Transcript 2,-3 - Math Forum

Quiz Bowl
All eight students will solve problems as part of a quiz
bowl.
Students will work together to answer questions
and compete head to head against other teams.
Teams will be seated at tables.
A moderator will ask a question using a
microphone.
Each team will write an answer to each question on
the paper provided.
Team captains will hand one answer to the table
runner.
The team captain should be seated in middle of
the team.
Teams should write answers clearly and neatly so that
judges can read answers. Teams with unclear
answers will not receive points.
Time will begin once the moderator finishes
reading the question.
Teams should NOT include computational or scratch
work on the paper; only answers should be written on
the paper.
Once a team turns in their paper, the team may
NOT change their answer.
Teams will be provided with scratch paper and pencils
as well as their answer sheets.
NO books, notes, calculators, or electronic
devices, such as cell phones, may be used.
Cell phones must be turned off.
Quiz Bowl
Round 1: All Groups Can Answer
Round 1 will consist of 10
questions.
All teams that provide the correct
answer to a question posed by a
moderator will earn 5 points.
You have 6 black socks, 12 white
socks, and 8 pink socks. It’s pitch
dark and you’re packing for a trip.
You reach into your drawer, blindly
choose an individual sock, and
pack it.
How many socks do you need to
choose to be sure you packed at
least one pair of pink socks?
Solution
• Think of the worst case scenario: choosing
no pink socks for the 1st, 2nd, 3rd, 4th, 5th,
6th, etc. socks. How long can you keep up
that string of bad luck?
• The worst case is you pull all 18 black and
white socks first.
• After that, 2 more socks will guarantee at
least one pair of pink socks.
• So if you pack 20 socks, you get at least
one pink pair.
Find the area of square BHIC,
given that ABC is a right triangle,
and ABFG and ACDE are both squares.
I
C
D
??
H
25 cm2
E
B
A
9 cm2
G
F
Solution
• According to the Pythagorean theorem,
the sum of the areas of squares
constructed on the legs of right triangles
equals the area of the square constructed
on the hypotenuse.
• A2 + B2 = C2
• 9 + 25 = 34
• No square roots needed!
Find the value of x and y
that make both equations
true:
3x – 5y = 17
y = 12 + 2x
Solution
•
•
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•
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3x – 5y = 17
y = 12 + 2x
Therefore, 3x – 5(12 + 2x) = 17
Distributing: 3x – 60 – 10x = 17
Combining Like Terms: -7x – 60 = 17
Addition Property of Equality: -7x = 77
Division Property of Equality: x = -11
Substitution: y = 12 + 2(-11)
y = -10
Two of the angles in a
triangle measure 48º
and 84º. What type of
triangle is this? Be as
specific as you can.
Solution
• If two of the angles are 48º and 84º, then
the third is 180º - 48º - 84º = 180º - 132º =
48º
• Since the triangle has two congruent
angles (the two 48º angles), it’s isosceles.
• The triangle is not equilateral or right, so
the most specific name is an isosceles
triangle.
Working together, Gertrude and
Bertha can clear their entire
driveway in 30 minutes. Working
alone it takes Gertrude 40
minutes to shovel the whole
driveway. How long does it take
Bertha to shovel the whole
driveway on her own?
Solution
Gertrude working alone:
Driveway Gertrude shovels in 40 minutes
Both
working
together
Driveway Gertrude shovels in 30 minutes
So Bertha working alone would
take 30 * 4 = 120 minutes
Driveway Bertha shovels in 30 minutes
If you roll two identical fair
dice, what is the probability
of rolling “snake eyes” – in
other words rolling a 2?
Solution
• With one fair die, the chance of rolling a 1
is 1 in 6 possibilities.
• With two independent events, the chance
of both happening is the product of their
probabilities.
• 1/6 * 1/6 = 1/36
• Or, list all the possibilities in a table:
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
A serving of grapes is 2/3
of a cup. How many
servings are in a 6 ½ cup
package of grapes?
Give your answer as a
mixed number of servings.
Solutions
• 6 ½ ÷ 2/3 = 13/2 * 3/2 = 39/4 = 9 ¾ servings
• 3 servings of grapes is 2 cups (2/3 + 2/3 + 2/3
= 2), so 6 ½ cups is 9 servings plus another
½ cup. A ½ cup of grapes is 3/6 of a cup, and
2/3 of a cup is 4/6 of a cup, so 3/6 of a cup is
¾ of what is needed to make a full serving.
So there are 9 full servings and another ¾ of
a serving.
Order the numbers
-1.125, ,- , 0.6,
from least to greatest.
Solution
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Clearly -1.125 and -7/5 are smallest
-7/5 = -14/10 = -1.4
-1.4 < -1.125
2/3 = 0.666666666…
0.6 < 0.666666666…
10/12 is 2/12 or 1/6 away from 1
2/3 is 1/3 away from 1, therefore 10/12 >
2/3
• -7/5, -1.125, 0.6, 2/3, 10/12
Graph all solutions to
3|-2x – 4| > 12
Solution
• 3|-2x – 4| > 12
• Therefore, |-2x – 4| > 4
• So -2x – 4 has to be greater (to the right
of) than 4 or less than -4 (to the left of)
• -2x – 4 > 4  -2x > 8  x < -4
• -2x – 4 < -4  -2x < 0  x > 0
-10
-8
-6
-4
-2
0
2
4
6
8
10
What is the sum of the
whole numbers from 1 to
100? In other words, 1 +
2 + 3 + 4 + 5 + … + 98 +
99 + 100 = ???
Solution
• 1 + 100 = 101. 2 + 99 = 101. 3 + 98 = 101.
4 + 97 = 101…. 50 + 51 = 101.
• There are 50 pairs of numbers that add up
to 101.
• 50 * 101 = 5050
• The sum of all the numbers from 1 to 100
is 5,050.
Quiz Bowl
Round 2: Multiple Answers–1 Minute
Round 2 will consist of 3 questions
that each have multiple answers.
Each team will earn 1 point for
each correct answer and 1 bonus
point if the team provides all of the
possible answers.
Mel, Pat, Ash, and Sal
meet for the first time
and all shake hands.
List all the handshakes.
Solution
• Mel – Pat, Mel – Ash, Mel – Sal
• Pat – Ash, Pat – Sal
• Ash – Sal
List all the prime
numbers less than 100
that are the sum of 3
(not necessarily
distinct) perfect cubes.
Solution
Cubes: 1, 8, 27, 64, 125…
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1 + 1 + 1 = 3 prime!
1 + 1 + 8 = 10
1 + 8 + 8 = 17 prime!
1 + 1 + 27 = 29 prime!
1 + 27 + 27 = 55
1 + 8 + 27 = 36
1 + 1 + 64 = 66
1 + 64 + 64 = too big
1 + 8 + 64 = 73 prime!
1 + 27 + 64 = even
•
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8 + 8 + 8 = 24
8 + 8 + 27 = 43 prime!
8 + 27 + 27 = even
8 + 27 + 64 = 99
8 + 8 + 64 = multiple of 8
8 + 64 + 64 = too big
27 + 27 + 27 = mult. of 3
27 + 27 + 64 = too big
27 + 64 + 64 = too big
64 + 64 + 64 = too big
A triangle has 3 sides,
the lengths of which are
whole numbers of
centimeters. Two sides of
the triangle are 3 cm and
5 cm, respectively. List all
the possible lengths for
rd
the 3 side.
Solution
If 5 is the longest side length, then
x + 3 > 5 and x ≤ 5, so x = 3, 4, 5.
If x is the longest side length, then x > 5 and
3 + 5 > x, so x = 6, 7.
x is any integer from 3 to 7 (including 3 and
7)
Quiz Bowl
Round 3: Speed Round with
Follow Up Questions
Round 3 will consist of 5 questions.
The first team to give a correct answer
to the runner will get 3 points.
The first team with the correct answer
will then receive a follow-up question
for 2 bonus points.
Q: A super tripledon is
found by taking an integer
rd
raising it to the 3 power,
then multiplying it by 3 and
then adding 3 to the result.
What is the super tripledon
of 3?
Solution
• 33 * 3 + 3 = 27 * 3 + 3 = 81 + 3 = 84
F: What number has a
super tripledon value
of 81,003 ?
Solution
•
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x3 * 3 + 3 = 81,003
x3 * 3 = 81,000
x3 = 27,000
x = 30
Q: Determine the slopeintercept equation of a
line that passes through
points (2,-3) & (5, 5).
Solution
• Slope:
– Change in y: 5 - -3 = 8
– Change in x: 5 – 2 = 3
– Ratio: 8/3
• Intercept:
– Going left 2 units results in going down 2 *
(8/3) units = 16/3 units
– Going left 2 and down 16/3 from (2, -3) yields
a y-intercept of (0, -25/3)
• y = (8/3)x – 25/3
F: What is the slope of
a line that passes
through the origin and
is perpendicular to the
previous line?
Solution
• Slopes of perpendicular lines are opposite
reciprocals.
• The opposite reciprocal of 8/3 is -3/8
• The slope is -3/8
Q: How many
distinguishable
arrangements are
there for the letters in
the word “Drexel”?
Solution
• 6 * 5 * 4 * 3 * 2 * 1 / 2 = 360 arrangements.
• If all the letters were distinct it would be 6 * 5
* 4 * 3 * 2 * 1 because there would be 6
choices for the first letter, times 5 for the
second, times 4 for the thirds, etc.
• But we divide by 2 because there are two
ways to arrange the indistinguishable e’s in
each of the words, so we double-counted, for
example xldree and xldree.
F: How many of those
arrangements start
with a “D”?
Solution
• 1 * 5 * 4 * 3 * 2 * 1 / 2 = 60 arrangements.
• One choice for the first letter (D), then 5
choices for the second, 4 for the third, etc.
• We again have to divide by 2 because we
are still double counting words like dlrxee
and dlrxee
Q: A right triangle has
side lengths of x, 5, and
12. What are the 2
possible exact values
for x?
Solution
Let’s say 5 and 12 are the legs, then
52 + 122 = x2.
I recognize that as a 5, 12, 13 triangle, but
could also solve: 25 + 144 = x2, so 169 = x2.
x = 13
The other case is that 12 is the hypotenuse.
52 + x2 = 122, or 25 + x2 = 144.
x2 = 144 – 25 = 119
x = √119
F: A triangle has sides
7, 9, 12. Classify the
triangle as acute, right
or obtuse.
Solution
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We can use the Pythagorean Theorem:
72 + 92 = 49 + 81 = 130
122 = 144
130 < 144
To be a right triangle 72 + 92 would have to
equal 122 but 122 is too big… so the
“hypotenuse” is too big for a right triangle
• That means this triangle is obtuse.
Q: Is the number of 2-digit
numbers in which both 2
digits are odd greater than,
less than, or the same as
the number of 2-digit
numbers in which both 2
digits are even?
Solution
• Half of the 2-digit numbers in the 10s, 30s,
50s, 70s, and 90s have both digits odd
• Half of the 2-digit numbers in the 20s, 40s,
60s, and 80s have both digits even.
• Because 00, 02, 04, 06, and 08 are not
considered 2-digit numbers, the number of
2-digit numbers with both digits odd is
greater.
F: How many 2 digit
numbers exist where
both 2 digits are odd?
Solution
• Half of the 2-digit numbers in the 20s, 40s,
60s, and 80s have both digits even.
• 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60,
62, 64, 66, 68, 80, 82, 84, 86, 88
• 20 two-digit numbers exist such that both
digits are even.