Math 6 High ( Period 3)

Discount & Sales Tax 7.7

Let’s go shopping!
A reduction in the price of an item is called a discount. Discounts are often calculated using percent of the original price.
To find the price of an item that is discounted:

1) find the amount of the discount

2) subtract the discount from the original price.

You buy a pair of hiking shoes. The original price of $65 is discounted by 35%. What is the sales price?

Solution:
Find the amount of the discount

Discount = 35% of $65

Discount = 65 ( .35)
=22.75

$22.75 is what you will SAVE!! It isn’t what you will pay to purchase them so

Remember

Sales price = Original price – discount
65- 22.75 = 42.25

$42.25

Which television’s sale price is less expensive? Let’s assume they are the same quality
Television A: Discount of 25% off the original price of $575
Television B: Discount of 30% off the original price of $585.

Television A:  Sales price = 575 – (0.25 · 575)
                        575 -143.75
                        431.75

Television B: Sales price = 585 –(0.30 · 585)
                        585- 175.5
                        409.50

Television B’s sales price of $409.50 is less expensive.

Finding Sales Tax

A sales tax is an amount added to the price of a purchased item. Sales tax is often calculated using a percent of the price.
To find the total price of an item including sales tax:

1) Find the amount of the sales tax

2) Add the sales tax to the price.

Example:
You buy an ipod shuffle. The price is $40.00 and the sales tax is 8.25% What is the total price you will pay?

Solution:
Find the amount of the sales tax.

Sales tax       = 8.25% of 40
                        =0.0825(40)
                        = 3.3

Which means the sales tax is $3.30

Add the sales tax to the price

Total price     = Purchase Price + Sales Tax
                        = $40 + $3.30 = $43.30

You will pay $43.30.

                       
Finding a Total Price

Shirts in a store are on sale for 30% off. The sales tax is 7.25%.  What is the total price of a shirt that has an original price of $29.95?

Solution: First find the discount and  the sale price.
Then find the sales tax and add that to the sale price to find the total price.
Careful—lots of steps here!

Finding the discount and the sales price.

Discount         = 30% of 29.95
                        = 0.3(29.95)
                        = 8.99
Sale Price     = Original Price – Discount
                        = 29.95 – 8.99 = 20.96

Sale Price is $20.96

That is the amount you calculate your sales tax upon so

Finding the Sales tax and the Total Price

Sales tax       = 7.25% of 20.96

                        = 0.0725(20.96)=1.52

Total Price     =  Sale Price + Sales Tax
                        = 20.96 + 1.52 = 22.48

The total price including sales tax is $22.48 

Math 6 High ( Period 3)

Simple Interest 7.6

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.

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:

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 discover that

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

Therefore 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 6 High ( Period 3)

 Finding a Percent of a Number 7.2

or… it’s just proportions!

Percent/100 = Percent of a number/number

Find 30% of 400
30/100 = x/400
Its a proportion!
30(400) = 100(x)
30(400)/100 = x
120 = x
Therefore 120 is 30% of 400

You drive 300 miles. the first 20% of the road is under construction. How many miles of the road are under construction?
20/100 = x/300
Its a Proportion!!
20(300) = 100(x)
20(300)/100 = x
60= x
therefore 60 miles of the road are under construction.

You can also change the percent to a fraction and multiply!
For example,
Find 50% of 46. 
[Remember: the word "OF" in a mathematical sentence means to multiply]

50% is just 1/2 
so
1/2(46) = 23

75% of 120
75% = 3/4
so 3/4 (120) = 90

You can also change the percent into a decimal and multiply
Find 35% of 150
Read this as
What is 35% of 150?
x= 0.35(150)
x = 52.5
Find 92% of 75
Read this as
What is 92% of 75?
x = 0.92(75)
x = 69

Are these reasonable?
Well 92% is pretty close to 90% and !0% of 75 would be 7.5 so 90% would be 75-7.5 or 67.5.   Since  69 is very close to 67.5  the answer is reasonable.

Math 6High ( Period 3)

Comparing Positive and Negative Numbers 4.8

Up until now we have focused on positive and negative Integers—now we will look at Rational Numbers – negative Fractions, negative Decimals and negative Mixed Numbers!

Please review Page 202 for graphing Negative Decimals and Negative Fractions

Comparing Positive and Negative Numbers

Complete the statements using < , >,  or =

-1/6  ? -0.6

-9/8  ?  -1.125

-2 1/3  ?  -2.1

To find the correct solution for each of the above, rewrite the fractions and mixed numbers as decimals. Then graph the numbers on a number line and compare.

-1/6 is    - 0.1666… or rounded   -0.17

-0.17 is to the right of -0.6 so

-1/6 > -0.6

-9/8 = -1 1/8 = -1.125

so -9/8 is at the same point on the number line as -1.125 so

-9/8 = -1.125

-2 1/3 = -2.33333 or rounded to -2.3 which is to the left of -2.1

so

-2 1/3 < -2.1

Ordering Positive and Negative Numbers

Write  1.3 ,     -2 ½ ,  - 2.6  and 3/8 in order from least to greatest.

Solution

1.  First rewrite the fractions as decimals ( or the decimals  as fractions)

-2½ = -2.5  and 3/8 = 0.375

2. Then draw a number line. You can estimate to plot the points.

(See our textbook page 203)

From least to greatest

-2.6,    -2 ½ ,   3/8,   1.3

Math 6 High (Period 3)

Integers & The Number Line 4.1

Integers are the Counting Numbers (also known as Natural Numbers), their opposites, and ZERO.

We have Negative Integers, Positive Integers and Zero. 

The symbol used to represent negative integers is a “negative sign.” 

-4 is normally read as negative 4 

BUT it is also understood to mean “the opposite of 4”

On a number line,
the negative numbers are to the LEFT of ZERO and
the positive numbers are to the RIGHT of ZERO.

Please turn to Page 167 in our textbook and look at the examples of graphing various negative integers on a number line. 

You will notice that -4 is to the left of -1… Therefore -4 < -1  That is -4 IS LESS THAN -1. 

You could also state that -1> -4. That is, -1 IS GREATER THAN -4.

Negative numbers are used in real-life situations to represent sub-zero temperatures as well as losses that occur in business.