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7th Grade Math
Review of 6th Grade
www.njctl.org
2015-01-14
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Table of Contents
Fractions
Number System
Decimal Computation
Click on the topic to go to that section
Expressions
Equations and Inequalities
Ratios and Proportions
Geometry
Statistics
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Fractions
Return to Table of Contents
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List what you remember about fractions .
Hin
t
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We can use prime factorization to find the greatest common factor (GCF).
1. Factor the given numbers into primes.
2. Circle the factors that are common.
3. Multiply the common factors together to find the greatest common factor.
Greatest Common Factor
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1 Find the GCF of 18 and 44.
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2 Find the GCF of 72 and 75.P
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3 Find the GCF of 52 and 78.
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A multiple of a whole number is the product of the number and any nonzero whole number.
A multiple that is shared by two or more numbers is a common multiple.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...
Multiples of 14: 14, 28, 42, 56, 70, 84,...
The least of the common multiples of two or more numbers is the least common multiple (LCM) . The LCM of 6 and 14 is 42.
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There are 2 ways to find the LCM:
1. List the multiples of each number until you find the first one they have in common.
2. Write the prime factorization of each number. Multiply all factors together. Use common factors only once (in other words, use the highest exponent for a repeated factor).
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EXAMPLE: 6 and 8
Multiples of 6: 6, 12, 18, 24, 30Multiples of 8: 8, 16, 24
LCM = 24
Prime Factorization:
2 3 2 4
2 2 2
2 3 23 LCM: 23 3 = 8 3 = 24
6 8
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4 Find the least common multiple of 10 and 14.
A 2
B 20C 70D 140
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5 Find the least common multiple of 6 and 14.
A 10B 30C 42D 150
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6 Find the LCM of 24 and 60.
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Which is easier to solve?
28 + 42 7(4 + 6)
Do they both have the same answer?
You can rewrite an expression by removing a common factor. This is called the Distributive Property.
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The Distributive Property allows you to:
1. Rewrite an expression by factoring out the GCF.
2. Rewrite an expression by multiplying by the GCF.
EXAMPLE
Rewrite by factoring out the GCF:
45 + 80 28 + 635(9 + 16) 7(4 + 9)
Rewrite by multiplying by the GCF:3(12 + 7) 8(4 + 13) 36 + 21 32 + 101
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7 In order to rewrite this expression using the Distributive Property, what GCF will you factor?
56 + 72
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8 In order to rewrite this expression using the Distributive Property, what GCF will you factor?
48 + 84
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9 Use the distributive property to rewrite this expression:
36 + 84
A 3(12 + 28)B 4(9 + 21)C 2(18 + 42)D 12(3 + 7)
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10 Use the distributive property to rewrite this expression:
88 + 32
A 4(22 + 8)
B 8(11 + 4)
C 2(44 + 16)
D 11(8 + 3)
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Adding Fractions...
1. Rewrite the fractions with a common denominator.2. Add the numerators.3. Leave the denominator the same.4. Simplify your answer.
Adding Mixed Numbers...
1. Add the fractions (see above steps).2. Add the whole numbers.3. Simplify your answer. (you may need to rename the fraction)
Link Backto List
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11 3 10 2 10
+
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12 5 8 1 8
+
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13 Find the sum.
5 3 10
+ 7 5 10
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14 Is the equation below true or false?
True False
1 8 12
+ 1 5 12
3 1 12
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Don't forget to regroup to the whole number if you
end up with the numerator larger than the
denominator.
ClickFor reminder
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A quick way to find LCDs...
List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator.
Ex: and
Multiples of 5: 5, 10, 15
Ex: and
Multiples of 9: 9, 18, 27, 36
2 5
1 3
3 4
2 9
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Common DenominatorsAnother way to find a common denominator is to multiply the two denominators together.
Ex: and 3 x 5 = 15
2 5
1 3
1 3
x 5
x 5 5 15
2 5
6 15
x 3
x 3
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15 2 5 1 3
+
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16 3 10 2 5
+
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17 5 8 3 5
+
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18
A
5 3 4
+ 2 7 12
=
7 1612
B 8 4 12
C
7 5 8
D
8 1 3
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19
A
2 3 8
+ 5 5 12
=
7 1924
7 8 20
B
7 8 12
C
8 7 12
D
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20
5 2 10
5 5 12
A
3 1 4
+ 2 1 6
=
B
5 1 2
C
6 5 12
D
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Subtracting Fractions...
1. Rewrite the fractions with a common denominator.2. Subtract the numerators.3. Leave the denominator the same.4. Simplify your answer.
Subtracting Mixed Numbers...
1. Subtract the fractions (see above steps..). (you may need to borrow from the whole number)2. Subtract the whole numbers.3. Simplify your answer. (you may need to simplify the fraction)
Link Backto List
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21 7 8 4 8
Pul
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22 6 7
4 5 P
ull
Pul
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23 2 3
1 5 P
ull
Pul
l
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24 Is the equation below true or false?
True False
4 5 9
3 9
3 2 9
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25 Is the equation below true or false?
True False
2 7 9
1 9
1 2 3
1
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26 Find the difference.
4 7 8 2 3
8
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27 6 7 3 5
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A Regrouping Review
When you regroup for subtracting, you take one of your whole numbers and change it into a fraction with the same denominator as the fraction in the mixed number.
3 3 5
= 2 5 5
3 5
= 2 8 5
Don't forget to add the fraction you regrouped from your whole number to the fraction already given in the problem.
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5 1 4
3 7 12
5 3 12
3 7 12
4 1212
3 7 12
3 12
4 1512
3 7 12
1 8 12
1 2 3
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28 Do you need to regroup in order to complete this problem?
Yes or No
3 1 2
1 4
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29Do you need to regroup in order to complete this problem?
Yes or No
7 2 3
3 46
Pul
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30 What does 17 become when regrouping? 3 10 P
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Pul
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31 What does 21 become when regrouping? 5 8 P
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32
2 1 12
A
1 2224
B
4 1 6 2 1
4=
1 1112
C
1 1 12
D
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33
A
3 1321
B
6 2 7 3 2
3=
3 8 21 2 2
3C
2 1321
D
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34
A
6 1 6
B
15 8 1012
=
7 5 6 7 1
6C
6 2 12
D
Pul
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Multiplying Fractions...
1. Multiply the numerators.2. Multiply the denominators.3. Simplify your answer.
Multiplying Mixed Numbers...
1. Rewrite the Mixed Number(s) as an improper fraction. (write whole numbers / 1)2. Multiply the fractions.3. Simplify your answer.
Link Backto List
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35
1 5
x 2 3
= Pul
lP
ull
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36
2 3
x 3 7
= Pul
lP
ull
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37
= 4 9
3 8( )
Pul
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38
True
False
x 1 2
=5 5 1
x 1 2
Pul
lP
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39
A
x 4 73
B
C
3 5 7
D
1221
12 7
1 5 7
Pul
lP
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40
True
False
x =2 1 4 3 1
8 6 3 8
Pul
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41
15 1 4
A
18 1 8
B
20 3 8
C
19 1 8
D
5 8( )5 2
5(3 ) Pul
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Dividing Fractions...
1. Leave the first fraction the same.2. Multiply the first fraction by the reciprocal of the second fraction.3. Simplify your answer.
Dividing Mixed Numbers...
1. Rewrite the Mixed Number(s) as an improper fraction(s). (write whole numbers / 1)2. Divide the fractions.3. Simplify your answer.
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To divide fractions, multiply the first fraction by the reciprocal of the second fraction. Make sure you simplify your answer!
Some people use the saying "Keep Change Flip" to help them remember the process.
3 5
x 8 7
= 3 x 8 5 x 7
= 2435
3 5
7 8
=
1 5
x 2 1
= 1 x 2 5 x 1
= 2 5
1 5
1 2
=
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42
True
False
8 10
= 5 4
x 8 10
4 5 P
ull
Pul
l
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43
True
False
2 7
= 3 4 2 7
8
Pul
lP
ull
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44
1A
3940
B
C
8 10
= 4 5
4042
Pul
lP
ull
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45
Pul
lP
ull
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To divide fractions with whole or mixed numbers, write the numbers as an improper fractions. Then divide the two fractions by using the rule (multiply the first fraction by the reciprocal of the second).
Make sure you write your answer in simplest form.
5 3
x 2 7
= 1021
2 3
=1 1 2
3 5 3
7 2
=
6 1
x 2 3
= 12 3
=6 1 2
1 6 1
3 2
= = 4
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46
= 1 2 2 2
31
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47
= 1 2 2 2
31P
ull
Pul
l
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48
= 1 2 52
Pul
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Decimal Computation
Return to Table of Contents
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List what you remember about decimals.
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Some division terms to remember....
· The number to be divided into is known as the dividend
· The number which divides the other number is known as the divisor
· The answer to a division problem is called the quotient
divisor 5 20 dividend
4 quotient
20 ÷ 5 = 4
20__5
= 4
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When we are dividing, we are breaking apart into equal groups
EXAMPLE 1
Find 132 3
Step 1 : Can 3 go into 1, no so can 3 go into 13, yes
4
- 12 1
3 x 4 = 1213 - 12 = 1Compare 1 < 3
3 132
3 x 4 = 1212 - 12 = 0Compare 0 < 3
- 12 0
2
Step 2 : Bring down the 2. Can 3 go into 12, yes
4
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EXAMPLE 2(change pages to see each step)
Step 1: Can 15 go into 3, no so can 15 go into 35, yes
2
-30 5
15 x 2 = 3035 - 30 = 5Compare 5 < 15
15 357
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2
-30 5
15 35715 x 3 = 4557 - 45 =12Compare 12 < 15
7 - 45 12
Step 2 : Bring down the 7. Can 25 go into 207, yes
3
EXAMPLE 2(change pages to see each step)
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2
-30 5
15 357.0
7 - 45 120 - 120 0
3
Step 3: You need to add a decimal and a zero since the division is not complete. Bring the zero down and continue the long division.
15 x 8 = 120120 - 120 = 0Compare 0 < 15
.8
EXAMPLE 2(change pages to see each step)
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49 Compute.P
ull
Pul
l
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50 Compute.
Pul
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51 Compute.
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If you know how to add whole numbers then you can add decimals. Just follow these few steps.
Step 1: Put the numbers in a vertical column, aligning the decimal points.
Step 2: Add each column of digits, starting on the right and working to the left.
Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1.
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C
52 Add the following:
0.6 + 0.55 =
A 6.1
B 0.115click
C 1.15
D 0.16
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53 Find the sum
1.025 + 0.03 + 14.0001 =
15.0551click
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54 Find the sum:
5 + 100.145 + 57.8962 + 2.312 = 165.3532click
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What do we do if there aren't enough decimal places when we subtract?
4.3 - 2.05
Don't forget...Line Them Up!
4.32.05
What goes here?
4.3 02.05
2.25
2 1
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55
5 - 0.238 =4.762click
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56
12.809 - 4 =8.809click
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57
4.1 - 0.094 = 4.006click
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58
17 - 13.008 = 3.992click
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If you know how to multiply whole numbers then you can multiply decimals. Just follow these few steps.
Step 1: Ignore the decimal points.
Step 2: Multiply the numbers using the same rules as whole numbers.
Step 3: Count the total number of digits to the right of the decimal points in both numbers. Put that many digits to the right of the decimal point in your answer.
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23.2x 4.04
928
92800 0000
93.728
}
There are a total of three digits to the right of the decimal points.
There must be three digits to the right of the decimal point in the answer.
EXAMPLE
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59 Multiply 0.42 x 0.032 0.1344click
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60 Multiply 3.452 x 2.1 7.2492click
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4.7383661 Multiply 53.24 x 0.089
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DividendDivisor
Step 1: Change the divisor to a whole number by multiplying by a power of 10.
Step 2: Multiply the dividend by the same power of 10.
Step 3: Use long division.
Step 4: Bring the decimal point up into the quotient.
Divide by Decimals
Quotient
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15.6 6.24
Multiply by 10, so that 15.6 becomes 1566.24 must also be multiplied by 10
156 62.4
.234 23.4
Multiply by 1000, so that .234 becomes 23423.4 must also be multiplied by 1000
234 23400
Try rewriting these problems so you are ready to divide!
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62 Divide
0.78 ÷ 0.02 = 39click
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63
10 divided by 0.25 = 40click
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64
12.03 ÷ 0.04 = 300.75click
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There are two types of decimals - terminating and repeating.
A terminating decimal is a decimal that ends.All of the examples we have completed so far are terminating.
A repeating decimal is a decimal that continues forever with one or more digits repeating in a pattern.
To denote a repeating decimal, a line is drawn above the numbers that repeat. However, with a calculator, the last digit is rounded.
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Examples:
6600 2342 2200 14200 13200 10000 8800 12000 11000 10000 8800 12000 11000
63 48 45 39 36 32 27 51 45 60 54 6
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65
click
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67
click
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Statistics
Return to Table of Contents
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List what you remember about statistics.
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Measures of Center Vocabulary:
· Mean - The sum of the data values divided by the number of items; average
· Median - The middle data value when the values are written in numerical order
· Mode - The data value that occurs the most often
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Measures of Variation Vocabulary:
Minimum - The smallest value in a set of data
Maximum - The largest value in a set of data
Range - The difference between the greatest data value and the least data value
Quartiles - are the values that divide the data in four equal parts.
Lower (1st) Quartile (Q1) - The median of the lower half of the data
Upper (3rd) Quartile (Q3) - The median of the upper half of the data.
Interquartile Range - The difference of the upper quartile and the lower quartile. (Q3 - Q1)
Outliers - Numbers that are significantly larger or much smaller than the rest of the data
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QuartilesThere are three quartiles for every set of data.
LowerHalf
UpperHalf
10, 14, 17, 18, 21, 25, 27, 28
Q1 Q2 Q3
The lower quartile (Q1) is the median of the lower half of the data which is 15.5.
The upper quartile (Q3) is the median of the upper half of the data which is 26.
The second quartile (Q2) is the median of the entire data set which is 19.5.
The interquartile range is Q3 - Q1 which is equal to 10.5.
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The mean absolute deviation of a set of data is the average distance between each data value and the mean.
Steps
1. Find the mean.2. Find the distance between each data value and the mean. That is, find the absolute value of the difference between each data value and the mean.3. Find the average of those differences.
*HINT: Use a table to help you organize your data.
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Let's continue with the "Phone Usage" example.Step 1 - We already found the mean of the data is 56.Step 2 - Now create a table to find the differences.
48 8
52 4
54 2
55 1
58 2
59 3
60 4
62 6
Data Value
Absolute Value of the Difference|Data Value - Mean|
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Step 3 - Find the average of those differences.
8 + 4 + 2 + 1 + 2 + 3 + 4 + 6 = 3.75 8
The mean absolute deviation is 3.75.
The average distance between each data value and the mean is 3.75 minutes.
This means that the number of minutes each friend talks on the phone varies 3.75 minutes from the mean of 56 minutes.
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FREQUENCY
8
6
4
2
030- 40- 50- 60- 70- 80- 90-39 49 59 69 79 89 99
GRADE
Grade Tally Frequency30-39 I 140-49 050-59 060-69 I 170-79 IIII 480-89 IIII III 890-99 III 3
TEST SCORES95 85 9377 97 7184 63 8739 88 8971 79 8382 85
SAMPLES:
Data
TEST SCORES87 53 9585 89 5986 82 8740 90 7248 68 5764 85
FREQUENCY
8
6
4
2
040- 50- 60- 70- 80- 90-49 59 69 79 89 99
GRADE
Grade Tally Frequency40-49 II 250-59 III 360-69 II 270-79 I 180-89 IIII II 790-99 II 2
FrequencyTable
Histogram
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A box and whisker plot is a data display that organizes data into four groups
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10 80 90 100 110 120 130 140 150
The median divides the data into an upper and lower half
The median of the lower half is the lower quartile.
The median of the upper half is the upper quartile.
The least data value is the minimum.
The greatest data value is the maximum.
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10 80 90 100 110 120 130 140 150
median
25% 25%25%25%
The entire box represents 50% of the data. 25% of the data lie in the box on each side of the median
Each whisker represents 25% of the data
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A dot plot (line plot) is a number line with marks that show the frequency of data. A dot plot helps you see where data cluster.
Example:
35 40 45 5030
xxxxxx
xxx
xxx
xxxx
xx
xxx
xxxxx
Test Scores
The count of "x" marks above each score represents the number of students who received that score.
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Number System
Return to Table of Contents
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List what you remember about the number system.
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{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}
Definition of Integer:
The set of natural numbers, their opposites, and zero.
Define Integer
Examples of Integers:
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-1 0-2-3-4-5 1 2 3 4 5
Integers on the number line
NegativeIntegers
PositiveIntegers
Numbers to the left of zero are less than zero
Numbers to the right of zero are greater than zero
Zero is neitherpositive or negative
`
Zero
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Absolute Value of Integers
The absolute value is the distance a number is from zero on the number line, regardless of direction.
Distance and absolute value are always non-negative (positive or zero).
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
What is the distance from 0 to 5?
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To compare integers, plot points on the number line.
The numbers farther to the right are greater.
The numbers farther to the left are smaller.
Use the Number Line
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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Comparing Rational NumbersSometimes you will be given fractions and decimals that you need to compare.
It is usually easier to convert all fractions to decimals in order to compare them on a number line.
To convert a fraction to a decimal, divide the numerator by the denominator.
4 3.00-28 020 -20 0
0.75
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The coordinate plane is divided into four sections called quadrants.
The quadrants are formed by two intersecting number lines called axes.
The horizontal line is the x-axis.
The vertical line is the y-axis.
The point of intersection is called the origin. (0,0)
0 x - axisy - axis
origin
(+, -)
(-, +)
(-, -)
(+, +)
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To graph an ordered pair, such as (3,2):· start at the origin (0,0)· move left or right on the x-axis depending on the first number· then move up or down from there depending on the second
number · plot the point
(3,2)
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List what you remember about expressions.
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Exponents
Exponents, or Powers, are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition.
These are all equivalent:
24 Exponential Form2∙2∙2∙2 Expanded Form16 Standard Form
In this example 2 is raised to the 4th power. That means that 2 is multiplied by itself 4 times.
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Powers of IntegersBases and Exponents
When "raising a number to a power",
The number we start with is called the base, the number we raise it to is called the exponent.
The entire expression is called a power.
You read this as "two to the fourth power."
24
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What does "Order of Operations" mean?
The Order of Operations is an agreed upon set of rules that tells us in which "order" to solve a problem.
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The P stands for Parentheses : Usually represented by ( ). Other grouping symbols are [ ] and { }. Examples: (5 + 6); [5 + 6]; {5 + 6}/2
The E stands for Exponents : The small raised number next to the larger number. Exponents mean to the ___ power (2nd, 3rd, 4th, etc.) Example: 2 3 means 2 to the third power or 2(2)(2)
The M/D stands for Multiplication or Division : From
left to right. Example: 4(3) or 12 ÷ 3
The A/S stands for Addition or Subtraction : From left to right. Example: 4 + 3 or 4 - 3
What does P E M/D A/S stand for?
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Watch Out!
When you have a problem that looks like a fraction but has an operation in the numerator, denominator, or both, you must solve everything in the numerator or denominator before dividing.
453(7-2)
453(5)
4515
3
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[ 6 + ( 2 8 ) + ( 42 - 9 ) ÷ 7 ] 3
Let's try another problem. What happens if there is more than one set of grouping symbols?
[ 6 + ( 2 8 ) + ( 42 - 9 ) ÷ 7 ] 3
When there are more than 1 set of grouping symbols, start inside and work out following the Order of Operations.
[ 6 + ( 16 ) + ( 16 - 9 ) ÷ 7 ] 3[ 6 + ( 16 ) + ( 7 ) ÷ 7 ] 3
[ 6 + ( 16 ) + 1 ] 3[ 22 + 1 ] 3
[ 23 ] 369
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What is a Constant?A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.
Example: 4x + 2
In this expression 2 is a constant.click to reveal
Example: 11m - 7
In this expression -7 is a constant.click to reveal
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What is a Variable?
A variable is any letter or symbol that represents a changeable or unknown value.
Example: 4x + 2
In this expression x is a variable.
click to reveal
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What is a Coefficient?
A coefficient is the number multiplied by the variable. It is located in front of the variable.
Example: 4x + 2
In this expression 4 is a coefficient.click to reveal
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If a variable contains no visible coefficient, the coefficient is 1.
Example 1: x + 4 is the same as 1x + 4
- x + 4 is the same as
-1x + 4
Example 2:
x + 2has a coefficient of
Example 3:
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Equations and Inequalities
Return to Table of Contents
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List what you remember about equations and inequalities .
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A solution to an equation is a number that makes the equation true.
In order to determine if a number is a solution, replace the variable with the number and evaluate the equation.
If the number makes the equation true, it is a solution.If the number makes the equation false, it is not a solution.
Determining the Solutions of Equations
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Why are we moving on to Solving Equations?
First we evaluated expressions where we were given the value of the variable and had which solution made the equation true.
Now, we are told what the expression equals and we need to find the value of the variable.
When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).
This will eliminate the guess & check of testing possible solutions.
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To solve equations, you must use inverse operations in order to isolate the variable on one side of the equation.
Whatever you do to one side of an equation, you MUST do to the other side!
+5+5
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An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:
Symbol Expression Words
< A < B A is less than B
> A > B A is greater than B
< A < B A is less than orequal to B
> A > B A is greater than orequal to B
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Remember: Equations have one solution.
Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
This would be read as, "The solution set is all numbers greater than or equal to negative 5."
Solution Sets
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Let's name the numbers that are solutions to the given inequality.
r > 10 Which of the following are solutions? {5, 10, 15, 20}
5 > 10 is not trueSo, 5 is not a solution
10 > 10 is not trueSo, 10 is not a solution
15 > 10 is trueSo, 15 is a solution
20 > 10 is trueSo, 20 is a solution
Answer:{15, 20} are solutions to the inequality r > 10
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Since inequalities have more than one solution, we show the solution two ways.
The first is to write the inequality. The second is to graph the inequality on a number line.
In order to graph an inequality, you need to do two things:
1. Draw a circle (open or closed) on the number that is your boundary.
2. Extend the line in the proper direction.
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Remember!
Closed circle means the solution set includes that number and is used to represent ≤ or ≥.
Open circle means that number is not included in the solution set and is used to represent < or >.
Extend your line to the right when your number is larger than the variable. # > variable variable < #
Extend your line to the left when your number is smaller than the variable. # < variable variable > #
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List what you remember about geometry .
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A = length(width)A = lw
A = side(side)A = s2
The Area (A) of a rectangle is found by using the formula:
The Area (A) of a square is found by using the formula:
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168 What is the Area (A) of the figure?
13 ft
7 ft
Pul
lP
ull
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169 Find the area of the figure below.
8
Pul
lP
ull
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A = base(height)A = bh
The Area (A) of a parallelogram is found by using the formula:
Note: The base & height always form a right angle!
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170 Find the area.
10 ft 9 ft
11 ft
Pul
lP
ull
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171 Find the area.
8 m
13 m 13 m
8 m
12 m
Pul
lP
ull
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172 Find the area.
13 cm
12 cm
7 cm
Pul
lP
ull
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The Area (A) of a triangle is found by using the formula:
Note: The base & height always form a right angle!
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173 Find the area.
8 in
6 in
10 in 9 in
Pul
lP
ull
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174 Find the area
14 m
9 m10 m 12 m
Pul
lP
ull
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The Area (A) of a trapezoid is also found by using the formula:
Note: The base & height always form a right angle!
10 in
12 in
5 in
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175 Find the area of the trapezoid by drawing a diagonal.
Pul
lP
ull
9 m
11 m
8.5 m
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176 Find the area of the trapezoid using the formula.
20 cm
13 cm
12 cm
Pul
lP
ull
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Area of Irregular Figures
1. Divide the figure into smaller figures (that you know how to find the area of)
2. Label each small figure and label the new lengths and widths of each shape
3. Find the area of each shape
4. Add the areas
5. Label your answer
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Example:Find the area of the figure.
12 m
8 m
4 m2 m
12 m6 m
4 m2 m #1
#2
2 m
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Pul
l177 Find the area.
4'
3'
1'
2'
10'
8'
5'
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178 Find the area.
12
101320
25
10 Pul
l
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179 Find the area.
8 cm 18 cm
9 cm
Pul
l
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Area of a Shaded Region
1. Find area of whole figure.
2. Find area of unshaded figure(s).
3. Subtract unshaded area from whole figure.
4. Label answer with units2.
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Example
Find the area of the shaded region.
8 ft
10 ft
3 ft3 ft
Area Whole Rectangle
Area Unshaded Square
Area Shaded Region
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180 Find the area of the shaded region.
11'
8'
3'4'
Pul
l
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181 Find the area of the shaded region.
16"
17"
15"7"
5"
Pul
l
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3-Dimensional SolidsCategories & Characteristics of 3-D Solids:
Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base
Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base
Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved
Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curved
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3-Dimensional Solids
Vocabulary Words for 3-D Solids:
Polyhedron A 3-D figure whose faces are all polygons (Prisms & Pyramids)
Face Flat surface of a Polyhedron
Edge Line segment formed where 2 faces meet
Vertex (Vertices) Point where 3 or more faces/edges meet
Solid a 3-D figure
Net a 2-D drawing of a 3-D figure (what a 3-D figure would look like if it were unfolded)
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182 Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD cylinderE coneF square pyramid
Pull
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183Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD cylinderE coneF square pyramid Pu
ll
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184Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD pentagonal prismE coneF square pyramid Pu
ll
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185Name the figure.
A rectangular prismB triangular prismC triangular
pyramidD pentagonal prismE coneF square pyramid Pu
ll
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186Name the figure.
A rectangular prismB cylinderC triangular
pyramidD pentagonal prismE coneF square pyramid Pu
ll
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187 How many faces does a cube have?
Pull
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188 How many vertices does a triangular prism have?
Pull
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189 How many edges does a square pyramid have?
Pull
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6 in
2 in7 in
7 in2 in
2 in6 in
A net is helpful in calculating surface area.
Simply label each section and find the area of each.
#2 #4
6 in
#1
#3
#5
#6
Surface AreaThe sum of the areas of all outside faces of a 3-D figure.
To find surface area, you must find the area of each face of the figure then add them together.
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7 in2 in
2 in6 in
#2 #4
6 in
#1
#3
#5
#6
#1 #2 #3 #4 #5 #6
Example
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190Find the surface area of the figure given its net.
7 yd
7 yd
7 yd
7 yd
Pul
l
Since all of the faces are the same, you can find the area of one face and multiply it by 6 to calculate the surface area of a cube.
What pattern did you notice while finding the surface area of a cube?
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191Find the surface area of the figure given its net.
9 cm
25 cm
12 cm
Pul
l
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Volume FormulasFormula 1
V= lwh, where l = length, w = width, h = height
Multiply the length, width, and height of the rectangular prism.
Formula 2
V=Bh, where B = area of base, h = height
Find the area of the rectangular prism's base and multiply it by the height.
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Example
Each of the small cubes in the prism shown have a length, width and height of 1/4 inch.
The formula for volume is lwh.
Therefore the volume of one of the small cubes is:
Multiply the numerators together, then multiply the denominators. In other words, multiply across.
Forget how to multiply fractions?
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192Find the volume of the given figure.
Pul
l
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193Find the volume of the given figure.
Pul
l
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194Find the volume of the given figure.
Pul
l
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Ratios and Proportions
Return to Table of Contents
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List what you remember about the ratios and proportions .
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Ratio- A comparison of two numbers by division
Ratios can be written three different ways:
a to b a : b a b
Each is read, "the ratio of a to b." Each ratio should be in simplest form.
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195 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes?
A 7 : 9
B 7 27
C 7 11
D 1 : 3
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196 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes?
A 20 16
B 11 7
C 5 4
D 16 20
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197 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes?
A 27 to 9
B 7 to 27
C 27 to 7
D 11 to 27
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Equivalent ratios have the same value
3 : 2 is equivalent to 6: 4
1 to 3 is equivalent to 9 to 27
5 35 6 is equivalent to 42
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4 125 15
x 3 Since the numerator and denominator were multiplied by the same value, the ratios are equivalent
There are two ways to determine if ratios are equivalent.1.
4 125 15
x 3
4 125 15
Since the cross products are equal, the ratios are equivalent.4 x 15 = 5 x 12 60 = 60
2. Cross Products
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198 4 is equivalent to 8 9 18
True
False
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199 5 is equivalent to 30 9 54
True
False
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Rate: a ratio of two quantities measured in different units
Examples of rates: 4 participants/2 teams
5 gallons/3 rooms
8 burgers/2 tomatoes
Unit rate: Rate with a denominator of one Often expressed with the word "per"
Examples of unit rates:
34 miles/gallon
2 cookies per person
62 words/minute
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Finding a Unit RateSix friends have pizza together. The bill is $63. What is the cost per person?
Hint: Since the question asks for cost per person, the cost should be first, or in the numerator.
$63 6 people
Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. $63 6 6 people 6 $10.50 1 person
The cost of pizza is $10.50 per person
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200 Sixty cupcakes are at a party for twenty children. How many cupcakes per person?
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201 John's car can travel 94.5 miles on 3 gallons of gas. How many miles per gallon can the car travel?
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202 The snake can slither 24 feet in half a day. How many feet can the snake move in an hour?
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