Download Mardi received an inheritance of 60,000. She invested part of at 10

Mardi received an inheritance of 60,000. She invested part of at 10% and deposited the
remainder in tax-free bonds at 11%. Her total annual income from the investments was
6100. Find the amount invested at 10%.
Let x represent the amount invested at 10%.
Let y represent the amount invested at 11%.
The total amount invested is x + y = 60,000.
The total income from the investments is 0.10x + 0.11y = 6,100
Solving the first equation for y gives:
x + y – x = 60,000 – x
y = 60,000 – x
Substituting this into the second equation and solving for x gives:
0.10x + (0.11)(60,000 – x) = 6,100
0.10x + 6,600 – 0.11x = 6,100
-0.01x + 6,600 = 6,100
-0.01x + 6,600 – 6,600 = 6,100 – 6,600
-0.01x = -500
x = -500 / -0.01
x = 50,000
The amount invested at 10% was 50,000.
The speed of a stream is 6mph. If a boat travels 42 miles downstream in the same time
that it takes to travel 21 miles upstream, what is the speed of the boat in still water?
Let s represent the speed of the boat in still water.
When the boat is traveling downstream, the total speed is s + 6.
When the boat is traveling upstream, the net speed is s – 6.
The time for the boat to travel 42 miles downstream is:
t=
distance traveled downstream
42
=
total speed downstream
s+6
The time for the boat to travel 26 miles upstream is:
t=
distance traveled upstream
21
=
total speed upstream
s−6
Since the two times are equal, we can write:
42
21
=
s+6 s−6
Multiplying both sides of the equation by (s + 6)(s – 6) gives:
42 ( s − 6 ) = 21( s + 6 )
Dividing both sides of the equation by 21 gives:
(42/21)(s – 6) = (21/21)(s + 6)
2(s – 6) = s + 6
2s – 12 = s + 6
2s – s – 12 = s – s + 6
s – 12 = 6
s – 12 + 12 = 6 + 12
s = 18
The speed of the boat in still water is 18 mph.
Andy has 26 coins made up of quarters and half dollars, and their total value is 11.50.
How many quarters does he have?
Let Q represent the number of quarters that Andy has.
Let H represent the number of half-dollars that Andy has.
Then the total number of coins is Q + H = 26
The total value of the coins is 0.25Q + 0.50H = 11.50
Solving the first equation for H gives:
Q – Q + H = 26 – Q
H = 26 – Q
Substituting this into the second equation gives:
0.25Q + (0.50)(26 – Q) = 11.50
Solving this for Q gives:
0.25Q + 13 – 0.5Q = 11.50
13 – 0.25Q = 11.50
13 – 13 – 0.25Q = 11.50 – 13
-0.25Q = -1.50
Q = -1.50 / -0.25
Q=6
Andy has 6 quarters.
Solve the equation:
x+2/x^2-16 + x-3/x^2-3x-4 = 2x-3/x^2+5x+4
x+2
x −3
2x − 3
+ 2
= 2
2
x −16 x − 3x − 4 x + 5x + 4
Factoring the denominators gives:
x+2
x −3
2x − 3
+
=
( x + 4) ( x − 4) ( x − 4) ( x +1) ( x + 4) ( x +1)
The LCM of the denominators is (x + 4)(x – 4)(x + 1).
Multiplying each term by (x + 4)(x – 4)(x + 1) gives:
( x + 2) ( x +1) + ( x − 3) ( x + 4) = (2x − 3) ( x − 4)
Multiplying out the products gives:
x2 + 2x + x + 2 + x2 + 4x – 3x – 12 = 2x2 – 8x – 3x + 12
2x2 + 4x – 10 = 2x2 – 11x + 12
4x – 10 = -11x + 12
4x + 11x – 10 = -11x + 11x + 12
15x – 10 = 12
15x – 10 + 10 = 12 + 10
15x = 22
x = 22/15
Perform the indicated operation and express in lowest terms.
x/x-4 + 8/x+4 -32/x^2-16
x
8
32
+
− 2
x − 4 x + 4 x −16
The LCM of the denominators is x2 – 16.
Multiplying the first term by
x+4
x−4
, and multiplying the second term by
, gives:
x+4
x−4
x " x + 4%
8 " x − 4%
32
$
'+
$
'− 2
x − 4 # x + 4 & x + 4 # x − 4 & x −16
x ( x + 4) 8 ( x − 4)
32
+ 2
− 2
2
x −16
x −16 x −16
Multiplying out the terms in the numerators gives:
x 2 + 4x 8x − 32
32
+ 2
− 2
2
x −16 x −16 x −16
Combining terms gives:
x 2 + 4x + 8x − 32 − 32
x 2 −16
x 2 +12x − 64
x 2 −16
Factoring both the numerator and denominator gives:
( x +16) ( x − 4)
( x + 4) ( x − 4)
Canceling the x – 4 terms gives:
x +16
x+4