5-QUESTION
CHALLENGE 12
Name
Calculators may NOT be used on this challenge.
1.�������� If b is positive, what is the value of b in the geometric sequence 9, a, 4,
b? Express your answer as a common fraction. weeks Noel earns $25 each week mowing lawns. Each week he spends
2.��������
of
his earnings on himself and has the rest of his earnings placed directly
into a savings account. Every third week he withdraws $20 from his
savings account to take his girlfriend to the movies. If he
continues this same spending pattern, how many weeks will
it take Noel to first exceed $125 in his savings account?
Express your answer as a whole number.
3.�������� Two complementary angles A and B have measures in the ratio of 13 to
17. What is the ratio of the measure of the supplement of angle A to the
supplement of angle B? Express your answer as a common fraction. 4.�������� Amy and Belinda each roll a sheet of 6-inch by 8-inch paper to form
a cylindrical tube. Amy tapes the two 8-inch sides together without
overlap. Belinda tapes the two 6-inch sides together without overlap. What is π times the positive difference of the volumes of the two tubes?
sq mi The scale of a certain map is
5.��������
inch = 16 miles. A square
park is represented on this map by a square with side
length inch. What is the actual area of this park?
Express your answer as a decimal to the nearest
hundredth. Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
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5-QUESTION
CHALLENGE 12
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Name
Calculators may NOT be used on this challenge.
1.�������� If b is positive, what is the value of b in the geometric sequence 9, a, 4,
b? Express your answer as a common fraction. In a geometric sequence, the numbers change by a factor “d.” Knowing this, we
can say that 9/a = a/4. Solving for a we see that a = 6. From there we can set
up the proportion 6/4 = 4/b. Solving for b we find that b = 8/3.
2.�������� Noel earns $25 each week mowing lawns. Each week he spends
of
his earnings on himself and has the rest of his earnings placed directly
13 weeks into a savings account. Every third week he withdraws $20 from his
savings account to take his girlfriend to the movies. If he
continues this same spending pattern, how many weeks will
it take Noel to first exceed $125 in his savings account?
Express your answer as a whole number. If Noel spends 2/5 of his earnings on himself each week, that means he is saving (3/5)(25) = $15 each week. Thus, that at the end of the second week he
will have $30. On the third week, he would have $45 but he withdraws $20 to go
to the movies which results in a net of -$5, leaving him with $25 in savings. If we
continue with this pattern (15, 30, 25, 40, 55, 50, 65, 80, 75, 90, 105, 100, 115,
130) we see that he exceeds $125 for the first time on week
3.�������� Two complementary angles A and B have measures in the ratio of 13 to
17. What is the ratio of the measure of the supplement of angle A to the
supplement of angle B? Express your answer as a common fraction. We are told that angles A and B are complementary, so we know that A + B =
90 degrees. Additionally, we are told that they are in the ratio of 13 to 17, so
we can say that 13x + 17x = 90. Solving for x we find that x = 3, thus A = 39
degrees and B = 51 degrees. Now we can see that the supplement of angle A
is 180 − 39 = 141 degrees and the supplement of angle B is 180 − 51 = 129 degrees. Thus, the ratio of the supplement of angle A to the supplement of angle B
is 141/129 = 47/43.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
24
4.��������
Amy and Belinda each roll a sheet of 6-inch by 8-inch paper to form a
cylindrical tube. Amy tapes the two 8-inch sides together without overlap.
Belinda tapes the two 6-inch sides together without overlap. What is π
times the positive difference of the volumes of the two tubes?
Amy’s cylinder has a height of 8 inches and a circumference of 6 inches. The
diameter of her cylinder must be 6/π inches and the radius 3/π inches. The
volume of a right cylinder is V = πr2h, thus the volume of Amy’s cylinder is π ×
(3/π)2 × 8 = 72/π cubic inches. Belinda’s cylinder has a height of 6 inches and a
circumference of 8 inches. The diameter of her cylinder must be 8/π inches and
the radius 4/π inches. Thus, the volume of Belinda’s cylinder is π × (4/π)2 × 6
= 96/π cubic inches. The difference in these two volumes is 96/π – 72/π = (96
– 72)/π = 24/π cubic inches, and π times this difference is 24.
156.25 sq mi The scale of a certain map is
5.��������
inch = 16 miles. A
square park is represented on this map by a square
with side length inch. What is the actual area of
this park? Express your answer as a decimal to the
nearest hundredth.
If the scale of the map is 4/5 inch = 16 miles, the equivalent scale of 1 inch = 20
miles would be easier to understand and easier to use. One square inch on the
map would then be 20 miles × 20 miles = 400 square miles. The square on the
map representing the park has an area of 5/8 × 5/8 = 25/64 square inches. The
actual area of the park is 25/64 × 400 = 156.25 square miles.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges