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Solving Percent (%) Problems with “I” Chart
When you see a percent problem, one way to solve it is using an “I” chart set up as
shown below.
fraction
side
Part
%
side
Whole
100%
Ex) 75% of what is 17? You can look at the question and remove unnecessary
wording (always the “of” phrase). You want to remove the preposition. Whenever
you read what you have left, it should be a complete sentence, so 75% is 17. This
information can be put into the “I” chart.
fraction part
17
N
whole
%
75%
100%
17 75
N = 100 cross multiply and 75N = 1700. Divide both sides by 75 and
N≈22.67
*completion statement* 75% of 22.67 is 17
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Solving Percent (%) Problems with the 10% Method
To find: (MEMORIZE)
10% - move the decimal one place to the left because you are dividing your whole by
10. 10% is 1 of the whole
10
5% - half of what you got for 10% (5 is half of 10)
1% - move the decimal two places to the left because you are dividing your whole by
100. 1% is 1 of the whole
100
.5% - half of what you got for 1% (.5 is half of 1)
When using the 10% method, think in terms of money. Round to the nearest penny.
Ex) 25% of 42 is what?
Ask yourself:
• How many dollars do I have? - 10 dollars
• How many dimes do I have:? - 5 dimes
• How many pennies do I have? - 0 pennies
10% = 4.20
10% = 4.20
+
5% = 2.10
(Add from left to right and estimate instead of from right to left.
25% = 10.50
This will MINIMIZE ERRORS!)
25% of 42 = 10.5.
Check using the “I” chart
N
25%
42
100%
N
25
=
100N=1050 N=10.5
42 100
*10% method is best on a NONCALCULATOR TEST.
You can also do the old way – change % to decimal and multiply by whole.
42
X .25
210
840
10.50
2-2
Consumer Problems
*Look at consumer problem hand out.
Step 1 – set up 3 columns Whole
%
?
Whole is the original amount. ? is what is being asked for. Note whether you need to
add/subtract after finding the percent. If the whole or percent are missing, use “I”
chart to solve, otherwise use 10% method.
Ex) Mary went to the store to buy shoes. The $30 pair of shoes was on sale for 40%
off the original price. What was the sale price?
?
Whole (original) %
30.00
40%
Sales price( - )Need to subtract
40% of 30.00
When I find 40% of $30.00, I am finding the
10% = 3.00
amount taken off the original price. Need
10% = 3.00
to subtract % off from original price to get
10% = 3.00
the sales price.
10% = 3.00
40% = 12.00
$30.00-$12.00=$18.00
Whole-discount=sales price
Common mistake – finding the percent of the percent instead of percent of the
whole.
Common mistake – students just find the tip, tax or discount when they sometimes
want you to find the new total. – Read carefully!
2-3
Frequency Distribution Tables
All graphs tell a story. The job is to interpret what that story is. The title is the main
part of the story. Graphs are visual representations of data sets.
Frequency is how often something occurs.
Table headings:
Intervals – Should not be too big or too small. They must be consecutive. They
cannot overlap. Intervals should be equal. (Sometimes the last interval is not equal
but this is not standard.) Frequency charts show where there are gaps or clusters of
data. Easier to visualize data.
Tally – must match number of data entries.
Frequency – number of tally marks in an interval.
Cumulative frequency – add consecutive frequencies. Total must equal number of
data entries.
Ex) High temperatures in the month of January in Amherst, VA. 25, 44, 40, 27, 30, 32,
39, 37, 44, 48, 52, 45, 32, 30, 35
Step one – order data from least to greatest and ensure all data is included. (Count
data entries to check.)25, 27, 30, 30, 32, 32, 35, 37, 39, 40, 44, 44, 45, 48, 48
High Temperatures in January in Amherst VA
Intervals
25-27
28-30
31-33
34-36
37-39
40-42
43-45
46-48
49-51
52-54
Tally
ll
ll
ll
l
ll
l
lll
l
l
Frequency
2
2
2
1
2
1
3
1
0
1
2-4
Cumulative Frequency
2
4
6
7
9
10
13
14
14
15
Histograms
Histograms are a type of bar graph. The bars in a histogram touch unless there is a
gap in the data. The categories (intervals) are consecutive, equal, and don’t overlap.
Intervals are equal so bar width should be the same. Histograms are similar to
frequency charts.
Gaps and clusters are easily seen in a histogram. Gaps are empty intervals (there is
no data). Cluster is where there is a group of data entries.
Scale – smallest number to largest number.
Ex) Create a histogram of 6th grade math test scores. Scores: 57, 68, 73, 73, 78, 78,
83, 83, 83, 85, 92, 100, 100
Order data least to greatest. Intervals usually go on the “x” axis and frequency
usually goes on the “y” axis.
Number of students
Gap
Cluster
Gap
5
4
3
2
1
56-60
61-65
66-70
71-75
76-80
81-85
86-90
91-95
96100
Intervals
Intervals in example are 5. Common mistake is to subtract smallest from biggest 6056=4 but interval is 56, 57, 58, 59, 60 (five). The interval will always be one more
than it looks.
Scale in the example is 56-100.
2-5
Line Plots
Line plots are another type of graph. With line plots you can find mean, median,
range and mode.
Measures of central tendency are:
Mean – average
Median – middle number (if there are two, find the average of the two)
Mode – the numbers that occur the most
Range – difference between the least and greatest numbers
Ex) Plot the amount of homework assignments students completed out of 20.
Data: 3, 7, 7, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 17, 20, 20, 20
Order data from least to greatest. Count the number of data entries. Make a number
line that includes the lowest and highest data.
X
2
4
6
Key X = 1 student
X
X
8
X
X
X
X
X X X
X X X
10
12
X
X
X
20
X
14
16
18
22
Common mistake – students don’t look at the key. (Sometimes X = more than one
entry.)
To find range – subtract lowest data recorded from highest data recorded. Common
mistake – student subtract lowest number on number line from greatest number on
number line.
For lines with big gaps, use a zigzag to show this.
5 10 15
2-6
100
Stem and Leaf
Look at questions for graphs handouts.
Stems must be consecutive. If no data, leave leaf blank. Columns of numbers should
line up in the leaves. Blanks in the leaves show gaps in the data.
Ex) Test scores: 37, 48, 68, 74, 75, 77, 77, 82, 82, 85, 88, 88, 90, 90, 97, 97, 97, 100,
100
Order from least to greatest and count number of data entries.
Stem (tens)
Leaf (ones)
3
7
4
8
5
6
8
7
4577
8
22588
9
00777
10
00
Key 3/7= 37
Columns of number should line up in the leaves. Blanks in the leaves show gaps in
the data.
1. How many students scored at least a 93? 5
2. How many more students scored in the 80’s than in the 50’s? 5
3. What is the mode? 97
4. How many students scored at most 85? 10
Know what these terms mean: (examples from chart above)
At most – that number and less (at most a 74 – 4)
At least – that number and more (at least an 82 – 12)
Greater than – just more than the number (greater than 90 – 5)
Less than – just less than the number (less than 68 – 2)
2-7
Probability – Experimental vs. Theoretical
*Look at probability handout (Math Discovery). Make sure you understand
questions.
Terms:
Probability – chance of something occurring.
Experimental – after trials
Theoretical – expected before trials
Theoretical probability is the expected probability of an event occurring.
Experimental probability is the probably of an event occurring determined by
carrying out a simulation or experiment (trials)
The law of large numbers – the more trials, the closer you get to the theoretical
probability.
Any time you do any problem
numerator
*Always think in terms of a formula:
with probability, think about
deno min ator
the formulas before you
Theoretical probability: # favorable
# possible
approach it.
Experimental probability:
# favorable
# trials
Ex) Using a six-sided number cube
1) What is the probability of getting a multiple of 2?
P(multiple of 2) = # favorable = 3 = 1 =0.5= 50% because # favorable is 3 (2,4,6) #
# possible
6
2
possible is 6 (6 sides to the cube)
2) What is the probability of not getting a 4?
P(not 4) - # favorable = 5 = .83 – 83.3% . because # favorable is 5 (1,2,3,5,6) # possible
# possible
6
is 6 (6 sides to the cube)
2-8
Probability and Predictions
When making a prediction, half of the proportion is going to be the theoretical probability.
Ex) Joe rolled a number cube 30 times. He got a 5, 17 out of the 30 times. If he rolled the number
cube 300 more times, how many times would you expect him to get a 5?
# favorable 1
n
= =
# possible 6 300 Cross multiply to get 6n=300. Then divide both sides by 6 to get n=50
Common mistakes:
#1)
17
n
=
in this case the experimental probability was used instead of the theoretical
30 300
probability.
1
6
#2) =
n
in this case the 30 trials were included in the formula and they should not be part of
330
the prediction.
The prediction could also be made using proportional reasoning, an “I” chart, or the 10%
method.
Prediction using proportional reasoning.
# favorable 1
n
= =
6•50=300 so 1•50=n. ( n=50) We
# possible 6 300
would expect to get a 5, 50 out of the next 300 trials or 16.66%
Prediction using “I” chart
fraction
part %
N
16.7% (rounded 16.66%)
300
whole 100% cross multiply and 100N≈5010. Divide both sides by 100
and N≈50.1
Prediction using 10% method 16.5% (rounded 16.66% to nearest half percent)
16.5%
300
10
=
30.00
5
=
15.00
1
=
3.00
.5 =
1.5
16.5 =
49.50
2-9
Tree Diagrams
*Tree diagrams are a tool used to show all possible outcomes of a situation
(theoretical probability). The list of these outcomes is the *sample space.
To set up the tree diagram first determine the number of items or categories, then the number
of possible outcomes for each. *Possible outcomes must be drawn for each possible outcome
from the first item/category. In other words, there are 6 possible outcomes in the 2nd category or
item below. You must draw each outcome from each outcome in the first category.
Ex) Using a 6 sided number cube and a coin determine all possible outcomes. Items/possible
outcomes Coin (2 possibilities)
Number cube (6 possibilities)
1
2
3
4
5
6
H
Possible Outcomes
1
2
3
4
5
6
T
If I get heads first, what
are my possible outcomes
for the number cube?
1,2,3,4,5,6
If I get tails first, what are
my possible outcomes for
the number cube?
1,2,3,4,5,6
Sample space: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. Count the branches to determine all
possible outcomes. There are six possible outcomes: 1-6 Heads and 1-6 Tails.
Problems:
# favorable 3 1
= = =.25=25%
# possible 12 4
# favorable 9 3
P(tails or even)
= = =.75=75%
# possible 12 4
P(tails and even)
Tree diagrams get more complicated if the categories are all the same. Try three coins.
2-10
Fundamental Counting Principal (FCP)
FCP is a way to find the total number of outcomes without using a tree diagram.
Multiply the possible outcomes in each category to get the total number of possible
outcomes in the sample space. You cannot get the list in the sample space using FCP.
Ex) Using a 6 sided number cube and a coin determine all possible outcomes.
Items/possible outcomes Coin (2 possibilities) Number cube (6 possibilities)
2
x
6
2•6 = 12
Common mistakes: When all the categories are the same do not just multiply the number of
items by possible outcomes to determine the sample space. You must multiply the number of
options in each category by the number of options for each category.
Ex) 3 coins with two options (heads/tails). It is not correct to multiple 3 times 2 and get a sample
space of 6. You must multiply 2• 2 • 2=8
Ex) What are all the possible numbers in a pick 3 lottery with 0-9 on each category.
3•10 = 30 WRONG (only counted number of items multiplied by options)
3•9 = 27 WRONG (same mistake as above and counted the wrong interval)
10•10•10 = 1,000 is correct sample space.
2-11
Compound Events – 2 or more simple events
Independent compound events – if the outcome of one event does not influence he
occurrence of the other event.
To solve independent compound events, break into simple events and mulitiply
them.
Ex) Using a 6 sided number cube and a coin what is the probability of flipping a head
and getting a 3 on the number cube?
P(heads, 3)=? is the same equation as P(heads and 3)=? Because the comma
represents “and.”
P(heads) = 1 and P(3) = 1 Multiply 1 • 1 =
2
6
getting a 3 on the number cube is
1
12
2 6
1
. This
12
The probability of flipping a head and
can be checked using a tree diagram and
determining the sample space.
Coin (2 possibilities) Number cube (6 possibilities)
1
2
3
H
4
5
6
Possible Outcomes
1
2
3
T
4
5
6
Sample space: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. The probability of
flipping a head and getting a 3 on the number cube is 1 .
12
Fair game means there is equal chance for everybody to win.
2-12