p. 1 of 105 p. 4 of 105 - sunysuffolk.edu · section 3.1/4.1 understanding ... the first 10 prime...
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MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
1 of 18
P. 1 of 105
Section 3.1/4.1
Understanding Fractions
Modules 3 & 4Fractions &
Mixed Numbers
P. 2 of 105
Representing Part of a Whole
A fraction represents parts of a whole.
The whole is the circle on the left.
14
numerator
denominator
The fraction represents the
shaded part of the circle. 1 out of
4 pieces is shaded. is read “one-
fourth.”
1
4
1
4
P. 3 of 105
Representing Part of a Whole
The fraction represents the
portion of the box that is shaded.
We can also think of a fraction as a division problem.
55 8
8 and
55 8
8
58
P. 4 of 105
CQ-3/4-01. Write a fraction that represents
the portion of the box that is shaded.
1.
2.
3.
4.
10
7
11
6
7
4
11
7
P. 5 of 105
Representing Part of a Whole
Example:
Use a fraction to describe the situation.
1. 11 out of 15 of the math students received an
“A” for the semester.
1115
2. Patty needed three-fourths of a yard of material
to make the doll.34
P. 6 of 105
CQ-3/4-02. Use a fraction to describe
the situation.
3 out of 25 students don’t have the
computer at home.
1.
2.
3.
4.
25
22
3
25
25
3
28
25
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
2 of 18
P. 7 of 105
Equivalent Fractions
and
Simplifying Fractions
Section 3.2/4.2
P. 8 of 105
Prime Numbers
A prime number is a whole number greater than 1
that cannot be evenly divided except by 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
The first 10 prime numbers
A composite number is a whole number greater
than 1 that can be evenly divided by whole numbers
other than 1 and itself.
24 = 2 12 24 = 3 8 24 = 4 6
P. 9 of 105
Divisibility Tests
1. A number is divisible by 2 if the last digit is 0, 2, 4,
6, or 8.
2. A number is divisible by 3 if the sum of the digits
is divisible by 3.
3. A number is divisible by 5 if the last digit is 0 or 5.
Example:
The number 450 is divisible by 2. (It ends in 0.)
The number 450 is divisible by 3. (4 + 5 + 0 = 9 and 9 is
divisible by 3.)
The number 450 is divisible by 5. (It ends in 0.)
P. 10 of 105
Prime Factorization
Example:
Write the number 24 as a product of primes.
Write 24 as the product of any two factors.24 = 4 6
If the factors are not prime, they must
be factored.2 2 2 3
When all of the factors are prime, the
number has been completely factored.24 = 2 2 2 3
Instead of writing 2 2 2 3, we can also write 23 3.
P. 11 of 105
The Fundamental Theorem of Arithmetic
24 = 4 6
2 2 2 3
The order of the prime factors is not important
because multiplication is commutative.
The Fundamental Theorem of Arithmetic
Every composite number can be written in exactly
one way as a product of primes.
24 = 3 8
2 4
2 2
24 = 2 2 2 3 24 = 3 2 2 2
P. 12 of 105
CQ-3/4-03. Write 54 as a product of primes,
i.e., write the prime factorization of 54.
1.
2.
3.
4.
96
932
3332
272
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
3 of 18
P. 13 of 105
Equivalent Fractions
Equivalent fractions can be written in more than one
way. The value of the fractions is the same.
is shaded.36
is shaded.12
Equivalent fractions
P. 14 of 105
Equal Fractions
If two fractions are equivalent, their diagonal products or
cross products will be equal.
1 8 8
2 4 8
Equality Test for Fractions
For any two fractions where a, b, and c are whole
numbers and b 0, d 0, if then a d = b c.,a cb d
1 4?
2 8=
The products are equal,
1 4therefore .
2 8
P. 15 of 105
CQ-3/4-04. Are the following fractions
equivalent?
1. Yes
2. No
63
32
9
4and
P. 16 of 105
Simplest Form (Lowest Term)
1 3 32 3 6
Any nonzero number divided by itself is equal to 1.
A fraction is in simplest form when the numerator
and denominator have no common factors (other
than 1).
2 431
2 43bb
1 4 42 4 8
1 50 502 50 100
are equivalent fractions.1 3 4 50
, , , and 2 6 8 100
We can multiply both the numerator and the
denominator by a non-zero number.
P. 17 of 105
Common Factors
When a fraction is not in simplest form, it can be reduced.
25 5 540 5 8
5 is the common factor.
To reduce a fraction, find a common factor in the
numerator and the denominator and divide it out.
25 540 8
A fraction is called simplified, reduced, or in lowest
terms if the numerator and the denominator only
have 1 as a common factor.
P. 18 of 105
CQ-3/4-05. Simplify to lowest
terms.
1.
2.
3.
4.
8
1
30
4
60
8
15
2
300
40
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
4 of 18
P. 19 of 105
CQ-3/4-06. Simplify to lowest
terms.
1.
2.
3.
4.
49
21
21
9
11
7
7
3
147
63
P. 20 of 105
Converting Between
Improper Fractions
and
Mixed Numbers
Section 3.3/4.3
P. 21 of 105
Proper and Improper Fractions
If the value of a fraction is less than 1 (the numerator is
less than the denominator), the fraction is proper.
If the value of a fraction is greater than or equal to 1 (the
numerator is greater than or equal to the denominator),
the fraction is improper.
3 5 1 27, , ,
4 9 2 40
9 5 18 27, , ,
5 2 13 27
The numerator is
greater than or equal
to the denominator.
P. 22 of 105
CQ-3/4-07. What is the name of a fraction
with a numerator that is greater than or
equal to its denominator?
1. Improper
2. Mixed
3. Proper
4. Unmixed
P. 23 of 105
CQ-3/4-08. The fraction is best
described to by which word.18
6
1. Proper
2. Improper
3. Simplified
4. Factored
P. 24 of 105
Mixed Numbers
A mixed number is the sum of
a whole number greater than zero and a proper fraction.
3 5 12 , 1 , 22
4 9 2
32
4
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
5 of 18
P. 25 of 105
Mixed Numbers to Improper Fractions
Changing a Mixed Number to an Improper Fraction
1. Multiply the whole number by the denominator of the fraction.
2. Add the numerator of the fraction to the product found in step 1.
3. Write the sum found in step 2 over the denominator of the
fraction.
Example: Change into an improper fraction.2
43
4 3 23
Multiply the whole number
by the denominator.
12 23
143
Add the numerator
to the product.
Write the sum over
the denominator.
P. 26 of 105
CQ-3/4-09. Write as an improper
fraction.
1.
2.
3.
4.
3
10
3
13
3
15
3
17
3
25
P. 27 of 105
Improper Fractions to Mixed Numbers
Changing an Improper Fraction to a Mixed Number
1. Divide the numerator by the denominator.
2. Write the quotient followed by the fraction with the
remainder over the denominator.remainder
quotientdenominator
Example: Change into a mixed number.214
4 21
5 R1 21 15
4 4
denominator
quotient
remainder
P. 28 of 105
CQ-3/4-10. Write as a mixed
number.
1.
2.
3.
4.
7
47
38
7
36
7
342
7
45
P. 29 of 105
Reducing a Mixed Number
Example: Reduce the improper fraction 125
.15
125 5 5 515 3 5
common factors
253
1
1
Example: Reduce the mixed number fraction 11
4 .66
11 11 14 4
66 11 6
1
1
14
6
3
18
P. 30 of 105
CQ-3/4-11. Simplify to lowest
terms. Write the result as a mixed
number if possible.
1.
2.
3.
4.
22
26
11
21
11
13
44
81
44
52
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
6 of 18
P. 31 of 105
Converting
Fractions to Decimals
&
Decimals to Fractions
Section 3.4/4.4
P. 32 of 105
Equivalent Fractions and Decimals
A number can be expressed in two equivalent forms: as a
fraction and as a decimal.
Fraction
three and one-fourth
13
4
Decimal
three and twenty-five hundredths
3.25
Same quantity,difference appearance
10.5
2
10.25
4
10.2
5
10.1
10
Common equivalent fractions and decimals
P. 33 of 105
Fractions & Equivalent Decimals
Note the relationship between fractions and their equivalent numbers’ decimal form.
30.3
10
490.049
1000
one zero
one decimal place
three zeros
three decimal places
P. 34 of 105
Changing from Fractions to Decimals
Example:
1. Write the fraction as a decimal.61
100
610.61
100
three zeros three decimal places
2. Write the fraction as a decimal.27
51000
275 5.027
1000
P. 35 of 105
Changing from Decimals to Fractions
Example:
1. Write the decimal 0.371 as a fraction.
3710.371
1000
Simplify when possible.
2. Write the decimal 4.0038 as a fraction.38
4.0038 410000
194
5000
P. 36 of 105
CQ-3/4-12. Write 0.155 in fractional
form and reduce, if possible.
2000
31.
2.
3.
4.
200
31
2000
31
20
3
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
7 of 18
P. 37 of 105
Converting a Fraction to a Decimal
Converting a Fraction to an Equivalent Decimal
Divide the denominator into the numerator until
a) the remainder becomes zero, or
b) the remainder repeats itself, or
c) the desire number of decimal places is achieved.
10.5
2
10.25
4
10.2
5
10.1
10
Common equivalent fractions and decimals
P. 38 of 105
Terminating and Repeating Decimals
30.375
8
10.333 0. 3
3
130.59090 0.5 90
22
10.0625
16
Terminating decimals(The remainder is zero when converting
the fraction into a decimal.)
Repeating decimals(When converting, the remainder is a
digit or group of digits that repeats.)
repeating digit repeating group of digits
P. 39 of 105
Converting a Fraction to a Decimal
Example:
Write as an equivalent decimal. 5
18
5 18 5
18
repeating
remainders
0.277
.00036140126
5 0.27
18
140
P. 40 of 105
CQ-3/4-13. Write as a decimal.
1. 1.
2. 2.
3. 3.
4. 4.
8
5
6.0
6.1
625.0
5125.0
P. 41 of 105
CQ-3/4-14. Write as a decimal.
1. 1.
2. 2.
3. 3.
4. 4.
38.0
2.1
6.5
1.1
6
5
P. 42 of 105
Ordering Fractions and Decimals
Example:
Fill in the blank with one of the symbols <, =, or >.
3 ____ 0.7
4
Change the fraction into a decimal
for easier comparison.
0.75 ___ 0.7>
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
8 of 18
P. 43 of 105
CQ-3/4-15. Select the correct statement.
1.1.
2.2.
3.3.
4. 4.
14.37
22
14.37
22
14.37
22
37
22
P. 44 of 105
Multiplying
Fractions
and
Mixed Numbers
Section 3.5/4.5
P. 45 of 105
Multiplying Fractions
Multiplication of fractions is used when we want to
take a fractional part of something.
1 5 52 9 18
12
59
1 5 of
2 9yields 5 out of 18
squares.
P. 46 of 105
Multiplying Fractions
To multiply two fractions, we
multiply the numerators and multiply the denominators.
3 2 67 5 35
In general, for all positive whole numbers a, b, c, and d,
a c a cb d b d
(when b and d are not 0).
3557
623
P. 47 of 105
Multiplying Proper or Improper Fractions
Example: Multiply12 3
.17 24
12 317 24
Simplify the
fraction.
12 317 24
12 317 24
36408
3
34
To make multiplying easier, the fractions may be
simplified before multiplying.
3 4 317 2 3 4
3
17 2
3
34
1 1
1 1
3 4 317 2 3 4
P. 48 of 105
CQ-3/4-16. Multiply:
1.
2.
3.
4.
56
15
35
24
15
8
24
35
8
5
7
3
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
9 of 18
P. 49 of 105
CQ-3/4-17. Multiply:
1.
2.
3.
4.
429
143
3
1
507
121
39
13
11
13
39
11
P. 50 of 105
CQ-3/4-18. Simplify:
1.
2.
3.
4.
9
7
72
52
14
10
49
25
2
7
5
P. 51 of 105
Multiplying Mixed Numbers
Example: Multiply4 4
2 1 .5 7
To multiply mixed numbers,
first change each mixed number into an
improper fraction.
14 115 7
22 2 or 4
5 5
2
1
4 42 1
5 7
=
P. 52 of 105
CQ-3/4-19. Simplify:
1.
2.
3.
4.
5
218
3
133
8
59
24
5
33
3
26
P. 53 of 105
CQ-3/4-20. Simplify:
1.
2.
3.
4.
42
19
9
15
3
119
42
118
9
13
14
36
P. 54 of 105
Dividing
Fractions
and
Mixed Numbers
Section 3.6/4.6
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
10 of 18
P. 55 of 105
Dividing Fractions
A cup of lemonade that is full must be divided
into -cup servings. How many cups of lemonade
will there be?34
34
14
How many
1 3's are in ?
4 4
14
14
14
14
three1 3
There are 's in .4 4
There will be three -cup servings in the cup.14
P. 56 of 105
Dividing Proper or Improper Fractions
When fractions are divided, we invert the second
fraction and multiply. [Multiplied by the reciprocal.]
3 14 4
3 44 1
1
1
Rules for Division of Fractions
a c a db d b c
(when b, c, and d are not 0).
To divide two fractions, we invert the second fraction and multiply.
31
3
P. 57 of 105
CQ-3/4-21. What do we call two
numbers which have a product of one?
1. Opposites
2. Factors
3. Quotients
4. Reciprocals
P. 58 of 105
CQ-3/4-22. Find the reciprocal of 8.
1.
2.
3.
4.
8
1
8.0
8
0
P. 59 of 105
CQ-3/4-23. Simplify:
1.
2.
3.
4.
25
6
8
3
6
14
3
23
10
3
5
4
P. 60 of 105
CQ-3/4-24. Simplify:
1.
2.
3.
4.
35
36
7
20
35
15
20
7
7
12
10
6
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
11 of 18
P. 61 of 105
Dividing Mixed Numbers
Example: Divide4 7
2 1 .5 10
To divide mixed numbers,
first change each mixed number into an improper
fraction.
4 7 14 172 1
5 10 5 10
14 105 17
2
1
28 11 or 1
17 17
P. 62 of 105
CQ-3/4-25. Simplify:
1.
2.
3.
4.
3
110
49
441
7
18
7
12
3
12
7
34
P. 63 of 105
CQ-3/4-26. Simplify:
1.
2.
3.
4.
8
4
19
36
2
140
4
1218
P. 64 of 105
CQ-3/4-27. Simplify:
1.
2.
3.
4.
8
12
4
32
2
12
8
112
64
312
P. 65 of 105
The Least Common
Denominator
and
Creating Equivalent
Fractions
Section 3.7/4.7
P. 66 of 105
A multiples of a number are the products of that
number and the numbers 1, 2, 3, 4, 5, …
The multiples of 3 are 3, 6, 9, 12, 15, …
3 13 2
3 3
The least common multiple, or LCM, of two
natural numbers is the smallest number that is
a multiple of both.
Least Common Multiple (LCM)
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
12 of 18
P. 67 of 105
Least Common Multiple (LCM)
Example: Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24 …
The multiples of 6 are 6, 12, 18, 24, 30, 36 …
The first number that appears on both lists is the
LCM.
12 is the least common multiple (LCM) of 4 and 6.
P. 68 of 105
Least Common Denominator (LCD)
A least common denominator (LCD) of two or more
fractions is the smallest number that can be divided
evenly by each of the fractions’ denominators.
7 3 and
12 4
Since 4 can be divided into 12, the LCD of
is 12.7 3
and 12 4
P. 69 of 105
Least Common Denominator (LCD)
3 4 and .
4 5
20 is also the smallest number that can be divided
by 4 and 5 without a remainder.
Example: Find the LCD for
4 5 = 20
The LCD of is 20.3 4
and 4 5
P. 70 of 105
Finding the Least Common Denominator(A more systematic/longer way)
Three-Step Procedure for Finding the LCD1. Write each denominator as the product of prime factors.
2. List all the prime factors that appear in either product.
3. Form a product of those prime factors, using each factor the greatest number of times it appears in any one denominator.
Example: Find the LCD for5 7
and .12 30
2 2 3 2 3 5Product of primes
Prime factors in either product: 2 2 3 5
The LCD is 60.
P. 71 of 105
CQ-3/4-28. Find the LCD for
1.
2.
3.
4.
36
56
32
72
.18
11 and
8
3
P. 72 of 105
Building Fraction Property
Building Fraction Property
For whole numbers a, b, and c where b 0, c 0,
1 .a a a c a cb b b c b c
Example:
Build to an equivalent fraction with a LCD of 20.34
3 ?4 20
cc
3 5 154 5 20
3 15and are
4 20
equivalent fractions.
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
13 of 18
P. 73 of 105
Creating Equivalent Fractions
Fractions with unlike denominators cannot be added.
To change the denominators and make them the same,
1) find the LCD and
2) build up the addends into equivalent fractions that
have the LCD as the denominator.
3 4 +
4 5The LCD is 20.
3 ?4 20
cc
4 ?5 20
cc
55 , 1
5c
44 , 1
4c
The building
fraction property
P. 74 of 105
CQ-3/4-29. Find the equivalent fraction
for the given fraction to have 48 as its
new denominator.
1.
2.
3.
4.
8
3
48
9
48
15
8
18
48
18
P. 75 of 105
Adding and Subtracting
Fractions
Section 3.8/4.8
P. 76 of 105
Fractions with Common Denominators
Fractions must have common denominators before
they can be added or subtracted.
24
+ =
14
34
2 1 34 4 4
P. 77 of 105
CQ-3/4-30. Simplify:
1.
2.
3.
4.
9
8
9
4
18
12
3
4
9
12
3
11
P. 78 of 105
Fractions with Different Denominators
If fractions have different denominators, find the
LCD and build up each fraction so that its
denominator is the LCD.
3 18 6
Example:
Add .
LCD = 24
3 3 98 3 24
1 4 46 4 24
9 4 1324 24 24
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
14 of 18
P. 79 of 105
CQ-3/4-31. Simplify:
1.
2.
3.
4.
10
5
21
19
2
1
21
4
7
4
3
1
P. 80 of 105
CQ-3/4-32. Simplify:
1.
2.
3.
4. 1
7
34
7
12
7
31
7
34
P. 81 of 105
Fractions with Different Denominators
5 712 30
Example:
Subtract .
LCD = 60
5 5 2512 5 60
7 2 1430 2 60
25 14 1160 60 60
P. 82 of 105
CQ-3/4-33. Simplify:
1.
2.
3.
4.
1
6
11
36
5
36
57
12
7
18
13
P. 83 of 105
CQ-3/4-34. Simplify:
1.
2.
3.
4.
40
5
8
5
40
19
3
5
5
2
8
7
P. 84 of 105
Comparing Two Fractions
When comparing two fractions, find the LCD
and build up each fraction so that its denominator
is the LCD. Then, the fraction with the larger
numerator is the larger fraction.
Example:
Compare .6
5
8
7 and
24
20
4
4
6
5
6
5
24
21
3
3
8
7
8
7
6
5
8
7,,
24
20
24
21 soBeacuse
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
15 of 18
P. 85 of 105
CQ-3/4-35. Compare
1.
2.
3.
4.
.5
4
7
4 and
5
4
7
4
5
4
7
4
5
4
7
4
35
16
5
4
7
4
P. 86 of 105
Adding and Subtracting
Mixed Numbers
&
the Order of Operations
Section 3.9/4.9
P. 87 of 105
Adding Mixed Numbers
When adding mixed numbers, it is best to add the
fractions together and then add the whole
numbers together.
1 29 2 .
6 3
Example:
Add
LCD = 61
9 6
42
6
2 2 42 2
3 2 6
19
6
22
3
5 6
11
Add the
fractions
first.
Add the whole
numbers.
P. 88 of 105
CQ-3/4-36. Simplify:
1.
2.
3.
4.
21
135
10
35
21
26
10
36
3
12
7
23
P. 89 of 105
CQ-3/4-37. Simplify:
1.
2.
3.
4.
36
5121
5
3120
216
65120
5
3121
18
1382
12
538
P. 90 of 105
Subtracting Mixed Numbers
Subtracting mixed numbers is like adding mixed
numbers.
2 1 9 7 .
3 8
Example:
Add
LCD = 24
169
24
37
24
1 3 37 7
8 3 24
29
3
17
8
13 24
2
Subtract the
fractions
first.
Subtract the whole
numbers.
2 8 169 9
3 8 24
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
16 of 18
P. 91 of 105
Subtracting Mixed Numbers
1 7 5 3 .
12 18
Example:
Add
LCD = 36
394
36
143
36
7 2 143 3
18 2 36
35
36
143
36
251
36
1 3 35 5
12 3 36
We borrow 1 from 5 to obtain
3 3 39 395 4 1 4 4 .
36 36 36 36
3 14We cannot subtract
36 36
so we need to borrow.
P. 92 of 105
CQ-3/4-38. Simplify:
1.
2.
3.
4.
3
26
8
16
8
15
4
16
2
16
8
512
P. 93 of 105
CQ-3/4-39. Simplify:
1.
2.
3.
4.
4
111
4
110
4
311
4
310
4
1718
P. 94 of 105
CQ-3/4-40. Simplify:
1.
2.
3.
4.
15
1126
15
825
20
1126
10
126
15
1387
5
2113
P. 95 of 105
Order of Operations
Order of Operations
1. Perform operations inside any parentheses.
2. Simplify any expressions with exponents.
3. Multiply or divide from left to right.
4. Add or subtract from left to right.
Do first
Do last
Example: Evaluate2
5 2.
8 5
5 48 25
1 12 5
1
10
Exponents
Multiplication1 1
2 5
P. 96 of 105
Order of Operations
Example: Evaluate3 1 5
.4 4 3
3 1 34 4 5
3 1 54 4 3
3 34 20
Express division
as multiplication.
Multiply.
15 320 20
Rewrite fractions
using the LCD.
18 920 10
Add and simplify.
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
17 of 18
P. 97 of 105
CQ-3/4-41. Simplify:
1.
2.
3.
4.
135
41
135
121
135
49
72
11
2
3
1
3
8
5
3
P. 98 of 105
CQ-3/4-42. Simplify:
1.
2.
3.
4.
85
93
5
41
85
462
85
24
8
12
5
36
P. 99 of 105
Solving Applied Problems
Involving Fractions
Section 3.10/4.10
P. 100 of 105
Problem Solving Steps1. Understand the problem.
a) Read the problem carefully.
2. Draw a picture if it is helpful.
a) Fill in the Mathematics Blueprint so that you have the facts and a method of proceeding in this situation.
3. Solve and state the answer.
a) Perform the calculations.
b) State the answer and include the unit of measure.
4. Check.
a) Estimate the answer.
b) Compare the exact answer with the estimate to see if
your answer is reasonable.
P. 101 of 105
Mathematics Blueprint
Mathematics Blueprint for Problem Solving
Key Points to Remember
How Do I Proceed?
What Am I Asked to Do?
Gather the Facts
The Mathematical Blueprint is simply a sheet of paper with four columns. Each column tells you something to do.
P. 102 of 105
All fractions must
use the LCD = 24.
Mathematics Blueprint
Example #1:
A carpenter is using an 8-foot length of wood for a frame. He needs to cut a
notch in the wood that is feet from one end and feet from the other
end. How long does the notch need to be?
74
8
213
Mathematics Blueprint for Problem Solving
Key Points to Remember
How Do I Proceed?
What Am I Asked to Do?
Gather the Facts
The board is 8 ft.
long. There is a
cut from each
side of the board.
Add the fractions
and subtract the
total from 8.
Find the length of
the notch.
Example continues.
MAT001 – MODULES 3 & 4
FRACTIONS & MIXED NUMBERS
18 of 18
P. 103 of 105
Mathematics Blueprint
Example #1:
A carpenter is using an 8-foot length of wood for a frame. He needs to cut a
notch in the wood that is feet from one end and feet from the other
end. How long does the notch need to be?
74
8
213
2 71 43 8
16 211 424 24
37 135 6
24 24
This is the part of the
board that is not
notched.
13 24 138 6 7 6
24 24 24
11124
The length of the
11notch is 1 feet.
24
P. 104 of 105
“of” means to
multiply.
Mathematics Blueprint
Mathematics Blueprint for Problem Solving
Key Points to Remember
How Do I Proceed?
What Am I Asked to Do?
Gather the Facts
The total income
is $450.There are
three deductions
to be subtracted
from the 450.
Multiply each
fraction by 450,
then subtract the
three products
from 450.
Find how much
money Patty has
after the
deductions.
Example continues.
Example #2:
Patty earns $450 per week. She has of her income withheld for federal
taxes, of her income withheld for state taxes, and of her income
withheld for medical coverage. How much money per week is left for Patty
after those three deductions?
115
15 1
25
P. 105 of 105
Mathematics Blueprint
1450
5
Example #2:
Patty earns $450 per week. She has of her income withheld for federal
taxes, of her income withheld for state taxes, and of her income
withheld for medical coverage. How much money per week is left for Patty
after those three deductions?
115
15 1
25
45090
5
1450
15
45030
15
1450
25
45018
25
The total of the deductions is 138.
450 – 138 = 312
Patty has $312 left
after deductions.`