SMT 2013
General Test Solutions
February 2, 2013
1. Robin goes birdwatching one day. He sees three types of birds: penguins, pigeons, and robins.
2
3
of the birds he sees are robins.
1
8
of the birds he sees are penguins. He sees exactly 5 pigeons.
How many robins does Robin see?
Answer: 16
Solution:
1

2
3

1
8
=
5
24
of the birds that Robin sees are pigeons.
Therefore, Robin sees
exactly
5
5
24
= 24 birds, so
2
3
·
24 = 16
robins.
2. Jimmy runs a successful pizza shop. In the middle of a busy day, he realizes that he is running
low on ingredients. Each pizza must have 1 lb of dough,
1
4
lb of cheese,
1
6
lb of sauce, and
1
3
lb of toppings, which include pepperonis, mushrooms, olives, and sausages. Given that Jimmy
currently has 200 lbs of dough, 20 lbs of cheese, 20 lbs of sauce, 15 lbs of pepperonis, 5 lbs of
mushrooms, 5 lbs of olives, and 10 lbs of sausages, what is the maximum number of pizzas that
Jimmy can make?
Answer: 80
Solution:
The limiting factor is the cheese. With only 20 lbs of cheese, the most pizzas that
can be made is 20
·
4 = 80
.
3. Queen Jack likes a 5card hand if and only if the hand contains only queens and jacks. Consid
ering all possible 5card hands that can come from a standard 52card deck, how many hands
does Queen Jack like?
Answer: 56
Solution:
There are a total of 8 queens and jacks, each of which is distinguishable from the
others. Thus, the number of hands that Queen Jack likes is
(
8
5
)
= 56
.
4. What is the smallest number over 9000 that is divisible by the first four primes?
Answer: 9030
Solution:
The first four primes are 2, 3, 5, and 7, so the number must be a multiple of
2
·
3
·
5
·
7 = 210. The least multiple of 210 that is greater than 9000 is 210
·
43 = 9030
.
5. A rhombus has area 36 and the longer diagonal is twice as long as the shorter diagonal. What
is the perimeter of the rhombus?
Answer: 12
√
5
Solution:
Let
d
be the length of the shorter diagonal and 2
d
the length of the longer diagonal. If
d
1
, d
2
are the lengths of the diagonals of a rhombus, its area will be
d
1
·
d
2
2
. Thus, we have
d
2
= 36
and
d
= 6. Therefore, the side length of the rhombus is
√
3
2
+ 6
2
= 3
√
5, so the perimeter is
12
√
5
.
6. Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of
10 mph. If he completes the first three laps at a constant speed of only 9 mph, what speed does
he need to maintain in miles per hour on the fourth lap to achieve his goal?
Answer: 15
Solution:
Let
d
be the length of one lap in miles. Then he needs to complete the four laps in
4
d
10
=
2
d
5
hours. He has already spent
3
d
9
=
d
3
hours on the first three laps, so he has
2
d
5

d
3
=
d
15
hours left. Therefore, he must maintain a speed of 15
mph on the final lap.