Document Text Contents
Page 1
visual math
Page 135
What we put into the A = P(1 + �n
r
�)nt formula:
� A is the total amount
� P is the original principal
� r is the rate
� n is the number of yearly compounds
� t is time (in years)
To find your “n” look out for these terms:
� compounded annually means interest is paid each year
� compounded semiannually means interest is paid two times per year
� compounded quarterly means interest is paid four times per year
� compounded monthly means interest is paid every month
� compounded daily means interest is paid every day
So, let’s look at an example. You open a savings account that pays 3% in-
terest semiannually. If you put in $1,000 initially, how much do you have af-
ter 2 years?
We use A = P(1 + �n
r
�)nt, and you substitute in the following values:
P = 1,000
r = 3%, or .03
n = 2 (compounded semiannually means twice a year)
t = 2
A = P(1 + �n
r
�)nt = 1000(1 + �.02
3
�)2•2
= 1,000(1 + .015)4
= 1,000(1.015)4
= 1,000(1.06)
= 1,061.36
–
= 1,061.37
Always round money to the nearest cent. Thus, you’d have $1,061.37.
127percents
Page 136
You ask: “What if I don’t want to memorize that scary formula?” Well, you have a
few options:
� You can do the calculation “the long way.” For example you would
know that after �12� a year, the $1,000 principal above would earn I =
PRT, or I = 1,000 × .03 × �12� = $15. Now the account has $1015. In
another �12� year you earn I = PRT = 1,015 × .03 × �
1
2� = 15.23, and you
would have $1,030.23. You would continue calculating in this manner
until you completed two years worth of money making.
� You can find out if there is a reference sheet that may contain this for-
mula (if you are taking a standardized test).
� You can use process of elimination on tests. Cross off any preposterous
answers and try to pick one that would make sense.
� TIP: In doing a compound interest test question, you know that a lot
of people would tend to accidentally solve it as if it were a simple inter-
est question. And you can bet the test designers know this! So, you can
cross off the answer that represents I = PRT (the simple interest for-
mula) and pick an answer that is greater.
Exercise 8: Evan opens a savings account that pays 5% interest quarterly. If
he put in $2,000 initially, how much does he have after six months?
algebraic percents
Let’s say that Jaclyn buys a printer for D dollars and gets a 20% discount. How
do you represent this mathematically?
Well, if Jaclyn is getting a 20% discount, she must be paying 80% of the orig-
inal price. What is the original price? D. So she is paying 80% of D. This is
just .8 • D, or .8D.
What if she was buying three items that cost D, E, and F dollars each, and
she was getting the same 20% discount on her entire order?
Well, without the discount, her cost would be (D + E + F ).
non-discounted = (D + E + F )
visual math128
Page 269
29. b. To figure out the probability for the given outcome, you need to cal-
culate the total possible outcomes. You know that the record com-
pany brought 300 hip-hop CDs, 500 alternative rock CDs, 200 easy
listening CDs, and 400 country CDs. The total possible outcomes
equal 300 + 500 + 200 + 400 = 1,400. The outcomes that fit the
criteria in the question = 200. This is because 200 easy listening CDs
will be given out. This means that the chance of getting an easy lis-
tening CD will be �1
2
,4
0
0
0
0�. This reduces to �
1
7�.
30. c. Given log10 10
5, you spiral through and say:
The power would be 5, so log10 10
5 = 5.
31. b. The area of the square is s2 = 72 = 49. If you double the side, the
new area equals s2 = 142 = 196. 49 times 4 equals 196, so you
quadrupled the area. Notice that you can “see” this effect below:
Algebraically, any square that has its side doubled will also have its
area quadrupled because when you compare s2 to (2s)2 you get:
s2 versus (2s)2
or
s2 versus 4s2
4s2 is obviously 4 times the s2.
261test your math skills
Page 270
visual math262
32. b. You set up the proportion as follows:
�7 day
4
s p
d
e
ay
r
s
week� = �1
x
00�
Thus, choice b is correct.
33. d. You need to take V and add 25% of V. 25% = .25 or �1
2
0
5
0� = �
1
4�. Thus,
choice a, V + �14� V is true. This is the same as V + .25 V, which is
choice c. If you actually add choice c, you get choice b.
34. c. Note the dimensions of the label (when peeled from the can):
The area of the label will be L × W = circumference × 4 in. C =
2πr = 2 • �27
2
� • 3.5 = 7 • �27
2
� = 22 in. Thus, the area = 22 in × 4 in =
88 in2.
35. d. You need to find the area of the big (outer) circle and subtract the
area of the small (inner) circle. You draw in the radius of the “big”
circle:
You can also draw in the radius of the small circle:
The big circle has an area equal to A = πr2 = π(12)2 = 144π. The
small circle has an area of A = πr2 = π(8)2 = 144π. Thus, the area of
the shaded region is 144π − 64π = 80π ft2.
visual math
Page 135
What we put into the A = P(1 + �n
r
�)nt formula:
� A is the total amount
� P is the original principal
� r is the rate
� n is the number of yearly compounds
� t is time (in years)
To find your “n” look out for these terms:
� compounded annually means interest is paid each year
� compounded semiannually means interest is paid two times per year
� compounded quarterly means interest is paid four times per year
� compounded monthly means interest is paid every month
� compounded daily means interest is paid every day
So, let’s look at an example. You open a savings account that pays 3% in-
terest semiannually. If you put in $1,000 initially, how much do you have af-
ter 2 years?
We use A = P(1 + �n
r
�)nt, and you substitute in the following values:
P = 1,000
r = 3%, or .03
n = 2 (compounded semiannually means twice a year)
t = 2
A = P(1 + �n
r
�)nt = 1000(1 + �.02
3
�)2•2
= 1,000(1 + .015)4
= 1,000(1.015)4
= 1,000(1.06)
= 1,061.36
–
= 1,061.37
Always round money to the nearest cent. Thus, you’d have $1,061.37.
127percents
Page 136
You ask: “What if I don’t want to memorize that scary formula?” Well, you have a
few options:
� You can do the calculation “the long way.” For example you would
know that after �12� a year, the $1,000 principal above would earn I =
PRT, or I = 1,000 × .03 × �12� = $15. Now the account has $1015. In
another �12� year you earn I = PRT = 1,015 × .03 × �
1
2� = 15.23, and you
would have $1,030.23. You would continue calculating in this manner
until you completed two years worth of money making.
� You can find out if there is a reference sheet that may contain this for-
mula (if you are taking a standardized test).
� You can use process of elimination on tests. Cross off any preposterous
answers and try to pick one that would make sense.
� TIP: In doing a compound interest test question, you know that a lot
of people would tend to accidentally solve it as if it were a simple inter-
est question. And you can bet the test designers know this! So, you can
cross off the answer that represents I = PRT (the simple interest for-
mula) and pick an answer that is greater.
Exercise 8: Evan opens a savings account that pays 5% interest quarterly. If
he put in $2,000 initially, how much does he have after six months?
algebraic percents
Let’s say that Jaclyn buys a printer for D dollars and gets a 20% discount. How
do you represent this mathematically?
Well, if Jaclyn is getting a 20% discount, she must be paying 80% of the orig-
inal price. What is the original price? D. So she is paying 80% of D. This is
just .8 • D, or .8D.
What if she was buying three items that cost D, E, and F dollars each, and
she was getting the same 20% discount on her entire order?
Well, without the discount, her cost would be (D + E + F ).
non-discounted = (D + E + F )
visual math128
Page 269
29. b. To figure out the probability for the given outcome, you need to cal-
culate the total possible outcomes. You know that the record com-
pany brought 300 hip-hop CDs, 500 alternative rock CDs, 200 easy
listening CDs, and 400 country CDs. The total possible outcomes
equal 300 + 500 + 200 + 400 = 1,400. The outcomes that fit the
criteria in the question = 200. This is because 200 easy listening CDs
will be given out. This means that the chance of getting an easy lis-
tening CD will be �1
2
,4
0
0
0
0�. This reduces to �
1
7�.
30. c. Given log10 10
5, you spiral through and say:
The power would be 5, so log10 10
5 = 5.
31. b. The area of the square is s2 = 72 = 49. If you double the side, the
new area equals s2 = 142 = 196. 49 times 4 equals 196, so you
quadrupled the area. Notice that you can “see” this effect below:
Algebraically, any square that has its side doubled will also have its
area quadrupled because when you compare s2 to (2s)2 you get:
s2 versus (2s)2
or
s2 versus 4s2
4s2 is obviously 4 times the s2.
261test your math skills
Page 270
visual math262
32. b. You set up the proportion as follows:
�7 day
4
s p
d
e
ay
r
s
week� = �1
x
00�
Thus, choice b is correct.
33. d. You need to take V and add 25% of V. 25% = .25 or �1
2
0
5
0� = �
1
4�. Thus,
choice a, V + �14� V is true. This is the same as V + .25 V, which is
choice c. If you actually add choice c, you get choice b.
34. c. Note the dimensions of the label (when peeled from the can):
The area of the label will be L × W = circumference × 4 in. C =
2πr = 2 • �27
2
� • 3.5 = 7 • �27
2
� = 22 in. Thus, the area = 22 in × 4 in =
88 in2.
35. d. You need to find the area of the big (outer) circle and subtract the
area of the small (inner) circle. You draw in the radius of the “big”
circle:
You can also draw in the radius of the small circle:
The big circle has an area equal to A = πr2 = π(12)2 = 144π. The
small circle has an area of A = πr2 = π(8)2 = 144π. Thus, the area of
the shaded region is 144π − 64π = 80π ft2.
