Math 7 ( Period 4)

Simple Interest 7.8

 Interest is $ paid for the use of money. The amount you deposit or borrow is called the principal. When you put money in a savings account at a bank, the bank pays you interest. When you borrow money from a bank, you must pay the bank interest to the bank in addition to paying bank the money that you borrowed.

The Percent of increase in principal is interest rate. When interest is paid only on the principal you have SIMPLE INTEREST

A principal is an amount borrowed, loaned or saved. An annual interest rate is the percent of the principal you earn or pay as interest for the year. SIMPLE INTEREST is the product of the principal, the annual interest rate,  and the time in years.

Interest = Principal ·rate ·time

I=Prt

Examples:

Example:

 $200 into the bank 6% per year for 1 year.

I = (200) (0.06)(1) = 12

I = (200)(6/100)(1) = 12

I like using fractions if I can simplify before I need to start multiplying

Example:

Deposit $ 500 in savings account for 9 months  simple interest is 1.5%

I = Prt

I = (500)(0.015)(.75)

I = (500)(15/1000)(3/4)

I = $ 5.625 which rounds to $ 5.63

Example:

You borrow $ 250 from your family. After 6 months yu pay back the $ 250 plus $10 in interest. What was the simple interest rate?

I = Prt

What do we know?
10 = (250)(r)(0.5)

10 = 125r

Now we have a simple one step equations

10/125 = r

We still need to put this as a %

simplify 10/125 to

2/25 and that easily changes to 8/100 so the interest rate is 8%

Balance à when you add interest to the principal the result is called the balance.

“A” is used to represent balance

A =  P + I

but then we could write it as

A = P + Prt

Example:
At June 1, a credit card balance was $2500. The cc co charges 21% interest ( annual rate). If no payments are made during  June, what is the balance on July 1?

A = P + Prt

A = 2500 + (2500)(.21)(1/12)  or A = 2500 + (2500)(21/100)(1/12)

A = 2500 + 43.75

A = $2543.75

Some additional examples:

You borrow $1500 from a friend for the down payment on a car. Your friend charges you an annual interest rate of 8% ( Nice friend!) Find the simple interest you will pay in 1 year.

Solution:
Remember   I = Prt

What do you know?

I = (1500) (0.08)(1) 
Most of the time change the interest rate to a decimal. Occasionally you will want to use fractions

Carefully multiply... and

I = 120

Therefore: The simple interest you will pay in 1 year is $120.

You deposit $300 in a savings account. The annual interest rate is 3% ( not much) Find the simple interest you will earn in 1 month.

What do you know?   You know the principal  ( or P in the formula) is $300. You know the annual interest rate ( or r in the formula)  is 3%  But.. wait… be careful…the time must be in years  this is only for one month. The time ( or t in the formula is 1/12)

Solution:

Remember   I = Prt

I = (300)(0.03)(1/12)

I would carefully simplify before I multiplied

I = 0.75

Again, think what does that mean? The simple interest you would earn on $300 in 1 month at that rate is $0.75 When you know the values of any three of the variables in the formula   I = Prt  you can use substitution to find the value of the fourth variable.

Your savings account earns $68 in simple interest in 1 year. The annual interest rate is 8% what is the principal?

Ask yourself, what do I know? What are the three variables  in the formula  I = Prt   that I do know?

$68 is the interest. The rate is 8% and it’s only for 1 year.

so

Remember   I = Prt

68 =P(0.08)(1)

68=0.08p

Solve this one-step equation

Divide both sides by 0.08 carefully

68/0.08 = 0.08p/0.08

850 = p

the principal is $850

You want to open an account with $100.
At Bank A you will earn $0.65 in 3 months.
At Bank B you will earn $2.25 in one year. At which bank will you open your account?
(Although banks do not pay simple interest – they use compound interest, we will assume that Bank A and B are paying only simple interest)

Figure out each bank’s simple interest rate.

Remember   I = Prt  

Bank A:   3 months is 3/12 or ¼  of a year so 

0.65 =  (100)(r) (¼)
0.65= 25r

Solve this one step equation
Divide both sides by 25

0.65/25= 25r/25

0.026= r
We need to change the decimal into a percent
That means Bank A’s rate is 2.6%

Bank B: Yes you can use the formula but look at what you know

2.25 =(100)(r)(1)
That means  r= 0.225
or that the rate for Bank B is 2.25%

You would earn more money if you deposit your savings into Bank A

Math 6A ( Periods 1 & 2)

Computing with Percents 9-3
The statement 20% of 300 is 60 can be translated into the following equations
20/100(300) = 60 or 0.20 •300 = 60

EQUATION METHOD:
Notice the following relationship between the words and the symbols
20% of 300 is 60
0.20 • 300 = 60

WRITE THE PROBLEM OUT AND THEN DIRECTLY UNDER THE "IS" WRITE AN EQUAL SIGN. DIRECTLY UNDER THE WORD 'OF" WRITE A MULTIPLICATION SIGN. iF YOU ARE GIVEN A % CHANGE IT FIRST TO A DECIMAL. THEN BRING DOWN ALL THE OTHER NUMBERS GIVEN IN YOUR PROBLEM. LET x OR n REPRESENT YOUR VARIABLE... THAT IS THE "WHAT " PART OF YOUR PROBLEM.

A similar relationship occurs whenever a statement or a question involves a number that is a percent of another number

What is 8% of 75?
Let n represent the number asked for
What number is 8% of 75?
n = 0.08 • 75

solve
What percent of 40 is 6?
let n represent the percent asked for.

What percent of 40 is 6?
n% • 40 = 6
n% • 40 = 6
n% (40)/40 = 6/40
n% = 6/40
n/100 = 6/40
(100) n/100 = (100) 6/40
n=15 so 15% of 40 is 6

140 is 35 % of what number?
let n represent the number asked for
140 is 35% of what number?
140 = 0.35 • n
140 = 0.35n
140/0.35 = 0.35n/0.35 divide carefully!! Watch those decimals!!
400 = n
so 140 is 35% of 400

Always check to see if your answer is logical.

PROPORTION METHOD

In these types of percent problems you are always know three parts of the following proportion

n/1oo = a/b
or better yet
n/100 = is/ of

The n represents the %

Read the problems carefully and you can easily determine which is the "is" and which represents the 'of"
For example:
What percent of 40 is 6?
What percent -- from the problem above indicates that we DO NOT know the n
of 40-- hmm... then 40 must be the 'of' and
similarly is 6 means that 6 represents the 'is'

n/100 = 6/40 solve as a proportion
and you get n= 15 but since it asked us to state the 5 your answer is 15%

140 is 35 % of what number?
In this problem I notice 35% right away so that is the n!!
Then I read the problem again and notice 140 is... hmmm.. THat says 140 must be the is

35/100 = 140/ x I do not know the 'of'

Solve again
x = 400

Algebra Honors ( Periods 6 & 7)

Work Problems 7-8

To solve work problems use the following formula
work rate × time = work done
or rt = w
Work rate means the fractional part of a job done in a given unit of time.
For example if it take you 3 hours to clean up your room, what part of the job can be done in 1 hour? That's easy... 1/3
To finish a job the sum of the fractional parts of the work done must be 1.
( for one whole job completed)

Josh can split a cord of wood in 4 days. His father can split a cord in 2 days. How long will it take them to split a cord of wood if they work together?
Let x = the number of days needed to do the job together.
Josh and his father will each work x days
Using those great tables from class fill in with the information you know
Since Josh can do the whole job in 4 days his work rate is 1/4 job per day.
His father's work rate is 1/2 job per day.

**posting the TABLE HERE**

Josh's part of the job = x/4
His father's part of the job = x/2
so the sum of that would equal the job completed
OR

Josh's part of the job + His father's part of the job = Whole JOB
x/4 + x/2 = 1
Clear the equation of fractions by multiplying by the LCD
4(x/4 + x/2) = 4(1)
x + 2x = 4
3x = 4
x= 4/3
It would take them 1 1/3 days to do the job together.

Robot A takes 6 minutes to weld a fender. Robot B takes only 5 1/2 minutes. If they work together for 2 minutes, how long will it take Robot B to finish welding the fender by itself?

Let x = the number of minutes needed for Robot B to finish the work.
Robot B's work rate is 1/5.5 or 1/(11/2) = 2/11

***posting the TABLE HERE***
Robot A's part is (1/6)(2)
Robot B's part is (2/11)(2 +x)
A's part of the job + B's part of the Job = Whole JOB

1/3 + (2/11)(2 + x) = 1
Multiply by the LCD, which is 33

(33)[1/3 + (2/11)(2 + x)] = 33(1)
11 + 6(2 + x) = 22
11 + 12 + 6x = 33
6x = 10
x = 5/3
It will take 1 2/3 minutes for Robot B to finish welding.
The charts or tables for work problems look similar to the charts and tables used for other problems. The following formulas show the similarities among some types of problems you have studied

Work done by A + work done by B = TOTAL work done
Acid in solution A + acid in solutions B = TOTAL acid in mixture
Interest from banks + Interest from Bonds = TOTAL Interest
Distance by bike + Distance by car = TOTAL distance traveled

Math 7 ( Period 4)

Percent of Increase or Decrease 7.7

A percent of change tells how much a quantity has increased or decreased relative to the original amount. you can use a ratio to find the percent of change.

change/ original

AGAIN—use the LADYBUG METHOD

When the new amount is greater than the old amount then you have a percent of increase. When the new amount is less than the old amount, you have a percent of decrease.

The enrollment in a middle school was 400 students for 1999, and 420 for 2000. Find the percent of change form 1999 to 2000.

LADYBUG-  find the difference 420 -200 = 20

20/400 = 2/40 = 0.05 = 5%

The percent of change was 5%

You place a 5 in by 7 inc photograph in a photo enlarger. You want to enlarge the photograph so it measures 7 inches by 9.8 inches. What enlargement setting should you use?

Use either the width or the length of each to find the percent of increase.

I want to use  the width

Enlarged width- original width
original width

7 -5 = 2

That’s the change

2/5 = 4/10 = 40%

The percent of increase is 40%. This means that the enlarged size  is 100% + 40% = 140%

Original grade =80 and new grade is 90

What’s your percent of increase

10/80 = 12.5%

What happens if your original grade was 90 and your new grae was 80  ( oh dear…)

10/90 = 11 1/9% decrease

Why are the %’s difference when the change is the same ( it was 10 in both cases) The c Percent of Change is based on WHAT YOU STARTED WITH.

Okay how about an original grade of 20 ( yes I said 20%-- Yikes…)

New grade is 30

30-20 = 10

What was the original? 20

so 10/20 = 50% Now, I know that this student is still failing but he ( she) has increase the grade 50% from what it had been.

This student can tell his parents that he increased his grade by 50% 
What his parents should ask is “From what?”

This is how a percent of change can be misused,  by advertisers.

Have you ever heard of  “ your teeth will be whiter by 50%.”  50% of what?

Or “Your gas mileage will increase by 50% if you use … gas..” Again, ask… 50% increase of what?

Math 7 ( Period 4)

Mark Up & Discount 7.6

A retail store buys items at wholesale prices. To cover expenses and make a profit, the store sells the items at high retail prices. The DIFFERENCE between the retail and wholesale prices is called the markup.

 Mark up = Retail price – Wholesale price

 A store buys a shirt at wholesale price of # 13.50 and sells it for $ 24.95. What is the amount of mark up?

Markup = Retail – Wholesale

24.95 – 13.50

$11.45

To find the percent of markup use the wholesale price as the original and the amount of markup as the difference or change! We use the LADYBUG Method…

Percent of markup =

Markup/wholesale price

11.45/13.50

0.85 or

85%

 Often times a store will use a fixed percent of markup. SO If a store bought a piece of jewelry fat a wholesale price of $ 180 and the percent of mark up was 150% what is the retail price?

First find the mark up

180(150%) = 180(1.50) = 270

$270 is the mark up. YOU MUST add that to the wholesale price

270 + 180 = $450

When an item is on sale, the difference between the regular price and the sale price is called the discount.

Discount = Regular price –sale price

The percent of discount is very similar to the percent of markup!

USE THE LADYBUG METHOD

$36book on sale for $ 27

Find the amount of discount

Find the percent of discount

36 – 27 = 9

9/36 = ¼  = 25%

Math 6A ( Periods 1 & 2)

Percents and Decimals 9-2

By looking at the following examples, you will be able to see a general relationship between decimals and percents

57% = 57/100
0.79 = 79/100 = 79%
113% = 113/100 = 1 13/100
0.06 = 6/100 = 6%

Rules

To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign

57% = 0.57
113% = 1.13

To express a decimal as a percent, move the decimal point two places to the right and add a percent sign

0.79 = 79%
0.06 = 6%


In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method

Rule
To express a fraction as a percent, first express the fraction as a decimal
and then as a percent

Express 7/8 as a percent

Divide 7 by 8
7/8 = 0.875 = 87.5%

Express 1/3 as a percent
divide 1 by 3
0.33333….. it’s a repeating decimal
express the decimal as a percent 0.333… = 33 1/3%
so, to the nearest tenth of a percent = 33.3% but it is much more accurate to keep the 1/3 and write 33 1/3%

Math 6A ( Periods 1 & 2)

Percents and Fractions 9-1

The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100

A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100

Our book’s example is as follows;
During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws. 17/25 or 7/10
We have calculated this type of problem before.. this time when we compare fractions use the common denominator 100, even if 100 is not the LCD of the fractions.
17/25 = 68/100 and
7/10 = 70/100

Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.
the ratio of a number to 100 is called a percent. We write percents by using the symbol %
so
17/25 =68% and
7/10= 70%

Rule
To express the fraction a/b
as a percent, solve the equation

a/b = n/100

for the variable n and write n%


Express 17/40 as a percent
n/100 = 17/40 multiply both sides by 100 100 ( n/100) = 17(100)/40
n = 17(100)/40 n = 85/2 n= 42½

Therefore, 17/40 = 42 1/2 %


Rule
To express n% as a fraction, write the fraction

n/100 in lowest terms


Express 7 ½ % as a fraction in lowest terms

7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal?
multiply the numerator and the denominator by 10
7.5(10)/100(10) simplify
Similarly, you could change a mixed numebr into an improper fraction
5 3/8% becomes 43/8 % and to change that to a fraction simple divide by 100
That looks messy but if you remember that to divide by 100 you are actually multiplying by 1/100
(43/8) (1/100) = 43/800 and you are finished with your calculations!! EASY!!

Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%

1 = 100/100 = 100%

165/100 = 165 %

Write 250% as a mixed number in simple form

250% = 250/100

250/100 = 2 50/100 = 2 1/2

The town of Wonderful spends 42% of its budget on education. What percent is used for other purposes?
the whole budget is represented by 100%. Therefore, the part used for other purposes is

100 - 42 or 58%

MAth 6A (Periods 1 & 2)

Graphs of Equations 11-9 (cont'd)

The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 - x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!