Download (1) The median of the set of numbers {12, 38, 45, x, 14}

(1)
The median of the set of numbers {12, 38, 45, x , 14} is five less than the
mean. If x is a negative integer, what is the value of x ?
(2)
The Asian elephant has an average gestation period of 609 days. How many
weeks is this gestation period?
What is the sum of the reciprocals of the positive integer factors of 28?
(3)
(4)
The circumference of a particular circle is 18 cm. In square centimeters,
what is the area of the circle? Express your answer as a common fraction in terms of π.
(5)
For what value of x is 23 × 3x = 72?
(6)
Based on this graph, what percent of viewers watched one hour or more of
television?
More than
2 hours
less than
1 hour
1-2 hours
(7)
(8)
Lee can make 18 cookies with two cups of flour. How many cookies can he
make with three cups of flour?
√
The hypotenuse and a leg of a particular right triangle are 97 inches and 4
inches, respectively. The area of this triangle is what common fraction of a square foot?
(9)
The measures of the interior angles of a particular triangle are in a 5:6:7
ratio. What is the measure, in degrees, of the smallest interior angle?
(10)
The ratio of girls to boys participating in intramural volleyball at Ashland
Middle School is 7 to 4. There are 42 girls in the program. What is the total number of
participants?
(11)
What is the median of all values defined by the expression 2x − 1, where x is
a prime number between 0 and 20?
(12)
What is the median of the prime numbers between 20 and 50?
(13)
Before taking his last test in a class, the arithmetic mean of Brian’s test
scores is 91. He has determined that if he scores 98 on his last test, the arithmetic mean
of all his test scores will be exactly 92. How many tests, including the last test, does Brian
take for this class?
(14)
A particular triangle has sides of length 14 cm, 8 cm and 9 cm. In
centimeters, what is the perimeter of the triangle?
(15)
One of the following four-digit numbers is not divisible by 4: 3544, 3554,
3564, 3572, 3576. What is the product of the units digit and the tens digit of that
number?
(16)
(17)
(18)
What is the mean of
1
2
and 87 ? Express your answer as a common fraction.
The set {5, 8, 10, 18, 19, 28, 30, x } has eight members. The mean of the
set’s members is 4.5 less than x . What is the value of x ?
Calculate: 9 − 8 + 7 × 6 + 5 − 4 × 3 + 2 − 1
(19)
There are eight furlongs in a mile. There are two weeks in a fortnight. The
British cavalry traveled 2800 furlongs in a fortnight. How many miles per day did the
cavalry average?
(20)
An environmental agency needs to hire a number of new employees so that
85 of the new employees will be able to monitor water pollution, 73 of the new employees
will be able to monitor air pollution, and 27 of the new employees will be able to monitor
both. (These 27 are included in the 85 and 73 mentioned above.) What is the minimum
number of employees that need to be hired?
(21)
Simplify:
√
2.52 −0.72
2.7−2.5 .
(22)
Think of a number. Double the number. Add 200. Divide the answer by 4.
Subtract one-half the original number. What is the value of the result?
(23)
The mean of three test scores is 74. What must a fourth test score be to
increase the mean to 78?
(24)
What is the greatest common factor of 154 and 252?
(25)
How many positive integer values of x are solutions to the inequality
10 < −x + 13?
(26)
Raquel has collected $3.80 in nickels and dimes. She has exactly 48 nickels.
How many dimes does she have?
(27)
The gauge of an oil tank indicated that the tank was 17 full. After 240
gallons of oil were added to the tank, the gauge indicated that the tank was 47 full. How
many gallons of oil will the tank hold, assuming the gauge is accurate?
(28)
A figure skater is facing north when she begins to spin to her right. She
spins 2250 degrees. Which direction (north, south, east or west) is she facing when she
finishes her spin?
(29)
When Cedric walked into a party, two-thirds of those invited had already
arrived. Six more people arrived just after Cedric, bringing the number at the party to 56 of
those invited. What was the total number of invited guests?
(30)
Bekah has three brass house number digits: 2, 3 and 5. How many distinct
numbers can she form using one or more of the digits?
Copyright MATHCOUNTS Inc. All rights reserved
Answer Sheet
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Answer
-14
87 weeks
2
81
π square centimeters
2
75 percent
27 cookies
1/8
50
66 participants
1087
37
7 tests
31 centimeters
20
11/16
22
37
25 miles per day
131
12
50
90
14
2
14
560 gallons
east
42 guests
15 numbers
Problem ID
A34C
014C
1003
1CC1
0CC1
BC31
CBC1
2CC1
414C
B1D2
A2AC
50B3
04A2
5C31
3222
B113
3102
22D2
B041
1BD3
D5D3
514C
00D4
0203
A14C
BC543
CAA2
2322
22A4
2443
Copyright MATHCOUNTS Inc. All rights reserved
Solutions
(1) -14
ID: [A34C]
Since x is negative, the median of the set is 14. Therefore, the mean of the set is
14 + 5 = 19, and its sum is 19 · 5 = 95. Since 12, 38, 45, and 14 sum to 109, the
remaining integer x must be 95 − 109 = −14 .
(2) 87 weeks
ID: [014C]
There are 7 days in one week, so 609 days equal 609/7 = 87 weeks.
(3) 2
ID: [1003]
The prime factorization of 28 is 22 · 7. Therefore, the positive integer divisors are 1, 2,
22 (= 4), 7, 2 · 7 (= 14), and 28 itself. The sum of their reciprocals is
1
1
1
1
1
1
1 + 2 + 4 + 7 + 14 + 28 . We can put this over a common denominator of 28, so that it
simplifies to
28 · 1 14 · 1 7 · 1 4 · 1
2·1
1·1
+
+
+
+
+
28 · 1 14 · 2 7 · 4 4 · 7 2 · 14 1 · 28
28 + 14 + 7 + 4 + 2 + 1
56
=
= 2.
28
28
(Notice that the original number, 28, can be used as a common denominator, and that
when the reciprocals of the divisors are put over this common denominator, the new
numerators are just the divisors of 28 listed in reverse order! Can you figure out why this
happens?)
=
(4)
81
π
square centimeters
ID: [1CC1]
If r is the radius of the circle, then the circumference is2πr
.2 Setting 2πr equal to 18 cm,
9
81
square centimeters.
we find r = 9/π cm. The area of the circle is πr 2 = π
=
π
π
(5) 2
ID: [0CC1]
Since the prime factorization of 72 is 72 = 23 · 32 , we have x = 2 .
(6) 75 percent
ID: [BC31]
The sector corresponding to “less than 1 hour” has a central angle of 90 degrees.
90
= 14 = 25% of viewers. It follows that 100% − 25% = 75% of
Therefore, it represents 360
viewers watch one hour of television or more.
(7) 27 cookies
ID: [CBC1]
Let x be the number of cookies that Lee can make with three cups of flour. We can set up
x
the proportion 18
2 = 3 . Solving for x , we find that x = 27 .
(8) 1/8
ID: [2CC1]
q √
2
The measure of the missing leg is
97 − 42 = 9 inches, by the Pythagorean theorem.
Converting 9 inches and
feet, we find that the area of the triangle is
4 inches
to 1 1
1
3
1
(base)(height) =
ft.
ft. =
square feet.
2
2 3
4
8
(9) 50
ID: [414C]
Choose k so that the smallest angle measures 5k degrees. Then the measures of the other
two angles are 6k degrees and 7k degrees. Since the measures of the angles in a triangle
sum to 180 degrees, we have 5k + 6k + 7k = 180 =⇒ 18k = 180 =⇒ k = 10. The
smallest angle measures 5k = 5(10) = 50 degrees.
(10) 66 participants
ID: [B1D2]
We have girls : boys = 7 : 4. Multiplying both parts of the ratio by 6 gives
girls : boys = 42 : 24, so there are 24 boys to go with the 42 girls. This gives a total of
42 + 24 = 66 participants.
(11) 1087
ID: [A2AC]
No solution is available at this time.
(12) 37
ID: [50B3]
The easiest way to attack this problem is to simply write out a list of the prime numbers
between 20 and 50, as follows: 23, 29, 31, 37, 41, 43, 47. The median is the middle number,
which is 37 .
(13) 7 tests
ID: [04A2]
Let S be the sum of all of Brian’s test scores up to this point, and let n be the number of
tests Brian has taken up to this point. Thus the arithmetic mean of his scores now is Sn
and the arithmetic mean of his scores after getting a 98 on the last test will be S+98
n+1 . This
gives the system of equations:
S + 98
S
= 91
= 92
n
n+1
From the first equation we have S = 91n. Substituting this into the second equation gives:
S + 98
= 92
n+1
S + 98 = 92(n + 1)
91n + 98 = 92n + 92
92n − 91n = 98 − 92
n=6
So Brian has to take n + 1 = 7 tests.
(14) 31 centimeters
ID: [5C31]
The perimeter of a polygon is defined to be the sum of the measures of the sides of the
polygon. Therefore, the perimeter of a triangle whose sides measure 14 cm, 8 cm, and 9
cm is 31 centimeters.
(15) 20
ID: [3222]
A number is divisible by 4 if its last two digits are divisible by 4. The only given number
that is not divisible by 4 is 3554 because 54 is not divisible by 4. The product of the units
digit and the tens digit of 3554 is 5 · 4 = 20 .
(16) 11/16
ID: [B113]
The sum of the two fractions is
1
2
+
7
8
=
11
8 .
So their mean is
1
2
11
8
(17) 22
ID: [3102]
Writing the information presented as an equation,
5 + 8 + 10 + 18 + 19 + 28 + 30 + x
= x − 4.5.
8
Solving, x = 22 .
=
11
.
16
(18) 37
ID: [22D2]
By the order of operations, we perform the multiplications before the additions and
subtractions:
9 − 8 + 7 × 6 + 5 − 4 × 3 + 2 − 1 = 9 − 8 + 42 + 5 − 12 + 2 − 1
= 1 + 42 + 5 − 12 + 2 − 1
= 48 − 12 + 2 − 1
= 36 + 1 = 37 .
(19) 25 miles per day
We have
ID: [B041]
14 days = 1 fortnight
and
8 furlongs = 1 mile,
and we are asked to convert a quantity whose units are furlongs per fortnight to miles per
day. We divide the first equation by 14 days to obtain a quantity which is equal to 1 and
has units of fortnight in the numerator.
1 fortnight
1=
.
14 days
Similarly,
1 mile
.
1=
8 furlongs
Since the right-hand sides of both of these equations are equal to 1, we may multiply them
by 2800 furlongs per fortnight to change the units without changing the value of the
expression:
1 mile
furlongs
miles
1 fortnight
= 25
2800
·
.
fortnight
14 days
8 furlongs
day
(20) 131
ID: [1BD3]
There are 85 + 73 = 158 jobs to be done. 27 people do two of the jobs, so that leaves
158 − 27 · 2 = 158 − 54 = 104 jobs remaining. The remaining workers do one job each, so
we need 27 + 104 = 131 workers.
We also might construct the Venn Diagram below. We start in the middle of the
diagram, with the 27 workers who do both:
Water
Air
85 − 27
73 − 27
27
This gives us 27 + (73 − 27) + (85 − 27) = 131 workers total.
(21) 12
ID: [D5D3]
We have
p
√
√
√
576/100
2.52 − 0.72
6.25 − 0.49
5.76
=
=
=
2.7 − 2.5
2.7 − 2.5
0.2
0.2
√
√
576/ 100
24/10
2.4
=
=
=
= 12 .
0.2
0.2
0.2
(22) 50
ID: [514C]
Let x be the number we think of. We double x to obtain 2x , add 200 to find 2x + 200,
divide by 4 to find
2x
200
x
2x + 200
=
+
= + 50.
4
4
4
2
After subtracting one-half of the original number, we are left with 50 .
(23) 90
ID: [00D4]
No solution is available at this time.
(24) 14
ID: [0203]
The prime factorizations of these integers are 154 = 2 · 7 · 11 and 252 = 22 · 32 · 7. The
prime factorization of their greatest common divisor (GCD) must include all of the primes
that their factorizations have in common, taken as many times as both factorizations allow.
Thus, the greatest common divisor is 2 · 7 = 14 .
(25) 2
ID: [A14C]
We first solve the inequality:
10 < −x + 13
−3 < −x
3 > x.
The only positive integers less than 3 are 1 and 2, for a total of 2 solutions.
(26) 14
ID: [BC543]
The 48 nickels are worth 48 · $0.05 = $2.40. The remaining dimes are worth
$3.80 − $2.40 = $1.40. Because each dime is worth $0.10, there are $1.40 ÷ $0.10 = 14
dimes.
(27) 560 gallons
ID: [CAA2]
No solution is available at this time.
(28) east
ID: [2322]
Each full circle is 360 degrees. Dividing 360 into 2250 gives a quotient of 6 with a
remainder of 90. So, she spins 90 degrees to her right past north, which leaves her facing
east .
(29) 42 guests
ID: [22A4]
Let P be the number of people invited to the party. Before Cedric arrived, there were 32 P
people at the party. After Cedric and six others arrived, there are 23 P + 7 people at the
party. Since this is the same as 56 P , we solve 23 P + 7 = 65 P to find that P = 42 .
(30) 15 numbers
ID: [2443]
If Bekah uses only one digit, she can form three numbers. If she uses two digits, she has
three choices for the tens place and two for the units, so she can form six numbers. Finally,
if Bekah uses all three digits, she has three choices for the hundreds place, two for the
tens, and one for the units, so she can form six numbers. Thus, Bekah can form
3 + 6 + 6 = 15 distinct numbers.
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