5-QUESTION
CHALLENGE 27
Name
Calculators may not be used.
cm
1.����������A
chain with two links is 13 cm long. A chain made from
three links of the same type is 18 cm long. How many
centimeters are in the length of a chain made from 25 such links?
2. ��������� The sum of the squares of two whole numbers is 160. The product of
the numbers is 48. What is the positive difference between the two
whole numbers? 3.���������� A building modeled after the Chicago Sears Tower consists
of nine square towers arranged in a 3x3 grid. They have
congruent bases, and the heights, in feet, are indicated in
the grid to the right. Which of the following is a side view of
the building from some direction?
A
B
C
D
% If x is 150% of y, what percent of 3y is x? 4.����������
100 200 100
300 300 200
200 200 100
E
sq units
5.����������Trapezoid
ABCD and trapezoid EFGH are congruent. What is the
difference between the area of triangle ABC and the area of triangle EFJ?
A
B
D
C
E
H
J
F
G
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
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5-QUESTION
CHALLENGE 27
lu
o
S
Name
Calculators may not be used.
128 cm chain with two links is 13 cm long. A chain made from
1.����������A
three links of the same type is 18 cm long. How many
centimeters are in the length of a chain made from 25 such links? If a chain with two links is 13 cm long, and a chain with three links is 18 cm long, we
see that each additional link adds 5 cm to the length of the chain. Adding 22 links to the
three-link chain will add 22 × 5 = 110 cm to its length, making its total length 18 + 110 =
128 cm.
8
2. ���������
The sum of the squares of two whole numbers is 160. The product of
the numbers is 48. What is the positive difference between the two
whole numbers? Starting with some factor pairs of 48, we can look at the sum of the squares of those
factors to see if their sum is 160. The factor pair 4 and 12 works, since 16 + 144 = 160.
The positive difference between the two numbers is 12 − 4 = 8.
D
3.���������� A
building modeled after the Chicago Sears Tower consists
of nine square towers arranged in a 3x3 grid. They have
congruent bases, and the heights, in feet, are indicated in W
the grid to the right. Which of the following is a side view of
the building from some direction?
A
B
C
D
N
100 200 100
300 300 200
E
200 200 100
S
E
If we look at the tower from the East or West (as labeled now in the figure), the only 300foot tower we see is in the middle. Option C matches this, but notice that we then also
would see 200-foot towers on the left and right, and option C then fails. Looking at the
tower from the North or South, there is a 300-foot tower in the middle and on either the
left (from the South) or the right (from the North). This could be option B or D if we are
looking from the South. Looking from the South we will see a 200-foot tower in front of
the 300-foot tower in the middle, and this is only the case with option D.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
50
% If x is 150% of y, what percent of 3y is x? 4.����������
If x must be 150% of y, let’s let x = 150 and y = 100. From here we can see 3y = 300, and x/(3y) = 150/300 = 1/2 = 50%.
Another solution is to turn the English into algebra. The first part tells us x = 1.5y or x = (3/2)y. We are trying to find x/(3y). Substituting (3/2)y for x, we need to simplify [(3/2)y]/(3y), which is [(3y)/2] × [1/(3y)] = 1/2 = 50%.
x
Visually, we can set up the first part.
Then when we set up the second part,
we can see the answer.
x
y
0 sq units
5.����������Trapezoid
ABCD and trapezoid A
EFGH are congruent. What is
the difference between the area
D
of triangle ABC and the area of triangle EFJ?
3y
B
C
E
F
H
J
G
Because the trapezoids are congruent, the two triangles have equal bases (AB and EF)
and equal heights, so they must have the same area. The difference between their area is,
therefore, 0 square units.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges