##### 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.