, or solving cool math problems.
INVESTIGATION & EXPLORATION
Problem #6
When a 2 00 × 2 00 × 2 00 cube was removed from the larger cube, the surface area remained the same. What effect
will removing a 3 00 × 3 00 × 3 00 cube from one of the corners have on the surface area? What about removing a
1 00 × 1 00 × 1 00 cube? Can you write a brief description to explain this phenomenon?
c MATHCOUNTS 19992000
WORKOUT 3
1.
A school organization consists of 5 teachers, 7 parents and
6 students. A subcommittee of this group is formed by choosing
2 teachers, 3 parents and 3 students. How many different
subcommittees can be formed?
1.
2.
A multivitamin contains 162 milligrams of calcium which
represents 16.2% of the recommended daily allowance for an adult.
How many milligrams of calcium are in the recommended daily
allowance for an adult?
2.
3.
According to an ancient belief, when a friend visits a sick person,
1
60 of his or her illness is taken away. How many friends need to
visit to take away at least 99% of a person’s illness?
3.
4.
For what value of x does 1 + 2 + 3 + 4 + 5 + . . . + x = 120?
4.
5.
In the diagram, each curve is an arc of a circle having a center at
a vertex of the square with edge length 4 cm. What percent of
the square is shaded? Express your answer to the nearest whole
number.
5.
6.
What is the positive difference between
7.
The purchase price of a bicycle, which includes 7% tax, is $374.50.
What is the number of dollars in the price before tax is added?
7.
8.
What is the least integer value of m such that 3m + 2m is a
three-digit number?
8.
9.
Thirty-six students took a final exam on which the passing score
was 70. The mean score of those who passed was 78, the mean
score of those who failed was 60, and the mean of all scores
was 71. How many students did not pass the exam?
9.
62 +62
6
and
62 ×62
6 ?
10. The lengths of the sides of a rectangular prism measuring
4 × 6 × 8 centimeters are each increased in length by 50%. What is
the percent of increase in the volume of the prism? Express your
answer as a decimal to the nearest tenth.
6.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 3
1.
7000
(FMP)
2.
1000
(FM)
3.
275
(GP)
4.
15
(FGP)
5.
43
(FM)
6.
204
(C)
7.
350
(CF)
8.
5
(G)
9.
14
(FMG)
10. 237.5
(FM)
SOLUTION
Problem #9
FIND OUT
What are we to find? The number of students who did not pass an exam, knowing the average
of those who passed, of those who failed, and of all students combined.
CHOOSE A
STRATEGY
There were 36 students in the class, and the total number of points earned was 2556
(36 students with average score of 71 points implies 36 × 71 = 2556 total points). Make an
educated guess as to the number of students who passed, see how close that is to the correct
answer, and then revise.
SOLVE IT
Because the overall average (71) is closer to the average of those who passed (78) than to the
average of those who failed (60), more students must have passed than failed. As a first guess,
then, assume that 21 students passed. That means that 15 students failed, and the total points
are 21(78) + 15(60) = 2538. That’s too low, so revise, increasing the number of students who
passed.
The total number of points increases by 18 for each additional student who passes, because the
difference in the average passing score and the average failing score is 78 − 60 = 18. Because
the first guess was 18 points too low, revise by increasing the number of students by 1 to 22.
That means 14 students failed, and the total points are 22(78) + 14(60) = 2556.
LOOK BACK The method used had an inherent check built in, so the correct numbers were certainly
attained. However, the question only asked how many students did not pass, so the answer
is 14.
MAKING CONNECTIONS. . . to Nutrition
Problem #2
A good estimate for the number of calories you require daily can be found, if you know your current weight. For
a moderately active person who is overweight, multiply your weight in pounds by 13.5 to find the approximate
number of calories you need a day. For a person of average weight, multiply by 16, and for an underweight person,
multiply by 18. A moderately active, 150-pound person would need about 2400 calories daily. A sedentary
person, however, needs to consume less, while a very active person has the luxury of requiring more calories.
INVESTIGATION & EXPLORATION
Problem #3
59 n
1
If algebra were used to write an equation for the situation in this problem, it would be ( 60
) < 100
. To solve this
problem with advanced mathematics would require logarithms, the exponent to which a number must be raised
to produce a given result.
A table of logarithmsor logs as they are commonly calledwas first generated by mathematician and
navigator John Napier (15501617), who used logarithms regularly when sailing. To create his table of logs,
Napier noticed that the arithmetic mean of two logarithms corresponded to the geometric mean of two numbers.
4
For instance, 102 = 100
√ and 10 = 1000; further, the arithmetic mean3 of 2 and 4 is 3, and the geometric mean
of 100 and 10,000 is 100 × 10,000 = 1000. The conclusion is that 10 = 1000.
Using a calculator, complete the chart below. How could you find the value of 101.375 ?
Logarithm (n)
Value of
10n
1.00
10
1.50
2.00
3.00
316.228
1000
c MATHCOUNTS 19992000
WORKOUT 4
1.
A compact disc player is loaded with 7 CDs. The CD player is
then programmed to play the CDs in random order. What is the
number of different ways that the compact discs can be ordered?
1.
2.
American Flyer model trains are built at 1:64 scale. If the actual
locomotive is 60 feet in length, what is the number of inches in the
length of the American Flyer model? Express your answer as a
mixed number.
2.
3.
Positive integers x, y and n are each less than 10. If x2 − y 3 = n,
find the median of all possible values for the sum x + y + n.
(Problem submitted by coach Joel Abrahamson.)
3.
4.
Notice the pattern in the first two figures below. Then, use that
same pattern to find the value of x in the third figure. What is x?
4.
5.
Jennifer puts 12 12 % of the monthly rent from a rental property
into an account for repairs and maintenance. She wants the annual
rent to be at least $9250 more than repairs and maintenance.
What is the minimum number of dollars she needs to charge for
monthly rent to accomplish these goals? Express your answer to
the nearest whole number.
5.
6.
If you begin counting two consecutive whole numbers each second,
starting on January 1, 2000, at 12:00 a.m., in what year will you
reach 1 billion?
6.
7.
In how many zeroes does
end?
7.
8.
Tishe deposited $200 at 8% interest compounded annually for
3 years. What is the number of dollars in the amount of interest
earned? Express your answer as a decimal to the nearest cent.
8.
9.
Find the least common multiple of 12, 16 and 18.
9.
20!
4(5!)
10. A cube has edge length 9 cm. What is the furthest distance
between any two vertices? Express your answer as a decimal to
the nearest hundredth.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 4
1.
5040
(FP)
2.
11 41
(CM)
3.
13
(EP)
4.
4
(FP)
5.
881
(FM)
6.
2015
(CP)
7.
3
(SP)
8.
51.94
(C)
9.
144
(CF)
10. 15.59
(FMS)
SOLUTION
Problem #6
FIND OUT
What are we asked to find? The year in which 1 billion numbers will have been counted if
two whole numbers are counted each second beginning on January 1, 2000.
CHOOSE A
STRATEGY
The original rate is given as numbers per second. Convert this to find how many numbers are
counted per year; then, divide 1 billion to find the number of years it will take.
SOLVE IT
Counting 2 numbers per second is equivalent to counting
2 × 60 = 120 numbers per minute,
120 × 60 = 7200 numbers per hour,
7200 × 24 = 172,800 numbers per day,
and 172,800 × 365 = 63,072,000 numbers per year.
Consequently, it will take
1,000,000,000
63,072,000
≈ 15.85 years to count to 1 billion.
Approximately 15.85 years from January 1, 2000, is roughly the beginning of October 2015.
The year will be 2015.
LOOK BACK When so many conversions are performed, it is possible to incorrectly identify the units of the
answer. When converting from numbers per second to numbers per year, multiply by unit
multipliersfractions that are equivalent to 1because multiplying by 1 doesn’t change a
value. For instance, 241 hours
day is a unit multiplier, because 24 hours = 1 day, and the fraction has
a value of 1. In this problem, unit multipliers were used:
63,072,000 numbers
2 numbers 60 seconds 60 minutes 24 hours 365 days
×
×
×
×
=
1 second
1 minute
1 hour
1 day
1 year
1 year
Notice that the units in each unit multiplier cancels with the units in the next multiplier, until
the desired result is found. This check verifies that the units are correct, so a reasonable level
of confidence can be had in the answer identified.
MAKING CONNECTIONS. . . to Interest
Problem #8
Investment analysts have a mantra that they continually preach to young investors: invest early. When planning
for retirement, they insist that investing when young is more important than how much is invested. If a 25-year
old recent college graduate puts $2000 into an account that earns only 6% interest a year, she’ll have earned
2000(1.06)40 = $20,571 by the time she retires at age 65. On the other hand, if a 50-year old invests $5000 at
9% interest, he’ll only have 5000(1.09)15 = $18,212 when he retires at the same age.
The reason for this is exponential growth. Because the money is able to earn interest for 40 years in the first case,
the interest accrued is significant. Consider depositing a penny into an account that doubles every day. After
one day, you’d have 2/c. And even after five days, you’d still only have 32/c. But after a month, you’d have over a
million dollars!
INVESTIGATION & EXPLORATION
Problem #3
In this problem, the values of x, y and n are restricted to being less than 10. What happens if this restriction is
lifted? Are there other integer values for which x2 − y 3 = n?
The values 32 and 23 have an interesting relationship. The numbers 2 and 3 differ by 1, and 32 and 23 differ by 1.
There are no other numbers which differ by 1 for which the square of one and the cube of the other also differs
by 1. But do you think there are any other positive integers for which the difference between the cube of one and
the square of the other is 1? If so, find them. If not, can you prove it?
c MATHCOUNTS 19992000
WORKOUT 5
1.
Three basins of a fountain are hemispherical. The top basin has a
diameter of 25 cm, the middle basin has a diameter of 50 cm, and
the bottom basin has a diameter of 100 cm. When the top basin
fills with water, it empties into the middle; and when the middle
basin fills with water, it empties into the bottom. Water begins
to fill the top basin at a rate of 1000 cm3 per minute. How many
minutes will it take to fill all three basins? Express your answer to
the nearest whole number.
1.
2.
A garage door opener has a ten-digit keypad. Codes to open the
door must consist of 5 digits with no adjacent digits the same.
How many codes are possible?
2.
3.
Congruent circles A and B intersect such that AB is a radius of
each circle. If AB = 6 cm, what is the number of square centimeters
in the area of the shaded region? Use 3.14 as an approximation
for π, and express your answer as a decimal to the nearest tenth.
3.
4.
Given that a ? b = ab − ba , and a∇b = (a + b)(a − b), what is the
value of a ? (a∇b) if a = 3 and b = 2?
4.
5.
An equilateral triangle is inscribed in a circle with diameter 9 cm.
How many square centimeters are in the area of the triangle?
Express your answer as a decimal to the nearest tenth.
5.
6.
Five positive integers less than 100 have 12 positive integer factors.
Which of these five integers is not divisible by 12? (Problem
submitted by coach Thomas Brown.)
6.
7.
One trillion is 10n times one-trillionth. What is the value of n?
7.
8.
Mrs. Quartro said the students in her math class could have a
pizza party when they completed 100,000 problems. Each of her
160 students did 25 problems a day. How many days did it take
them to earn the pizza party?
8.
9.
School portraits come in two sizes, 5 00 × 7 00 and 3 00 × 5 00 . The area
of the smaller portrait is what percent of the area of the larger
portrait? Express your answer to the nearest whole number.
9.
10. A three-digit number consisting of three different digits has its
digits reversed, resulting in a smaller number. The smaller number
is subtracted from the original number. The answer is a number
composed of the same three digits in a different order. What is
the original number?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 5
1.
299
(FM)
2.
65,610
(FT)
3.
44.2
(FMS)
4.
118
(C)
5.
26.3
(FM)
6.
90
(TEP)
7.
24
(P)
8.
25
(C)
9.
43
(CM)
10. 954
(TGP)
SOLUTION
Problem #10
FIND OUT
What are we asked to find? A three-digit number, with three different digits, that adheres to
several criteria.
CHOOSE A
STRATEGY
A process is described in the problem. Represent each of the digits with a variable and use
algebra and logic to determine each of the digits.
SOLVE IT
The original number has hundreds digit a, tens digit b and units digit c; hence,
represent the number with the expression 100a + 10b + c. When the digits
are reversed, the result is 100c + 10b + a. The difference between these is
100a + 10b + c − (100c + 10b + a) = 99a − 99c = 99(a − c). Consequently, this final result will be
a multiple of 99. There are only nine multiples of 99 which are three-digit numbers, so checking
each of them should not be terribly difficult.
The first is 99 × 2 = 198. Rearranging these digits and subtracting as required gives
three possibilities: 981 − 189 = 792, 918 − 819 = 99, and 891 − 198 = 693. None of these works,
though, so check the next value.
The next value is 99 × 3 = 297. Again, rearranging the digits gives three possibilities, none of
which work.
Continuing in this manner, it isn’t until the fifth multiple, 99 × 6 = 594, that a rearrangement
of the digits is found that works; namely, 954 − 459 = 495. Consequently, the original number
is 954.
LOOK BACK Using a combination of algebra and logic, the answer was identified. Note that using algebra
alone doesn’t guarantee an answer, and there isn’t enough information that logic alone could
generate the answer without the use of some algebraic manipulation. A combination of the two
resulted in finding a solution to the problem.
MAKING CONNECTIONS. . . to the NBA Draft and Combinatorics
Problem #2
Combinatorics is the mathematical field that investigates how elements can be arranged. In some cases, the
elements arranged are the digits in a security code. In other cases, the elements are numbers written on
fourteen balls, four of which are randomly drawn from a bin, as in the NBA lottery.
At the end of the NBA season, 16 of the 29 teams make the playoffs. The other 13 teams are then entered in a
lottery to see which team gets the first pick in the draft that year. The method for determining which team gets
the first pick is somewhat obscure, though fairly interesting.
In reverse order of their records, the 13 non-playoff teams are respectively given 250, 200, 157, 120, 89, 64, 38,36,
18, 11, 7, 6 and 5 chances to win the number one pick. For each chance, a team is given a distinct combination of
four of the numbers 114. Then, 14 ping-pong balls are placed in a drum, 4 balls are chosen at random, and the
team which has been assigned the combination chosen is given the first pick.
Now, you may have noticed that the total number of chances assigned is 1000, but 14
4 = 1001. What happens to
the remaining combination? Quoting the NBA press release on the lottery, Fourteen ping-pong balls numbered 1
through 14 will be placed in a drum and four will be drawn to determine the number one pick. . . There are
1001 possible combinations when four balls are drawn out of 14, without regard to their order of selection. Prior
to the lottery, 1000 combinations will be assigned to the 13 lottery teams. If the one unassigned combination is
drawn, the drawing will be repeated.
c MATHCOUNTS 19992000
WORKOUT 6
1.
If one liter equals approximately 1.06 quarts, and one quart
contains 32 ounces, how many ounces are in 3 liters? Express your
answer as a decimal to the nearest hundredth.
1.
2.
What is the smallest counting number divisible by each of the first
ten counting numbers?
2.
3.
What is the number of square
centimeters in the area of the
triangle shown?
3.
4.
The hemispherical dome of a state capital needs to have its
interior painted. The interior diameter of the dome is 60 feet.
If a one-gallon can of paint covers 400 square feet, what is the
minimum number of cans needed to cover the dome with one coat
of paint?
4.
5.
The speed of light is 186,000 miles per second. How many miles
per hour is the speed of light? Express your answer in scientific
notation to one decimal place.
5.
6.
Two-hundred students at Hypatia Middle School were surveyed
about their after-school activities.
6.
57 participated in basketball
113 participated in MATHCOUNTS
46 participated in neither activity
How many students participated in both activities?
7.
Compute 40% of 30% of 20% of 10% of 160,000.
7.
8.
What value of x will give the minimum value for x2 − 10x + 24?
8.
9.
Brian has 100 feet of fencing. He will use the fencing to enclose a
play area for his puppy. What is the maximum number of square
feet he can enclose? Express your answer to the nearest whole
number.
9.
10. What is the sum of the series
1 + 2 − 3 + 4 + 5 − 6 + 7 + 8 − 9 + . . . + 101 − 102?
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 6
1.
101.76
(FM)
2.
2520
(FTG)
3.
234
(FM)
4.
15
(FM)
5.
6.7 × 108
(C)
6.
16
(FM)
7.
384
(C)
8.
5
(TEP)
9.
796
(FM)
10. 1683
(FP)
SOLUTION
Problem #10
FIND OUT
What do we wish to know? The sum of a series with repeated operations.
CHOOSE A
STRATEGY
Because the pattern of operations is regular, the expression can be simplified before computing
its value. The first three terms of the series are 1 + 2 − 3, which have a value of 0. The second
set of three terms has a value of 4 + 5 − 6 = 3. Note that each group of three terms is greater
than the previous set by 3. The last set of three terms is 100 + 101 − 102 = 99. Hence, the
series can be rewritten as 0 + 3 + 6 + 9 + . . . + 99, or 3(0 + 1 + 2 + 3 + . . . + 33).
SOLVE IT
The value of the series 0 + 1 + 2 + 3 + . . . + 33 can be found with the formula for the sum of
the first n positive integers, which was discovered by Karl Friedrich Gauss. The formula for the
. Because the sum of the first 33 integers is wanted,
sum of the first n positive integers is n(n+1)
2
the value is
33(34)
2
= 561. Consequently, the value of the sequence is 3 × 561 = 1683.
LOOK BACK A spreadsheet is an excellent way to verify our answer. By using the first column to list the
integers 1102, using the second column to compute the value of each set of three terms, and
then using the sum function to find the total, it can be found that the answer of 1683 checks.
MAKING CONNECTIONS. . . to Architecture
Problem #4
Christopher Williams, in Origins of Form, praised the dome. Very exciting possibilities arise when the
structural potential of the dome is joined with the rigidity of the triangle. The dome is one of the best ways to
enclose large spaces, for it resists the pull of gravity evenly over its surface, and gives stability with minimum
material.
The first domes were built from rock as early at 12,000 B.C. Today, geodesic domes, like Spaceship Earth
at Epcot Center, are fairly common. In fact, companies now specialize in dome homes, and for good reason.
Spheres are geometrically efficient; they maximize volume while minimizing surface area. Thus, any spherical
dome has the least surface through which to lose heat or intercept damaging winds. Domes are an example of
ephemeralization, as Buckminster Fuller, the inventor of the geodesic dome, liked to say. The best domes are
proportionally thinner than a chicken egg shell is to the egg.
INVESTIGATION & EXPLORATION
Problem #4
All second-degree functions take the shape of a parabola when graphed, and parabolas have a unique maximum
or minimum that occurs at its vertex. Although calculus can be used to find the vertex, there is a simple formula
for finding the minimum or maximum of a second-degree function. The formula is based on the coefficients of the
function.
Use the chart below to find a formula for the maximum or minimum value of a second-degree function.
Function
Value of x at Min (Max)
x2 + 4x + 7
−2
x2 + 4x − 16
2x2 + 4x + 8
2x2 + 4x
x2 − 10x + 24
ax2
5
+ bx + c
c MATHCOUNTS 19992000
WORKOUT 7
1.
A best-of-five series ends when one team wins three games. The
probability of team A defeating team B in any game is 94 . What
is the probability that team A will win the series? Express your
answer as a common fraction.
1.
2.
Compute: (654,321)(654,321) − (654,326)(654,316).
2.
3.
Alexandra received scores of 96, 90, 84 and 88 on her first
four exams. What average score does she need on the next
two exams to achieve a final average score of 92, if all six scores
are weighted equally?
3.
4.
Express the ratio of
5.
If x ⊕ y = (xy )x , what is the units digit of 7 ⊕ 5?
5.
6.
In a given sample, the number of microbes doubled every second.
After 9 seconds, there were 1,000,000,000 microbes per cm3 in
the sample. A second sample, which also doubled every second,
started with four times as many microbes per cm3 as the original
sample. How many seconds passed before the second sample had
1,000,000,000 microbes per cm3 ?
6.
7.
The volume of a cube is twice the volume of another smaller
cube. If the edge length of the smaller cube is 1 inch, what is the
number of inches in the edge length of the larger cube? Express
your answer as a decimal to the nearest hundredth.
7.
8.
9.
2
5
to
4
7
as a decimal to the nearest tenth.
√
In the figure shown, AC = CD = DE = EB, and AE = 4 5 in.
What is the number of square inches in the area of 4 ADB?
On a circular dartboard with concentric scoring areas, it is possible
to score 15, 16, 17, 18 or 19 points with one dart. What is the
fewest number of darts necessary to score exactly 100 points on
this dart board?
10. Find the sum of the two smallest positive integers, each of which
is a perfect square, a perfect cube, and a perfect fourth power.
(Problem submitted by coach Chris Goodrich.)
4.
8.
9.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 7
1.
7808
19,683
(FMT)
2.
25
(P)
3.
97
(FG)
4.
0.7
(C)
5.
3
(FP)
6.
7
(TP)
7.
1.26
(MG)
8.
16
(FM)
9.
6
(TEP)
10. 4097
(TEP)
SOLUTION
Problem #7
FIND OUT
What are we asked to find? The edge length of a cube, given that its volume is twice the
volume of a cube with edge length 1 inch.
CHOOSE A
STRATEGY
The volume of the smaller cube is obviously 1 × 1 × 1 = 1 cm3 , so the volume of the larger cube
is 2 cm3 . Using the formula for the volume of a cube, we can compute the edge length of the
larger cube.
SOLVE IT
The volume of the larger cube is 2 cm3 , and the
√ formula for the volume of a cube based on its
edge length is V = e3 . Hence, 2 = e3 , or e = 3 2.
We now need to find the number that, when cubed, equals 2. A calculator will be very helpful
in finding this number. Because 1.412 ≈ 2, we know the value for which we are searching must
be less than 1.41.
Try 1.33 . Its value is 2.197. Too much.
So, try 1.253 . Its value is 1.953. Too low, but just a little.
A bit more playing with the calculator shows that 1.263 = 2.000, to three decimal places.
LOOK BACK The decimal portion of the answer we found, 0.26, is roughly one-fourth. Imagine that a cube
is divided into four congruent slices, and arrange three of them around a whole cube as shown
below. Notice how the three unfilled areas could each be filled with one-third the fourth slice.
This makes it seem reasonable that our answer is correct.
MAKING CONNECTIONS. . . to Advertising
Problem #6
A billion no doubt seems like a lot, and it is. But considering the size of bacteria, it’s not unreasonable to have a
billion microbes in a cubic centimeter. When working with such large numbers, it’s necessary to be extremely
careful. Some time ago, a commercial advertised that a mouthwash could eliminate 99% of the bacteria cells in a
person’s mouth. However, it’s common for more than a billion bacteria cells to be present in a person’s mouth, so
killing 99% still leaves 0.01 × 1,000,000,000 = 10,000,000 cells! Just another example of truth in advertisingthe
math is technically correct, but unless it is interpreted, a false impression can be given.
INVESTIGATION & EXPLORATION
Problem #2
Algebra has often been called the generalization of arithmetic. The methods of symbolic manipulation used in
algebra manipulate quantities based on the properties of real numbers.
For instance, take any positive integer, and square it. Now, take the same number; add 5 to it, subtract 5 from it,
and multiply those two results.
The difference between the square of a number and the product of 5 more and 5 less than the number is 25.
Always. No doubt about it. Why is that?
What happens if you add and subtract 6 from a number, multiply those values, and compare it to the square of
the same number? What do you think will happen if you add and subtract 18 instead?
c MATHCOUNTS 19992000
WORKOUT 8
1.
A car that originally sold for $12,000 depreciates at a rate of 20%
per year. What is the number of dollars in the value of the car at
the end of 3 years?
1.
2.
A rectangular corral is to be built using 160 feet of fence. One
side of the corral will be part of a straight 100-foot wall of an
adjacent building. What is the maximum number of square feet
possible for the area of the corral?
2.
3.
A rental company charges $45 per day and 35/c per mile to rent a
car. What is the maximum whole number of miles that can be
driven in one day and still keep the cost less than $125?
3.
4.
In the diagram,
AB = AD = CD = BD, and A, D
and C are collinear. How many
square centimeters are in the area
of 4 ABD? Express your answer
to the nearest whole number.
4.
5.
Twenty 11 00 × 17 00 sheets were printed on both sides so that when
the sheets were stacked and folded down the center, an 8 12 00 × 11 00
booklet with the pages numbered 1-80 would be formed. When
stacked and folded, however, the sheets containing pages 12 and 53
were accidentally omitted. What is the sum of the other numbers
on the pages that were missing?
5.
6.
The exterior wall of a building forms a right angle with the ground
at its base. A 25-foot ladder is placed against the wall so that the
foot of the ladder is 7 feet from the base of the wall. The ladder
slips and its upper end slides 4 feet down the wall. How many feet
did the foot of the ladder slide along the ground?
6.
In trapezoid ABCD, AB k DC, AB = BC = 5 cm, BD = 12 cm, and
DBC is a right angle. What is the number of square centimeters
in the area of 4 ABD? Express your answer as a decimal to the
nearest tenth.
7.
8.
The number 34,459,425 is the product of several consecutive
positive odd numbers. What is the greatest of these numbers?
8.
9.
The basketball team at Parker Middle School consists of
seven students from the 6th, 7th and 8th grades. If the product of
their ages is 35,335,872, what is the sum of their ages?
9.
7.
6
10. What is the value of (1110 )(115 + 115 )−1 ? Express your answer as
a decimal to the nearest tenth.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 8
1.
6144
(FS)
2.
3200
(TG)
3.
228
(FM)
4.
28
(FM)
5.
259
(TEP)
6.
8
(FM)
7.
11.5
(FM)
8.
17
(TE)
9.
84
(GE)
10. 80,525.5
(CS)
SOLUTION
Problem #6
FIND OUT
We are looking for the amount a ladder slips along the ground when we know how far from the
base of the wall the ladder touches the ground and how far up the wall the ladder touches the
house.
CHOOSE A
STRATEGY
The wall forms a right angle with the ground, so a right triangle is formed with the ladder as
the hypotenuse. It seems obvious, then, to repeatedly invoke the Pythagorean theorem to solve
this problem.
SOLVE IT
Initially, the right triangle formed by the ladder and the wall has legs of length 7 feet
and
Using the Pythagorean theorem, the length of the ladder can be found to be
√ 24 feet. √
72 + 242 = 625 = 25 feet.
When the ladder slides down the wall 4 feet, the new height along the wall is 20 feet. The
hypotenuse of this new right triangle is still 25 feet, since the hypotenuse is the ladder. The
distance from
√ the ladder√to the base of the wall can be found by again using the Pythagorean
theorem: 252 − 202 = 225 = 15 feet. The ladder moved 15 − 7 = 8 feet.
LOOK BACK Because the hypotenuse is the ladder in both cases, the sum of the squares of the lengths of the
legs in both cases should be equal. Indeed, 72 + 242 = 152 + 202 , so the answer is likely correct.
MAKING CONNECTIONS. . . to Physics
Problem #6
The center of gravity is the average location of an object’s weight. The Earth pulls on each part of an object
with a gravitational force. This force is what we call weight. Although all of the individual parts of an object
contribute weight in this way, the net effect is as if the total weight of the object were concentrated in a single
point. Less technically, the center of gravity is the place where you could balance an object on just one finger.
When a person climbs a ladder, the center of gravity needs to be balanced over the ladder. If a person leans back,
for example, the center of gravity may move beyond the base of the ladder, in which case the ladder will tip over.
And that’s never a good thing.
Try this. Stand with your heels and the back of your legs against a wall. Now without bending your legs or
pulling them away from the wall, try to reach down and touch your toes. If you don’t cheat, you’ll likely find it
hard to do without falling over. When you stand against the wall, your center of gravity is near your stomach,
about two inches behind your belly button, inside you. But when you bend over, your center of gravity moves
out to a point in front of your stomach and in front of your feet, which causes you to lose your balance and fall
forward.
INVESTIGATION & EXPLORATION
Problem #2
The fence in this problem dictates the perimeter, though the area is variable. In general, which quadrilateral
yields the largest enclosed area when the perimeter is fixed? Is it a rectangle? A trapezoid? A square? A
rhombus? A parallelogram? Which of these shapes will yield the smallest enclosed area?
Regardless of the perimeter, when the wall of an existing building is used as one side, there is a fixed relationship
between the length and width of the area enclosed. What is the ratio of the width to length in every case?
c MATHCOUNTS 19992000
WORKOUT 9
1.
A rectangle is divided into five congruent regions. Sunil removes
one-fourth of the first region, one-fifth of the second region,
one-sixth of the third region and one-eighth of the fourth region.
What percent of the original area remains? Express your answer
to the nearest whole number.
1.
2.
A stone is dropped into a well, and the splash is heard 8.9 seconds
after it is dropped. The stone falls at a rate of 16t2 feet in
t seconds, and the speed of sound is 1120 feet per second. What is
the number of feet from the well to the top of the water? Express
your answer to the nearest whole number.
2.
3.
The first term of a geometric sequence is 7, and the 7th term
is 5103. What is the 5th term?
3.
4.
Two Algebra classes at Western View Middle School took the same
test. The first class of 22 students had an average score of 74,
and the second class of 30 students had an average score of 79.
What was the average score for all of the students in both classes?
Express your answer as a decimal to the nearest tenth.
4.
5.
What is the greatest possible product of a two-digit number and a
three-digit number obtained from five distinct digits?
5.
6.
What is the number of square inches in the area of the shaded
region in Stage 10 if the original square has side length 8 inches?
Express your answer as a decimal to the nearest tenth.
6.
7.
What is the remainder of 191999 divided by 25?
7.
2012 −1992
52
8.
Compute:
9.
If the hundreds and units digits of a three-digit number are
switched, the new number is 693 greater than the original number.
If the tens and units digits are switched, the new number is 27
greater than the original number. The sum of the digits is 14.
What is the original number?
.
10. Brian and Anna each have a spinner with the integers 14 being
equally likely. Brian spins first. Then Anna spins, and she wins
if her number matches Brian’s number. If it doesn’t match, then
Brian spins, and he wins if his number matches Anna’s number. If
Brian doesn’t win, then Anna spins again. Play continues in this
manner until one of their spins matches the most recent spin by
the other. What is the probability that Brian will win the game?
8.
9.
10.
c MATHCOUNTS 19992000
ANSWER KEY
WORKOUT 9
1.
85
(CM)
2.
1021
(FMTG)
3.
567
(TP)
4.
76.9
(F)
5.
84,000
(GP)
6.
60.4
(TSP)
7.
4
(SP)
8.
32
(C)
9.
158
(GS)
10.
3
7
(FMS)
SOLUTION
Problem #2
FIND OUT
What are we asked to find? The depth of a well when the length of time the sound took to
return after a stone was dropped into it is known.
CHOOSE A
STRATEGY
The distance the stone falls is equal to the distance the sound travels. Represent the distance
the stone travels and the distance the sound travels in terms of t, and set these expressions
equal.
SOLVE IT
Call the amount of time it takes the stone to fall t seconds; then, the amount of time it takes
the sound to travel up the well is (8.9 − t) seconds. Consequently, 16t2 = 1120(8.9 − t), or
16t2 + 1120t − 9968 = 0. A guess-and-check strategy can be used.
Assume that it takes as long for the stone to fall as it takes for the sound to return from
the water. Then, t = 4.45. In this case, the rock falls 16t2 = 16(4.45)2 ≈ 317 feet, and the
sound rises 1120(8.9 − t) = 1120(4.45) = 4984 feet. The distances aren’t equal, and because
317 < 4984, the value of t must be greater.
Try t = 7. That gives 16(7)2 = 784 and 1120(1.9) = 2128. Closer, but still off by quite a bit.
Continually revising the guess will eventually show that when t = 7.98, 16t2 = 1019 and
1120(8.9 − t) = 1030. Although those numbers aren’t equal, they are fairly close. Further
refinements to the guess for t, with accuracy beyond two decimal points, will show that the
depth of the well is 1021 feet.
LOOK BACK A spreadsheet can be used to check the values. A segment of a spreadsheet is shown below,
verifying the answer when t = 7.9883, accurate to four decimal places:
Time (t)
Distance stone fell (16t2 )
Distance sound traveled (1120(8.9 − t))
7.9881
1020.955
1021.328
7.9882
1020.981
1021.216
7.9883
1021.006
1021.104
7.9884
1021.032
1020.992
√
2
b −4ac
The quadratic formula, t = −b± 2a
, could also be used to find two possible solutions,
t = 7.98 and t = −77.98. The first of these agrees with the answer found.
MAKING CONNECTIONS. . . to the Speed of Sound
Problem #2
When planes move farther than 1120 feet every second, they break the sound barrier and a sonic boom is the
result. Likewise, when the end of a bullwhip exceeds 761 mph, it breaks the sound barrier, and a crack of the
whip occurs.
Sound travels 1120 feet per second, or approximately 1 mile in 5 seconds. Amateur meteorologists use this
knowledge when watching a thunderstorm. When lightning leaves the sky, they begin counting. When they
finally hear the sound made by the lightning striking earth, they can divide the number to which they counted
by 5 to get a rough approximation of how many miles away the lightning hit.
INVESTIGATION & EXPLORATION
Problem #5
This problem asked for the greatest product possible with a three-digit and a two-digit number. Find the greatest
product with a two-digit and a one-digit number, or with two two-digit numbers. How does the problem change?
What is the greatest product of a two-digit and a four-digit number? Is there a definite pattern emerging?
Slightly more predictable is the pattern when searching for the greatest possible sums. What is the greatest
possible sum of a three-digit and a two-digit number? What pattern emerges when looking for sums?
c MATHCOUNTS 19992000