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Carlisle Math Team
Meet #1 – Category 4
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M1C4
Arithmetic
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Self-study Packet
1.
2.
3.
4.
Mystery: Problem solving
Geometry: Angle measures in plane figures including supplements and complements
Number Theory: Divisibility rules, factors, primes, composites
Arithmetic: Order of operations; mean, median, mode; rounding; statistics
5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown
including identities
For current schedule and information,
See http://imlem.org
M1C4
ARITHMETIC
Operations
Operations means things like add, subtract, multiply, divide, squaring, etc. If it
isn't a number it is probably an operation.
When you see something like 7 + (6 × 52 + 3), what part should you calculate
first? Start at the left and go to the right? Or go from right to left?
Warning: Calculate them in the wrong order, and you will get a wrong answer !
So, long ago people agreed to always follow certain rules when doing
calculations, and they are:
Order of Operations
Do things in Parenthesis First. Example:
6 × (5 + 3)
= 6×8
= 48
6 × (5 + 3)
= 30 + 3
= 33 (wrong)
Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract.
Example:
5 × 22
= 5×4
= 20
2
2
5×2
= 10
= 100 (wrong)
Multiply or Divide before you Add or Subtract. Example:
2+5×3
= 2 + 15
= 17
2+5×3
= 7×3
= 21 (wrong)
Otherwise just go left to right. Example:
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30 ÷ 5 × 3
= 6×3
30 ÷ 5 × 3
= 30 ÷ 15
1
8
= 2
=
(wrong)
ARITHMETIC
How Do I Remember ? PEMDAS !
P
E
MD
AS
This can
Parenthesis first
Exponents (ie Powers and Square Roots, etc.)
Multiplication and Division (left-to-right)
Addition and Subtraction (left-to-right)
also be remembered as "Please Excuse My Dear Aunt Sally".
How to Find the Mode or Modal Value
The mode is simply the number which appears the most. To find the mode or
modal value requires you to put the numbers you are given in order.
Look at these numbers:
3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56,
23, 29
In order these numbers are:
3, 5, 7, 12, 13, 14, 20, 23, 23, 23, 23, 29, 39,
40, 56
This makes it easy to see which numbers appear the most.
In this case the mode or modal value is 23.
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ARITHMETIC
How to Find the Mean
The mean is just the average of the numbers.
It is easy to calculate: Just add up all the numbers, then divide by how many
numbers there are. (In other words it is the sum divided by the count)
Example 1:
What is the Mean of these numbers?
3, 10, 5
Add the numbers: 3 + 10 + 5 = 18
Divide by how many numbers (ie we
added 3 numbers): 18 ÷ 3 = 6
The Mean is 6
Example 2:
Look at these numbers:
3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56,
23, 29
The sum of these numbers is equal to 330
There are fifteen numbers.
The mean is equal to 330 ÷ 15 = 22
The mean of the above numbers is 22
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ARITHMETIC
How to Find the MEDIAN
To find the Median, place the numbers you are given in value order and find
the middle number.
Look at these numbers:
3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56,
23, 29
If we put those numbers in order we have:
3,
5, 7, 12, 13, 14,
21, 23, 23, 23, 23, 29, 39, 40, 56
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39,
40, 56
The median value of this set of numbers is 23.
BUT, if there are an even amount of numbers things are slightly
different.
In that case we need to find the middle pair of numbers, and then
find the the value that would be half way between them. This is easily done by
adding them together and dividing by two. An example will help:
3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23,
29
If we put those numbers in order we have:
3, 5, 7, 12, 13, 14,
21, 23, 23, 23, 23, 29, 40, 56
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40,
56
In this example the middle numbers are 21 and 23.
To find the value half-way between them, add them together and divide by 2:
And, so, the Median in this example is 22.
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ARITHMETIC
“Missing quiz grade to make average be …” problems
۞ You got 91 and 85 on the first two quizzes. What do you need on the 3rd quiz to get
a mean of 92? Solution: For a 3-quiz mean of 92, you need a total of 3x92=276
points. So far, you have 91+85=176 points. You need 100 on the 3rd quiz.
Another way to think about this problem: “We want an average of 92. 91 is one below,
or -1. 85 is seven below, or -7. That’s 8 below. We need 8 above, or 92+8=100.
This saves a lot of work.
۞ Sam got an average of 93 on the first 3 quizzes. What is the lowest grade he could
get on the 4th quiz and still have a 90 average? Solution: A 4-quiz average of 90
means 4x90=360 total points. So far, Sam has 3x93=279 points. 360-279=81, so 81
is the lowest he could get.
Another approach: “We want an average of 90, but we got 9 ‘extra’ points (three 93’s,
essentially.) So, we can afford to be 9 below 90, or 81.
“Pile of X’s” problems (histograms)
Example:
Grades on Quiz
6
7
8
9
10
Number of Students
X
XX
XXXX
X
XX
Here, the lowest grade is a 6, and one student got that (one X.) The most common
grade was an 8 with 4 students. That’s the mode. To find the median, you can count
all the X’s (1+2+4+1+2 = 10) and then find the “middle” values, which would be the
5th and 6th both of which got 8, so 8 is the median. To find the mean, you find the total
and divide by 10. The total is 6+2x7+4x8+9+2x10 or 81, so the mean is 81/10 = 8.1.
(There are usually quicker ways to find the total or the mean.)
Rounding Decimals
First you need to know if you are rounding to tenths, or hundredths, etc. Or
maybe to "so many decimal places". That tells you how much of the number will
be left when you finish.
Examples
3.1416 rounded to hundredths is 3.14
1.2635 rounded to tenths is 1.3
1.2635 rounded to 3 decimal places is
1.264
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Because ...
... the next digit (1) is less than 5
... the next digit (6) is 5 or more
... the next digit (5) is 5 or more
ARITHMETIC
Rounding Whole Numbers
You may want to round to tens, hundreds, etc, In this case you replace the
removed digits with zero.
Examples
Because ...
134.9 rounded to tens is 130
... the next digit (4) is less than 5
12,690 rounded to thousands is 13,000
... the next digit (6) is 5 or more
1.239 rounded to units is 1
... the next digit (2) is less than 5
“Evaluate the following expression” problems
There’s one of these every meet. If you truly know order of operations, you can do it,
but many have trouble. For example, 1+19*3. Sometimes we think “19 times 3 is
hard, so I’ll do the 1+19 first, giving 20, and 20 times 3 is easy.” Oops! The correct
answer is 58.
Another common mistake is to write on top of the problem, scribbling out parts as
they’re done, so we can’t read it any more, and we can’t check our work. One trick is
to write ABOVE the problem the pieces as you do them, so you can check:
25
–
5
21
Answer: 4
3
9– 4
5 x (32 – 22) – 7 x (2 + 1)
What is 2 + 98 x 3 ?
It sure is tempting to say 300, but it’s 296.
What is 2+3*42 ? Answer: 50.
42 is 16, 3*16=48, 2+48=50.
What is 2 x (5 + 90) ? Answer: 190.
5+90=95, 2x95 = 190.
What is 2 x [1 + 2 x {3 + (1x3)} + 1]? Answer: 28. 2 x (1+2x6+1)=2x14.
Sometimes there are other “groupings” besides parentheses, such as the bar or
3
vinculum:
(of course, you do the 1+2 before doing the division!)
1 2
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ARITHMETIC
Category 4
Arithmetic
Meet #1, October 2004
1. Evaluate the following expression.
100  48  2  3
2
Hint: Remember order of operations – PEMDAS. Should you start with 48÷2 ?
2. Leslie scored 98, 88, 92, 86, 88, and 98 on the 
first six math quizzes. Larson scored 89, 95,
78, 80, 96, and 95 on the first six math quizzes. What is the positive difference between the
median of Leslie’s quiz scores and the median of Larson’s quiz scores.
Hint: A median is the ‘middle’ number – but what if you have TWO middle numbers?
3. A teacher gave her students little packages of m&m’s. Each X in the line plot below
represents the number of m&m’s in a student’s package. As you can see, there were not
enough packages for everyone. To make it fair, the people who had more than the average
number of m&m’s gave some to those people who had less than the average, until everyone
had the same number of m&m’s. If Roger is the student who got the package with 26 m&m’s,
how many m&m’s did Roger have to give away?
X
2 students got X
21 m&m’s
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
0
20
21
22
23
24
25
26
Answers
1. _______________
2. _______________
3. _______________
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Hint: Three students got 0, one student got 20, two got 21, etc. How many
students are there total (how many X’s)? What is the total number of m&m’s?
How many m&m’s should each student have to make it fair?
ARITHMETIC
Solutions to Category 4
Arithmetic
Meet #1, October 2004
Answers
1. Using the order of operations, we get:
1. 14
100  48  2  3 100  24  3 100  72 28



 14
2
2
2
2
2. 2
3. 6
2. To find the median of a set of data, we must first arrange the data
from least
to greatest. Leslie’s quiz scores are 86, 88, 88, 92, 98, 98
and Larson’s quiz scores are 78, 80, 89, 95, 95, 96. The median of
Leslie’s scores is half way between 88 and 92, which is 90. The
median of Larson’s scores is half way between 89 and 95, which is
92. The positive difference between these medians is 2.
3. The total number of m&m’s is 3  0 + 1  20
+ 2  21 + 5  22 + 4  23 + 5  24 + 2  25 + 1  26 = 0 + 20 + 42
+ 110 + 92 + 120 + 50 + 26 = 460. There are 23 X’s on the line plot,
which means there are 23 students in the class. Everyone should get
460 ÷ 23 = 20 m&m’s each. This means Roger will have to give
away 26 – 20 = 6 m&m’s.
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ARITHMETIC
Category 4
Arithmetic
Meet #1, October 2003
1. Evaluate the following expression:
1 5 2  2 4   2  5 2  2 4   3  5 2  2 4 
2
2
2
Hint: Remember order of operations – PEMDAS. Start inside the (parentheses)! What is 5 2-24?
given out as follows:
2. The prize money at the raffle was
20 people received $5
10 people received $10
5 people received $20
2 people received $50
1 person received $100
1 person received $500
1 person received $1000
How much greater was the mean (average) of the prize money than the median?
Hint: How many people won something? What is the total prize money, starting with 20x$5=$100.
For the median, imagine a list of twenty 5’s, ten 10’s, etc. and find the “middle” number(s).
3. The upper quartile is the median of the upper half of a set of data and the lower quartile is
the median of the lower half. The interquartile range is the difference between the upper
quartile and the lower quartile. Find the interquartile range of the data shown in the line plot
below.
Answers
1. _______________
2. _______________
3. _______________
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Hint: First find the median, so you know what the upper half and lower half are.
Then find the median of each half. These are the quartiles.
ARITHMETIC
Solutions to Category 4
Arithmetic
Meet #1, October 2003
1. It is helpful to notice that the same expression, 5 2  2 4  , appears
in three places in the larger expression. Evaluating this by itself,
according to the order of operations, we get:
2
5 2  2 4   25 162  9 2  81

Substituting this value in all three places in the original expression,
we get:
181 2 81 381

From here we can proceed according
to the order of opperations and
get 81 – 162 + 243 = 162.
2
Answers
1. 162
2. $42.50
3. 6

2. A total of 40 people received a total of $2000 in prize money, for a mean (average) payout
of $50. Since half the people received only $5, this figure can be misleading. Actually, half
the prize money went to one person. The median payout falls halfway between $5 and $10,
which is $7.50. The mean was thus $50 – $7.50 = $42.50 more than the median.
3. The median of the data is 17.5, which means that half the data is greater than 17.5 and half
the data is less than 17.5. The upper quartile is 21 and the lower quartile is 15. The
interquartile range is 21 – 15 = 6.
M1C4
ARITHMETIC
Category 4
Arithmetic
Meet #1, October, 2002
1. Evaluate the expression below. Express your result as a decimal rounded to the nearest
tenth.
38 3 22  543 38
72 4  2
Hint: Remember order of operations – PEMDAS. What is 43-38? What is 22? Then, what is 3x22?
2. The line plot below shows the ages of the students who went on the ski trip. Each X
represents a person with the age indicated by the number below it. If A is the mode of the
data, B is the median of the data, and C is the number of students on the trip, find the value of
2
2
B A
C
. Express your answer as a mixed number in simplest form.
Two 15-year-olds
went on the trip
Hint: You DON’T need the average. Mode: most common age (most X’s). Median: just as many X’s on each side.
3. Max’s average on his first six quizzes was 88. After two more quizzes, his average was 90.
What is the average of his 7th and 8th quiz scores?
Answers
1. _______________
2. _______________
3. _______________
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Hint: What was his TOTAL score on his first six quizzes? (Total / 6 = 88)
What will the new total of all 8 quizzes need to be? (New Total / 8 = 90)
ARITHMETIC
Solutions to Category 4
Arithmetic
Meet #1, October, 2002
Answers
1. 1.4
2.
1
3
20
1. Following the order of operations to evaluate the expression, we
get:
38 3 22  543 38 3812 55 26 25


72 4  2
18 2
36
51 17


36 12 about 1.4 to the nearest tenth.
3. 96
2. The line plot shows a mode of 11, a median of 12, and the
2
2
B A
C
number of students is 20. Thus, the expression
can be
2
2
12 11 144 121 23
3


1
20
20
20
20
evaluated as follows:
3. If Max’s average on his first six quizzes was 88, we know that the
sum of the six quiz scores was 88 6  528. Similarly, after
eight quizzes, when his average was 90, the sum of his 8 quiz scores
must have been 90 8  720. The difference, 720 528 192, is
the sum of the seventh and eighth quiz scores. The average of these
two is thus 192 ÷ 2 = 96.
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ARITHMETIC
Category 4
Arithmetic
Meet #1, October, 2001
1. Jason and Clara found different decimal values for the following expression. Jason rounded each quotient
to the nearest tenth and then added the results. Clara added the exact values of the quotients and then rounded
the result to the nearest tenth. What is the positive difference between their answers?
15  8  17  20
Hint: The trick is not getting confused. First, do the divisions exactly and add and round (Clara’s way.)
2. Find the value of the expression


100  3 6  7  8  5 2  2  1  3  6
Hint: Remember order of operations – PEMDAS. Start on the inside with 2+2 and work your way out.
3. Find the mean (average) value of the following expressions. Round your answer to the nearest whole
number.



32  4  5 
3  42  5 
3  4  52 
Answers
1. _____________
2. _____________
Hint: Remember PEMDAS. Do each and then add the results.
3. _____________
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ARITHMETIC
Solutions to Category 4
Arithmetic
Meet #1, October, 2001
Answers
1. 0.1
2. 25
1. Jason would round 15  8  1875
to 1.9 and
.
17  20  0.85 to 0.9, obtaining a sum of 2.8. Clara
would round the sum 1875
.
 0.85  2.725 to 2.7.
The positive difference between their answers is
2.8  2.7  0.1 .
3. 44
2. Simplifying the expression a few steps at a time,
according to the order of operations, we get:


100  3 6  7  8  5 2  2  1  3  6
100  36  56  20  1  18
100  36  37  18
100  325
100  75  25
3. First we need to find the value of each expression.
2
 3  4  5  9  4  5  36  5  41
2
 3  4  5  3  16  5  48  5  53
2
 3  4  5  12  25  37
Now, to find the mean we add up these values and divide
by 3. 41  53  37  131 and 131  3  43.6 , which
rounds to 44.
M1C4
ARITHMETIC
Category 4
Arithmetic
Meet #1, October, 2000
1. Sly knows that his quiz scores were 92, 85, 96 82, 88, and 91 for the first quarter, but he can’t remember
what he earned on the test. His teacher told him that his combined average for the quizzes and the test is
exactly 90 and that she counts the test as two quizzes. What score did Sly get on his test?
Hint: What is the total of the 6 quizzes? What must the grand total be of those 6 plus the test (counted twice)?
Grand Total / 8 = 90.
2. Old McDonald has 7 cats and 4 dogs. The average weight of the cats is 12 pounds and the average weight
of the dogs is 33 pounds. What is the average weight of Old McDonald’s eleven animals? Round your
answer to the nearest tenth of a pound.
Hint: If the average cat is 12 pounds, and there are 7 cats, what is the total cat weight?
3. The line plot shows the data Mr. Jones collected on his peas. Each X represents a pod that had the given
number of peas in it.
A = the mode of the data set
B = the range of the data set
C = the median of the data set
0
X
X
X X
X X
X X X
X X X X X X
X X X X X X X X X X
1 2 3 4 5 6 7 8 9 10 11
Number of Peas in the Pod
Find the value of .
Answers
1. _____________
2. _____________
3. _____________
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Hint: The “Range” is the difference between the lowest and highest
values, or 10-1 here.
A=
B=
C=
ARITHMETIC
Solutions to Category 4
Arithmetic
Meet #1, October, 2000
Answers
1. 93
2. 19.6
3. 13
1. Sly’s overall average of 90 comes from six quizzes
and one test, which counts as two quizzes. We multiply
90 by 8 to find the sum of these eight scores, which is
720. Since the six quiz scores add up to 534, the
difference , or 186, must be the test score counted twice.
Dividing by 2, we find that Sly must have earned a 93 on
the test.
2. The total weight of the animals can be found without
knowing the weights of any of the individual cats or
dogs. We multiply the number of cats by their average
weight and the number of dogs by their average weight
and then add these together. Thus we have pounds in
cats and pounds in dogs, which is a total of pounds of
animals. Dividing 216 by 11 and rounding to the
nearest tenth, we find that the average weight of Old
McDonald’s animals is 19.6 pounds.
3. A = 4 (the mode of the data set); B = 9 (the range of
the data set); C = 6 (the median of the data set).
Substituting these values into gives us:
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ARITHMETIC