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Rounding and Estimation
Rounding a number provides an approximate value of the number. Estimation is
often used when the final result does not need to be an exact number.
To round a number:
Step 1: Find the place to which you are rounding.
Step 2: Look at the digit immediately to the right of that place.
Step 3: If this digit is greater than or equal to 5, round the digit in the rounding place up 1,
If the digit is less than 5, the digit in the rounding place is unchanged.
Step 4: If you are rounding to a decimal place or to the nearest whole number, drop the
digits to the right of the rounding place. If you are rounding to tens, hundreds, or
any larger place value, fill in zeros as needed.
Example: Round 14.638 to the nearest tenth.
Step 1:
Step 2:
Step 3:
Step 4:
Answer:
The digit to be rounded is 6.
3 is to the right of 6.
3 < 5, so 6 is unchanged.
Drop the digits 3 and 8.
14.6
Practice on Your Own
Round each number to the indicated place value.
1. 146.3892; hundredth
2. 235.7; whole number
3. 47; tens
4. 15.275; tenth
5. 0.0048; thousandth
6. 3.99; tenth
Estimate the sum or product by rounding each number to the
indicated place value.
7. 76 + 148; tens
8. 9.46 x 18; tens
10. 412 + 709 + 99; hundreds 11. 6.62 x 1.89; tens
9. 10.2 •*• 1.975; ones
12. 780.5 ; 7.88; tens
Check
Estimate each number by rounding it to the indicated place value.
13. 0.0748; hundredth
14. 1324.8; whole number
15. 18.996; tenth
Estimate the sum or product by rounding each number to the
indicated place value.
16. 11.8 x 47; tens
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All rights reserved.
17. 862 + 59.4; hundreds
'Jfl
UU
18. 9.2 -s- 2.866; ones
Hnif Ronmotru
null UGUIIIGU y
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Simplify Fractions
Definition: a fraction is in simplest form when the numerator and the
denominator do not share any common factors, other than the factor of 1.
An improper fraction should be written as a mixed number.
To write a fraction in simplest form:
Step 1: List all the factors of the numerator and the denominator.
Step 2: Identify the greatest common factor (GCF).
Step 3: Divide both the numerator and the denominator by the GCF.
Example: Write
in simplest form.
Factors of 18: {1, 2, 3, 6, 9, 18}
Factors of 45: (1, 3, 5, 9, 15, 45}
GCF: 9
1 ft — Q
P
ft
1
7° ' „ = f
4b — 9
o
O
7— written in simplest form is ~.
45
5
Practice on Your Own
Identify the greatest common factor of the numerator and
denominator of the fractions given.
>. |
| Factors of 36:
1. ~| Factors of 16:
DO
Factors of 24:
Factors of 63:
GCF:
GCF:
Write each fraction in simplest form.
3. 6 - D _ D
R
On
60 72 +
D
D
8. 24
49
56
121
66
5. 45 54 +
26
Check
Identify the greatest common factor of the numerator and
denominator of the fractions given.
Q
—t. UUInrF =
9. 12
10. I GCF =
11. I GCF =
' 180
\j
Write each fraction in simplest form.
13.
D
2i-D~D
16. 32
40
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Al! rights reserved.
14.
15. 14 -f-
48+n
18 -
D
18. Z2
17. 20
81
16
32
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^SKILL
"' '
mim Multiply
MU
and Divide Fractions
General Operation Reminders
Multiplying Fractions
Dividing Fractions
Step 1: Multiply the numerators. Multiply the
denominators.
Step 2: Write the answer in simplest form.
Divide by the greatest common
factor if needed.
Step 1: Find the reciprocal of the divisor
(the second fraction) and rewrite the
problem as a multiplication problem.
Step 2: Multiply the numerators. Multiply the
denominators.
Step 3: Write the answer in simplest form.
Divide by the greatest common
factor if needed.
Example 1: Multiply | x ~.
1 3
Example 2: Divide g ^ T-
2
5
6 ' 4
f\
3_
4~
X
X
O
^ = ~K GCF of 6 and 20 is 2.1 ^ 3 = 1 X 4 = A GCF of 4 and 18 is 2.
^
20 = 2 0 - 2 = TO'
The product is -r^r.
18
6 3
18-2
18
2
9'
2
The product is .
Practice on Your Own
Multiply or divide. Give your answer in simplest form.
2. 4 • 4
4 1
7X4
2
- 9
7-
X
1
2
10 X 5
4. 3 . 2
8. 8 -
Check
Multiply or divide. Give your answer in simplest form.
q lv 1
9-
9X 2
13.^x1
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Al! rights reserved.
, 2^3
' • 5 ' 5
14.
-6
19 _2_ j. JL
12' 11 ' 11
15.
106
x 2
16. 12+
Holt Geometry
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SKILL
Add and Subtract Fractions
General Operation Reminders
Adding and Subtracting Fractions
Like Fractions (same denominators)
Step 1: Add or subtract the numerators.
Step 2: Write the sum or difference of the
numerators over the denominator.
Step 3: Write the answer in simplest form.
Unlike Fractions (different denominators)
Step 1: Find the least common denominator
(LCD) and then rewrite each fraction
so that its denominator is the LCD.
Step 2: Follow the steps for adding or
subtracting like fractions.
1
3
Example 2: Subtract 1p - T-
Example 1: Add ~ + |.
1 , 5 _ 1+5
f-^ = |
8 + 8=
(GCF of 6 and 8 isRewrite
2.) 1 as . The LCD of 2 and 4 is 4.
— ~ ® """ ^ ___ 3
8~8-^2~4
• 3
4"
Th
3v 2_6
2 2 4
6
4
3
4
3
4
The difference
jis
-3
4.
Practice on Your Own
Add or subtract. Give your answer in simplest form.
1
5
1.
5. 1]
3.
2
9
7
8
10
4
9
_
3
7
3
7.
4
Check
Add or subtract. Give your answer in simplest form.
9.
11
A
11
8~8
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All rights reserved.
10 3 - 2
1Ul 9
9
14.
10
11 -3- + 1
"' 14 + 7
12.
_ _
12
4
1
4
3
' 5 + T 5 + 10
3
5
108
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Simplify Radical Expressions
Definition: A radical expression is in simplest form when all of the following conditions are met.
1. The number, or expression, under the radical sign contains no perfect square
factors (other than 1).
2. The expression under the radical sign does not contain a fraction.
3. If the expression is a fraction, the denominator does not contain a radical expression.
How to Simplify Radical Expressions
Look for perfect square
factors and simplify these
first. If the radical expression
is preceded by a negative
sign, then the answer is
negative.
If the expression is a
product, simplify then
multiply, or multiply then
simplify, whichever is most
convenient.
If the expression is (or
contains) a fraction, simplify
then divide, or divide then
simplify, whichever is most
convenient.
Example 1: Simplify Y81.
Since 81 is a perfect
square factor, simplify the
expression to 9.
Example 2: Simplify V25V16.
Since both numbers are
perfect squares, simplify then
multiply: VS^SV^ • ~4 =
5 • 4 = 20
Example 3: Simplify - /_4__
V8T_= VsTl) = 9
-V81 = -V9" • 9 =
_
49
V2 • 2
2
-9
Practice on Your Own
Simplify each expression.
81
1. V25
2. V9V36
3.
5. V'TOOV'4
6.
7. V169
V 121
4. -V8f
~V625
Check
Simplify each expression.
9. Vl6
13. V2V50
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All rights reserved.
10. V81V64
11. -V49
12. y 25_
14. -Y144
15. -V9V4
16 -Vi
118
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Solve One-Step Equations
To solve a one-step equation, do the inverse of whatever operation is being done to
the variable. Remember, because it is an equation, what is done to one side of the
equation must also be done to the other side.
Solve an addition equation
using subtraction.
Solve a subtraction equation
using addition.
x + 5 = 15
-5 -5
x
=10
x - 8 = -3
+8 +8
x
= 5
Solve a multiplication equation
using division.
Solve a division equation
using multiplication.
X
7x = 42
o
12
7x 42
7
7
x= 6
HO
\C.
'
X
~Tr)
.
~
Q
~G
HO
l<£
x = -36
Practice on Your Own
Solve.
1. m- 5 = 9
2•
5. 4y = -12
6. k+ 9 = -3
q9- ^
4 = —-
/-v —
"~ —w
—
^
D
3. 6x = 54
4. b + 15 = 25
7. p - 7 = -2
10. 5 +h= 16
11. -12x= -24
12. r - 2 = -9
13. 3x= 15
14. c- 11 = 1
15. d+ 9 = 5
16.
17. z - 2 = -17
18.
19. -10/7= 120
20. x + 99 - 100
Check
Solve.
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148
Holt Geometry
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Solve Multi-Step Equations
To solve an equation, you need to isolate the variable on one side of the equals
sign. Follow the order of operations in reverse to solve a multi-step equation. That
is, add and subtract before you multiply or divide. Sometimes, you need to use the
Distributive Property before you use inverse operations.
Example: Solve 9x - 6 = 21.
9x - 6 = 21
+6 +6
9x
= 27
9x
9
x
27
9
3
Add 6 to both sides.
Divide both sides by 9.
Practice on Your Own
Solve.
1. 3x- 2 = 10
2. 7m + 3 = 45
3. 12 +
, p _ 3« =
5. -12 - 9y = -20
6. 26 = 5c - 4
7. 3(|+ 2) = -9
8. 5(2n- 1) = -5
9. 2(h + 3) + 5/? = -3
4.
4
= 17
Check
Solve.
10. 8x+ 1 = 17
11. -3 +
13. 18 = |+15
14. -3(x + 4) = -5
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= -7
150
12. -12 + 6fif = 14
15. 5(z- 2) + 3z= 14
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LLII Solve Equations with Variables on Both Sides
To solve an equation with variables on both sides:
Step 1 : Move all the variable expressions to one side of the equals sign by adding
or subtracting.
Step 2: Isolate the variable by adding or subtracting any constants and then
multiplying or dividing by any coefficients.
Example: Solve 7x - 12 = 3x + 8.
Step 1: Subtract Sxfrom both sides.
7x - 12 = 3x + 8
3x
3x
~4x- 12 =~
8
Step 2: Add 12 to both sides, then divide both sides by 4. 4x - 12 =
+12
8
+12
4x
4
x-
20
4
5
Practice on Your Own
Solve.
1. 3y + 5 = 9 y - 1 3
2. 4 + 11m = 7 m - 1
3. 9 x + 1 5 - 7
4. -1 - 3f = 9 - 8f
5. x - 11 = 7x - 8
6. 3b + 5 = -2b + 5
7. -k - 7 = -5/c - 6
8. 16 - 3x = 5x - 8
9. -6a - 4 = -7a - 5
Check
Solve.
10. 15x+ 1 = 13x+ 17
11.9-4f=2f-1
13. p + 1 3 = 5 p + 5
14. -56+ 11 = 7 - 6 b
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"1 *1?
' »"•
12. 6 y - 5 = 7 y + 1
15. - x - 8 = -5x - 10
Hfllt
Mul1
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Solve Equations with Fractions
You solve multi-step equations with fractions just like you solve multi-step equations
with integers.
Step 1: Use inverse operations to undo any addition or subtraction.
Step 2: When the coefficient of the variable is a fraction, multiply each side of the
equation by the reciprocal of the fraction and then simplify. If the coefficient
is not a fraction, but there are other fractions in the equation, multiply by the
reciprocal of the coefficient rather than dividing.
Example: Solve x - 12 = 8.
|x - 1 2 + 12 = 8 + 12
Add 12 to both sides.
fx = 2 0
5 .4
2
5
4 5X ~ ^ 4
#
/r
Multiply by the reciprocal.
Simplify.
Y —
.A
Practice on Your Own
Solve.
1. fx + 5 = 17
2.
A 9y +
4.
IX +
_
5. x
7
7. -
1_ 5
-
7
-7
3 —
1
9
3
8
8
8. 6x == 3x
25
3.4y-
=
" 4
9. 4y = 9y - §
Check
Solve.
10. |x+ 11 = 21
10
5
TT = TT
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All rights reserved.
11.y-8=-7
1 -3
2 ~2
154
12. 5x-
10
15. 8x = 5x
Holt Geometry