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Park Forest Math Team
Meet #2
Algebra
Self-study Packet
Problem Categories for this Meet:
1. Mystery: Problem solving
2. Geometry: Angle measures in plane figures including supplements and
complements
3. Number Theory: Divisibility rules, factors, primes, composites
4. Arithmetic: Order of operations; mean, median, mode; rounding; statistics
5. Algebra: Simplifying and evaluating expressions; solving equations with 1
unknown including identities
Meet #2 – Algebra
Ideas you should know:
! Common Fraction:
! Money Answers:
23¢
“What is One-Quarter of a dollar, minus two cents?”
$0.23
! Square Root of a Product:
$.23
Not 0.23¢
Not 23
“What is the square root of 14x21 x 6?”
! Slow hard way – “Multiply it out first”:
14x21=294,
294x6=1764,
“Um, Are we allowed to
use calculators? How are we supposed to do this?”
" Faster way – “Factor it first”:
From question:
14
21
6
Factor:
2x7
3x7
2x3
Regroup factors:
2x2
3x3
7x7
=3
=7
Now do !
Answer:
2x3x7=42
! “Five consecutive multiples of 5 have a sum of 250 …” problems:
40
45
50
55
60
Average=50
If 5 numbers add to 250, the average is 50. That’s also the middle number,
so the five numbers are 40,45,50,55,60. Often the problem will ask for the
2nd number times the 4th number (45 x 55 here).
Meet #2, Algebra
! “Four times the sum of a number and one is two more than
seven times an amount one less than the number.
Find the number.”
So confusing!
Make N be the number you don’t know, then translate the
words to Algebra:
Four times the sum of a number and one is two more than
4
x
(
N
+
1)
-2
=
Seven times an amount one less than the number
7
Or:
x
(
N
-
1)
4 (N+1) – 2 = 7 (N-1)
Find the number.
Solve for N.
First, distribute, then combine like terms:
4N + 4 – 2 = 7N – 7
4–2+7
9
(Add 7 – 4N to each side)
= 7N – 4N
= 3N
3=N
Then check this answer in the original problem!
“Four times (3+1) is two more than 7(3-1)”
Meet #2, Algebra
Yes, 14=14
!
Relatively Prime (review): Having no common factors besides 1. 100 and
99 are relatively prime, since the only prime factors of 100 are 2 and 5, and
99 has 3 and 11.
Are 6 and 10 relatively prime?
No, both share 2 as a factor.
Are 27 and 111 relatively prime?
No, both share 3 as a factor.
Are 35 and 66 relatively prime?
Yes, 35 is 5x7, 66 is 2x3x11.
How many natural numbers less than 10 are relatively prime to 10, counting
1 as relatively prime to everything?
Answer: 4: 1,3,7,9
How many whole numbers less than 17 are relatively prime to 17?
P3
! Subscripts like P3 : pronounced “P sub three”
If P = {2, 3, 5, 7, 11, 13, 17, 19, 23, …} then
P1 = 2
P2 = 3
P3 = 5
If Pn=19, what is n?
P3 just means the 3rd P in a list. For example, if M is the set of how much
money Anna, Bridget, and Caroline have, MAnna (pronounced M sub Anna) is
how much money just Anna has.
If T is the set of multiples of 3: T = {3, 6, 9, 12, …} then T2=6, and Tn=3n
Answer to Pn=19: n=8.
! Time for some real problems from previous meets.
Meet #2, Algebra
Category 5
Algebra
Meet #2 - November, 2013
1) Sam has three more eggs than Sham but twice as many eggs as Faro.
If there are 62 eggs in all, then how many eggs does Sham have?
2) If X + 3Y - 4A = 17 and X + 7Y - 4A = 45, then what is the
value of 3X - 12A ?
3) The formula that converts a temperature in Celsius (C) degrees to
Fahrenheit (F) degrees is
A roasted turkey is sufficiently cooked when a thermometer inserted
into the thickest part of the thigh registers 180 degrees Fahrenheit.
Priscilla only has a Celsius thermometer. How many minutes longer
must she roast the turkey if its current temperature is 70 degrees
Celsius and every five minutes in the oven produces a rise in
temperature of one degree Celsius? Round your answer to the
nearest minute.
ANSWERS
1) ______
2) ______
3) ______
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Solutions to Category 5
Algebra
Meet #2 - November, 2013
Answers
Answers
1)
1)
23
1) Let
2) - 12
3)
3)
61
X = the number of Faro's eggs
2X - 3 = the number of Sham's eggs
2X = the number of Sam's eggs
X + (2X - 3) + 2X
5X - 3
5X
X
2X - 3
=
=
=
=
=
62
62
65
13 (Faro)
23 (Sham)
2) Compare the two equations and find their difference:
Difference:
X + 3Y - 4A
X + 7Y - 4A
4Y
Y
=
=
=
=
17
45
28
7
Substituting 7 for Y into the first equation: X + 3(7) - 4A
X + 21 - 4A
X - 4A
Multiply both sides by 3:
3X - 12A
Done!!
=
=
=
=
17
17
-4
-12
Therefore, it would take an additional 61 minutes to roast the
turkey to perfection! Scrumptious!
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Meet #2
December 2011
Category 5 – Algebra
1. If you add
to the number , you’d get a number that’s
times one third of
. What is ?
2. Inheriting a large sum of money, Mr. Lazy decided he does not need to work
and can simply live off his fortune. He spends the same amount of money
every year. After
years, he realized he has
of the money he had
years
earlier. How many years will his fortune last overall?
3. A car drives from point
to point , then turns around and drives back to point
at twice the original speed.
The average speed for the round trip was
mph (miles-per-hour).
What was the car’s original speed?
Answers
1. _______________
2. __________ years
3. __________ mph
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Meet #2
December 2011
Solutions to Category 5 - Algebra
Answers
1. Writing this algebraically:
1.
2.
3.
. To solve we can multiply both sides by :
or
Note:
The original problem said “ times greater than one third of .”
Some students observed that this could be interpreted as “4/3 of A MORE than
1/3 of A,” in other words, 5/3 of A. In this case, A+30 = 5 A / 3, giving A=45.
This was judged a reasonable interpretation of the problem and so both answers
were allowed.
One former mathlete wrote: The issue is whether "4 times greater" means 4 times as big or 5 times
as big. The 5 interpretation makes sense since "400% greater" means 500% as much.
2. If we note the number of years the money will suffice for as
he spends
, then each year
of his initial fortune, and we know that:
(
) . Multiplying by
we get:
or
3. If we note the car’s original speed as , and the distance between points
as , then the trip from
to
took time
Overall the round-trip took time
and the trip back took time
to cover a distance of
so the average speed is
and
.
, and
and so
Note that the distance D does not affect the answer: For any distance, if we
drive it once at
mph, and then back at
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mph, the average speed is
mph.
Category 5 - Algebra
Meet #2, December 2009
1. During a basketball game, your team scored three times as many 2-point field goals
than it did 3-point field goals, and scored a total of 90 points.
How many field goals did your team score?
(There were no 1-point free-throws.)
2. The force by which an object in space is pulled by the Earth’s gravity is proportional
to
𝑚
𝑅2
where m is the mass of the object, and R its distance from the center of the
Earth.
Satellite #1 is orbiting Earth at a distance (from its center) of 5,200 miles.
Satellite #2 is orbiting Earth at a distance of 20,800 miles, and has half the mass of
satellite #1.
What is the ratio of the gravitational force on satellite #1 to that on satellite #2?
3. The product of 3 consecutive even natural numbers divided by their sum is 64.
What is the middle number?
Answers
1. _______________
2. _______________
3. _______________
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Solutions to Category 5 - Algebra
Answers
Meet #2, December 2009
1.
2.
3.
40
32 or 32:1
14
1. If we call the number of 3-point goals made G, then there were (3 ∙ 𝐺) 2-point goals
made, and the total score would be:
3𝑝𝑜𝑖𝑛𝑡𝑠 ∙ 𝐺 + 2𝑝𝑜𝑖𝑛𝑡𝑠 ∙ 3 ∙ 𝐺 = 9 ∙ 𝐺 = 90 𝑝𝑜𝑖𝑛𝑡𝑠.
So 𝐺 = 10, and the total number of field goals made is 4 ∙ 𝐺 = 40 (Ten 3-pointers
and thirty 2-pointers).
𝑚1
2. The ratio we seek is 𝑚 2
𝑅1 2
𝑅2
2
𝑅
𝑚1
𝑅1
𝑚2
, or ( 2 )2 ∙
and we know that
𝑚1
𝑚2
= 2 𝑎𝑛𝑑
𝑅2
𝑅1
=4
so that the ratio is 42 ∙ 2 = 32
3. If we call the middle number x then the problem is
If we simplify this a bit we get
𝑥∙(𝑥 2 −4)
3∙𝑥
= 64 =
𝑥−2 ∙𝑥∙(𝑥+2)
𝑥−2+𝑥+𝑥+2
𝑥 2 −4
3
= 64
Or 𝑥 2 = 3 ∙ 64 + 4 = 196
Therefore 𝑥 = 14. (𝑥 = −14 is a solution too, but we’re looking for a natural
number).
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Category 5
Algebra
Meet #2, November 2007
1. Two years ago, Bob was
2
as old as he will be in 6 years. In how many years
3
from now will Bob be 40 years old?
4 3
π r and the formula for Surface
3
Area of a sphere is SA = 4π r 2 . If the Volume of a given sphere is 972π , what
is the Surface Area of the same sphere? Express your answer in terms of π .
(note: 972π is an example of a number given "in terms of π ".)
2. The formula for Volume of a sphere is V =
3. The sum of seven consecutive multiples of 7 is 1078. What is the sum of the
second smallest and the second largest of these seven numbers?
Answers
1. _______________
2. _______________
3. _______________
Solutions to Category 5
Algebra
Meet #2, November 2007
Answers
1. 22
2. 324π
3. 308
1.
2
(b + 6)
3
2
b−2= b+4
3
1
b=6
3
b = 18
b−2=
So if Bob is 18 years old now, he
will be 40 in 22 more years.
2. 972π = 4 π r 3
3
4
972 = r 3
3
3
972 ⋅ = r 3
4
729 = r 3
9=r
SA = 4π 92
SO
SA= 4π 81 = 324π
3. If we call the middle of the seven numbers x, we could use
this equation :
( x − 21) + ( x − 14 ) + ( x − 7 ) + x + ( x + 7 ) + ( x + 14 ) + ( x + 21) = 1078
7 x = 1078
x = 154
Since we want the sum of the 2nd largest and 2nd smallest,
we are looking for :
( x − 14 ) + ( x + 14 ) = 2 x = 2(154) = 308
Category 5
Algebra
Meet #2, December 2005
1. Sammy Squirrel, Sally Squirrel, and Sydney Squirrel have stored a total of 219
acorns for the winter. Sammy then gives 12 acorns to Sally and 12 acorns to
Sydney. Now Sammy and Sally have the same number of acorns, but Sydney has
18 fewer than each of them. How many acorns did Sydney have originally?
2. Each shape in the picture below represents a number. Same shapes have the
same number. If the triangle has a positive value, what number goes in the
hexagon?
÷5=
3×
12 +
–
=
×
=
= 49
(-2)3 =
3. The formula for the sum of the squares of the numbers from 1 to n is:
12 + 2 2 + 3 2 +
+ n2 =
n(n + 1)(2n + 1)
6
Find the sum of the squares of the numbers from 1 to 25.
Answers
1. _______________
2. _______________
3. _______________
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Solutions to Category 5
Algebra
Meet #2, December 2005
Answers
1. 49
2. 13
3. 5525
1. Let x, y, and z represent the original number of acorns
stored by Sammy, Sally, and Sydney respectively. Then
x + y + z = 219. After Sammy gives 12 acorns to Sally
and 12 acorns to Sydney, he has x – 24 acorns, Sally has
y + 12 acorns, and Sydney has z + 12 acorns. Let’s
imagine that Sydney receives another 18 acorns from an
anonymous squirrel. Sydney would then have z + 12 +
18 = z + 30, which is the same amount as the other
squirrels. The new total would be 219 + 18 = 237
acorns, and we would know that x – 24 = y + 12 = z + 30.
Dividing 237 by 3, we find that each squirrel would have
79 acorns. Now we can solve z + 30 = 79 for z. Sydney
must have had 79 – 30 = 49 acorns originally.
2. The value of the pentagon can be computed directly:
3
(−2) = −8. Also, the triangle must be 7, since 7 × 7 =
49. Now we can find that the circle equals 12 + 7, which
is 19. The value of the square must then be
3 × 19 – (-8) = 57 + 8 = 65. Finally, the value of the
hexagon is 65 ÷ 5 = 13.
3. Substituting 25 in place of n in the formula, we get
12 + 2 2 + 3 2 +
=
+ 25 2 =
25(25 + 1)(2 ⋅ 25 + 1)
6
25 ⋅ 26 ⋅ 51
= 25 ⋅13⋅17 = 5525
6
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Category 5
Algebra
Meet #2, November 2003
1. Pick’s formula gives the area of any polygon whose vertices are on the lattice
points of a square grid. Such polygons can be made with rubber-bands on a peg
board such as the Geoboard. Pick’s formula can be stated as follows:
1
A = B + I −1
2
where A is the area, B is the number of pegs touching the border of the figure (at
vertices and along sides), and I is the number of pegs in the interior of the
polygon. A certain polygon is made with rubber bands on a peg board and has an
area of 70.5 square units. If there are 60 pegs in the interior of the polygon, how
many pegs must be on the border of the figure?
2. Four times the sum of a number and two is five more than seven times an
amount six less than the number. Find the number.
3. The sum of three consecutive multiples of 120 is the same as the sum of five
consecutive multiples of 96. If the least of the three multiples of 120 is 840, then
what is the value of the greatest of the multiples of 96?
Answers
1. _______________
2. _______________
3. _______________
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Solutions to Category 5
Algebra
Meet #2, November 2003
Answers
1. 23
2. 15
3. 768
1. Substituting the value 70.5 for A and 60 for B in
Pick’s formula, we get:
1
1
70.5 = B + 60 −1 or 70.5 = B + 59
2
2
1
Solving for B, we get 11.5 = B, then 23 = B .
2
There must be 23 pegs on the border of the figure.
2. Translating the English to Algebra, we get:
4 (x + 2) = 5 + 7(x − 6)
4 x + 8 = 5 + 7x − 42
4 x + 50 = 5 + 7x
45 = 3x
x = 15
3. If the least of the three multiples of 120 is 840, then the first multiplier must be
7, since 7 × 120 = 840. The next two multiples of 120 are 8 × 120 = 960 and 9 ×
120 = 1080. The sum of these is 840 + 960 + 1080 = 2880. Now, if the sum of
five multiples of 96 is also 2880, then we can let n be the first multiplier and write
the equation: 96n + 96(n + 1) + 96(n + 2) + 96(n + 3) + 96(n + 4 ) = 2880
This can be simplified by factoring out the common 96 and then solving for n:
96(n + n + 1 + n + 2 + n + 3 + n + 4 ) = 2880
96(5n + 10) = 2880
480n + 960 = 2880
480n = 1920
n=4
We want the greatest multiple of 96, which is 96(n + 4 ) = 96(4 + 4 ) = 96 × 8 = 768.
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