Download Math40Lesson9

Translating Verbal Expressions into Mathematical Expressions
Addition
Subtraction
Verbal
Expressions
added to
more than
the sum of
increased by
the total of
minus
l
less
th
than
subtracted from
decreased by
the difference
between
times
ti
M lti li ti
Multiplication
of
the product of
Division
Examples
6 added to y
8 more than x
the sum of x and z
t increased by 9
the total of 5 and y
x minus 2
7 lless th
than t
5 subtracted from 8
m decreased by 3
the difference between y and 4
10 ti
times 2
one half of 6
the product of 4 and 3
Math
Translation
6+y
8+x
x+z
t+9
5+y
x-2
t7
t-7
8-5
m-3
y-4
10 X 2
(1/2) X 6
4X3
multiplied by
y multiplied by 11
11y
divided by
the quotient of
x divided by 12
the quotient of y and z
x/12
y/z
the ratio of
the ratio of t to 9
t/9
8 pieces distributed
Distributed (or
split)
lit) equally
ll equally
ll among 2 people.
l
8/2
x2
z3
y2
Exponents
the square of
the cube of
squared
the square of x
the cube of z
y squared
Equivalency
equals
l
is
is the same as
1+2
1
2 equals
l 3
2 is half of 4
½ is the same as 2/4
1 2=3
1+2
2 = (½)4
yields
represents
3+1 yields 4
y represents x+1
3+1 = 4
y=x+1
Three consecutive integers
n,n+1,n+2
Consecutive Consecutive
integers
1
2
=
2
4
Three consecutive even
n,n+2, n+4
integers
(where n is even)
Three consecutive odd integers
n,n+2, n+4
(where n is odd)
Expressions and Equations
Expressions (contain no “=” sign) :
An expression is one or numbers having some mathematical operations done on them.
Numerical Expressions:
3+5
3(4)
6/2
5-1
4
Expressions can be evaluated or simplified: 3 + 5 can be simplified to 8
This just means
means, “Whatever
Whatever you see
see, do”
do
Algebraic Expressions:
x+5
3x Í in Algebra, it is implied that 3x means 3 times x. The “proper” way to write the product of a number
and a variable is to always write the number to the left of the variable. x times 5 = 5x
When numbers are multiplied
p
byy variables,, theyy are ggiven a special
p
name,, “coefficient”. 5 is the coefficient
of 5x.
Algebraic expressions can be simplified by using the associative and distributive properties.
-4m(-5n) can be simplified by rearrange the terms (we can do this when only operation is multiplication) so
that all the constants are grouped together and all the variables are grouped together(alphabetically))
-4m(-5n) = (-4)(-5)mn = 20mn
2(-4z)(6y)=2(-4)(6)yz = -48yz
3(s+7) can be simplified by using the distributive property
3s + 3(7) = 3x + 21
-6(-3x -6y + 8) can also be simplified. Change any subtraction to adding a negative.
-6(-3x + -6y + 8)
Now distribute the -6 and make sure to glue the negative sign on that -6 wherever you distribute it!
-6(-3x) + -6(-6y) + -6(8) = 18x + 36y + -48 = 18x + 36y + 48.
Algebraic expressions can only be simplified if they have “like terms,” which are terms with the same
variable and same exponent.
x+5+2
Only 5 + 2 can be simplified. x + 5 + 2 becomes x + 7
3x + x + 2
can be simplified to 4x + 2
2
x +x
cannot be simplified because they don’t have the same exponents and are therefore not like terms.
Algebraic expressions can be evaluated if you are given the value for the variables
variables.
Example: Evaluate
−x −15
6
for x = 3. Substitute 3 for x in the expression.
− (3) − 15 − 3 − 15 − 3 + ( −15) − 18
=
=
=
= −3
6
6
6
6
Example 2 p.168
If a=-2, evaluate -3a + 4a2
-3(-2) + 4(-2)2 Do exponents first
\= -3(-2) + 4(4) Multiplication left to right
= 6 + 4(4)
= 6 + 16
Addition
= 22
Example 3
E l t (8hg
Evaluate
(8h + 6g)
6 )2 for
f h=
h -11 andd g =55
Can we simplify what’s on the inside of the parentheses?
No, because the first term is 8hg and the second is 6g. They
are not like terms because 6g does not have an h.
h
Just go directly to substituting h= -1 and g =5 into the
expression. Surrounding them by parentheses reminds you
that 8hg actually means “8 times h times g.”
2
(8(-1)(5)
8( 1)(5) + 6(5) )
Inside the parentheses, we do multiplication first.
=(-40 + 30)2 We’re still simplifying what’s inside the
parentheses before we can do the exponent.
p
p
So do addition next.
= (-10)2
= 100
We now we can do the exponent
FORMULAS, FORMULAS, FORMULAS!
Formulas are just equations for finding certain types of things (distance, time, rate,
price, temperature, etc..)
Examples:
Distance Formula
Time units must correlate with speed units.
If you want speed in mph, then we need the time in hours.
Speed
Distance traveled = Rate X Time
d = rt
Example:
How fast were you driving if you
traveled 44 miles in 2 hours?
How fast? = speed = r
r = d/t
D= 44 miles, t=2 hours
R = (44 miles)/(2 hours)=22 mph
d
t = d/r
r
r = d/t
t
Distance an Objects Falls (due to the force of gravity)
The acceleration of gravity is 16 ft per sec2, so
Distance Fallen = 16 ft × (time fallen in seconds )2
sec 2
D = 16t2
Example 6 p.172:
Find the distance a camera fell in
seconds if it was dropped
overboard by a vacationer taking
a hot-air balloon ride.
Temperature Formula
Celsius =
5
(Farenheit − 32)
9
Example 5 p.171 Convert 77˚F to degrees Celsius
5
(F − 32)
9
9
F = C + 32
5
C=
Price Formulas
$ Sale Price= $ Original Price – $ Discount
Do pp.175 #66
$ Retail Price = $Cost + $ Markup
Selling price for an item
Amount the store paid for the item
p.175#64
A
Average
F
Formula
l (M
(Mean))
Mean =
Sum of Values
Number of Values
Amount the store raises the
price to make a profit
Example 7 p.172
Conversion Formulas
i h 
 12 inches
Inches = Feet × 

1
Foot


 1 Yard 
Yards = Feet × 

 3 Feet 
 1 Foot 
Feet = Inches × 

12
inches


 3 Feet 
Feet = Yards × 

 1 Yard 
 5280 ft 
Feet = Miles × 

1
Mile


 1 Mile 

Miles = Feet × 
5280
ft


 52 Weeks 
× 
W k = Years
Weeks
Y
 1 year 
 1 yyear 
Y
Years
= Weeks
W k ×
 52 Weeks 
 12 Months 

Months = Years × 
 1 year 
 60 seconds 
 1 minute 
Seconds = Minutes × 
 Minutes = Seconds × 

 1 minute 
 60 seconds 
 1 hour 
 60 minutes 
Hours = Minutes × 
Minutes = Hours × 


 60 minutes 
 1 hour 
 1 year 
Years = Months × 

 12 Months 
Example:
Express f feet in inches, and then in yards
 12 inches 
f feet × 
 = 12 f inches
1
foot


 1 yard  f
f feet × 
= yards
3
feet

 3
Express annual salary, s, into weekly salary. s ×  1 year  = s per week
Annual salary = s dollars per year = year  52 weeks  52
Sec. 3.1 #1--4 all, 9-19 odd,
21 67 odd
21-67
dd
Sec. 3.2 #17-77 EOO (Every
Other Odd)
Sec. 3.3 #19-87 EOO