5-QUESTION
CHALLENGE 14
Calculators may be used.
Name
1.�������� If x − y = 6 and x + y = 12, what is the value of y?
degrees When the temperature goes up 3˚ on the Cantor scale,
2.��������
Fronbenius it goes up 8˚ on the Frobenius scale. On both scales,
18˚ is the same temperature. How many Frobenius
degrees are equal to 30˚ Cantor? feet When a car’s brakes are applied, it travels 5 feet less in each second
3.��������
than the previous second until it comes to a complete stop. A car goes 35 feet in the first second after the brakes are applied. How
many total feet does the car travel from the time the
brakes are applied to the time the car stops?
sq in An 8-inch by 8-inch square is folded along a diagonal creating a
4.��������
triangular region. This resulting triangular region is then folded so that the
right angle vertex just meets the midpoint of the hypotenuse. What is the
area of the resulting trapezoidal figure? citizens In Sumville, 40% of the citizens are at least
5.��������
6 feet tall and 25% of the citizens have red
hair. Height and hair color have no correlation;
the people with red hair are proportionally
distributed among the people at least 6 feet tall
and the people less than 6 feet tall. Sumville
has exactly 500 citizens. How many citizens of
Sumville have red hair and are at least 6 feet
tall?
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
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5-QUESTION
CHALLENGE 14
lu
o
S
Calculators may be used.
Name
3
1.��������
If x − y = 6 and x + y = 12, what is the value of y?
By solving the first equation for x, we find that x = 6 + y. We can now
substitute our new expression for x into the second equation to get 6 + y + y = 12. Now we can solve for y as follows:
6 + 2y = 12
2y = 6
y=3
50 degrees When the temperature goes up 3˚ on the Cantor scale,
2.��������
Fronbenius it goes up 8˚ on the Frobenius scale. On both scales,
18˚ is the same temperature. How many Frobenius
degrees are equal to 30˚ Cantor? The problem tells us that both scales are the same at 18˚, so let’s start
from there. At 30˚ Cantor, the temperature is 30˚ − 18˚ = 12˚ Cantor above
the 18˚ mark. Knowing that every 3˚ increase on the Cantor scale is equal
to an 8˚ increase on the Frobenius scale, we can set up the proportion 3/8 = 12/x → 3x = 96 → x = 32˚ Frobenius. Thus, when the Cantor scale
reads 30˚, the Frobenius scale reads 18˚ + 32˚ = 50˚. 140 feet When a car’s brakes are applied, it travels 5 feet less in each second
3.��������
than the previous second until it comes to a complete stop.
A car goes 35 feet in the first second after the brakes are
applied. How many total feet does the car travel from the
time the brakes are applied to the time the car stops?
We can solve this problem by actually subtracting the 5 feet for each
second until
we reach 0. This will show us that he traveled the following
distances in the seconds from the time the brake was applied until the
car stopped:
35, 30, 25, 20, 15, 10 and 5 feet. Now we can add those
find the
total distance traveled, which is distances up to
35 + 30 + 25 + 20 + 15 + 10 + 5 = 140 feet.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
24 sq in An 8-inch by 8-inch square is folded along a diagonal creating a
4.��������
triangular region. This resulting triangular region is then folded so that the
right angle vertex just meets the midpoint of the hypotenuse. What is the
area of the resulting trapezoidal figure? The area of the full square is 8 × 8 = 64 in2. After the first fold, the shaded
triangular region is 1/2 the area of the square therefore, the shaded triangle
in the second figure is (1/2)(64) = 32 in2. In the third figure we see that we
are subtracting a portion of these 32 in2. We can see that portion is half of
a 4 × 4 square, which means it is (1/2)(16) = 8 in2. The area of the resulting
shaded region in the third figure is 32 in2 – 8 in2 = 24 in2.
50 citizens In Sumville, 40% of the citizens are at least
5.��������
6 feet tall and 25% of the citizens have red
hair. Height and hair color have no correlation;
the people with red hair are proportionally
distributed among the people at least 6
feet tall and the people less than 6 feet tall.
Sumville has exactly 500 citizens. How many
citizens of Sumville have red hair and are at
least 6 feet tall?
Let’s first determine how many people have red hair in Sumville.
0.25(500) = 125 people
Since the problem states that the height and hair color have no correlation
we know that 40% of the people with red hair are also at least 6 feet tall.
Thus, there are 0.40(125) = 50 citizens with red hair that are at least 6 feet
tall.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges