Math 6A (Periods 2 & 4)

Commission And Profit 9-6 

Some sales jobs pay an amount based on how much you sell. This amount is called a commission.

Like a discount, the commission can be expressed as a percent or as an amount of money.

amount of commission = percent of commission X total sales.

Using the examples from our textbook,

Maria sold $42,000 word of insurance in January. If her commission is 3% of the total sales, what was the amount of her commission in January?

amount of commission = percent X total sales
0.03 X 42,000 = 1260
Her commission was $1,260.

Profit is the difference between total income and total operating costs.

profit = total income – total costs

The percent of profit is the percent of total income that is profit

percent of profit = profit/total income
A shoe store had an income of $8600 and operating costs of $7310. What percent of the store's income was profit?

profit= income- total costs = 8600 -7310 = 1290

percent of profit = profit/total income = 1290/8600 = 0.15
So the percent of profit was 15%.

Practice finding 10%-- its easy--- just move the decimal over one place.
We practiced finding 20%. Just double what you got for 10%.



MATH AT WORK:
Caterer
A caterer provides food for parties, weddings, bar/bat mitzvahs, and other events. Caterers plan the menu, buy the ingredients, and cook the food. Often they provide seating and music as well. For each event, a caterer determines the cost per guest. The catering business requires a thorough knowledge of ratios, proportions, and percents.

Algebra Honors (Periods 5 & 6)

Solving Problems with Two Variables 9-3

Although the textbook uses charts and tables for these word problems, I think they work easily without the charts...

John has 15 coins -- all dimes and quarters , worth $2.55 How many dimes and quarters does he have?
let d = the number of dimes and let q = the number of quarters
we know d + q = 15 and we know 10d + 25q = 255
using a system of equations and the substitution method
since d + q = 15 we know d = 15- q
10d + 25q - 255
10(15-q) + 25 q = 255
150 -10q + 25q = 255
150 + 15q = 255
15q = 105
q = 7
He has 7 quarters and 8 dimes


Ann and Betty together have $ 60. Ann has $9 more than twice Betty's amount. How much money does each have?
Let a = the amount Ann has and let b = the amount Betty has.
we know
a + b = 60
and we know
a= 2b + 9
so using a + b = 60
(2b+9) + b = 60
3b = 51
b = 17
Betty has $17 and Ann has (60-17) = $43

Joan Wu invested $8000 in stocks and bonds. the stocks pay 4% interest and the onds pay 7% interest . The annual interest from the stocks and bonds is $500.
How much is invested in bonds?
let s = the amount invested in stocks
let b = amount invested in bonds
s + b = 8000
0.04s + 0.07b = 500
clear the decimals
4s + 7b = 50000
but we know s + b = 8000 or s = 8000 -b
4(8000 -b) + 7b = 50000
32000 - 4b + 7b = 50000
3b = 18000
b = 6000
She invested $6000 in bonds.

Math 6A (periods 2 and 4)

 Discount & Markup 9-5

A discount is a decrease in the price of an item. A markup is an increase in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A store may announce a discount of $3 off the original price of $30 basketball, or a discount of 10%

A warm-up suit that sold for $42.50 is on sale at a 12% discount. What is the sale price?

Method 1: Use the formula

amount of change = percent of change X original amount
= 12% X $42.50

therefore the discount is 0.12 X 42.50 or 5.10
The amount of discount is $5.10
The sale price is 42.50 – 5.10 = $37.40

Method 2: Since the discount is 12%, the sale price is 100% - 12% = 88%.
The sale price is 0.88 X 42.50 = $ 37.40

When you know the amount of discount you subtract to find the new price. When dealing with a markup you add to find the new price.

The price of a new car model was marked up 6% over the previous year’s model. If the previous year’s model sold for $7800, what is the cost of the new car? {and what kind of a car could that be?}
Method 1: Use the formula
amount of change = percent of change X original amount

= 6% X 7800

Therefore the markup is 0.06 X7800= $468
The new price is 7800 + 468 = $8268

Method 2: Since the markup is 6% the new price is 100% + 6% or 106% of the original price. so the new price is 1.06 X7800 = $8268

This year a pair of ice skates sells for $46 after a 15% mark up over last year’s price. What was last year’s price?

This year’s price is 100 + 15 or 115% of last year’s price. Let n present last year’s price
46 = (115/100)n

46 = 1.15n
46/.15 = 1.15n/1.115
40 = n

So last year’s price was $40.

A department store advertised eclectic shavers at a sale price of $36.
If this is a 20% discount, what was the original price?
The sale price is 100 - 20 or 80% of the original price. Let n represent the original price.
36 = (80/100)n
36 = .8n
36/.8 = .8n/.8

45 = n
The original price was $45.
Check to see that your answers are logical and reasonable.

Try these: A service station (that’s gas station, now—they no longer provide service!!) give cash customers a 5% discount on the price of gasoline. If gasoline regularly sells for $3.00 a gallon, what is the discounted price?


A store marks up the price of a $5 item to $12. What is the percent of markup?

Algebra Honors ( Periods 5 & 6)

Solving Systems of Linear Equations
The Graphing Method 9-1
Two or more equations in the same variables form a system of equations. The solution of a system of two equations in two variables is a pair of values x and y that satisfies each equation in the system. The point corresponding to the ordered pair (x, y) must lie on the graph of both equations.
Solve the system by graphing
2x - y = 8
x + y = 1

Solution:
Graph both 2x - 7 = 8 and x + y = 1 in the same coordinate plane.
We did this in class by transforming both equations to slope-intercept form (y = mx +b)
and then graphed them. We noticed that the only point on BOTH lines is the intersection point ( 3, -2)
The only solution of both equations is (3, -2).
You can check that ( 3, -2) is a solution fof the system by substituting x = 3 and y = -2 in BOTH eqquations.

Solve the system by graphing
x - 2y = -6
x -2y = 2

When you graph the equations in the same coordinate plane, you see that the lines have the same slope but different y-intercepts. The graphs are parallel lines. SInce the lines do not intersect, there is no point that represents a solution of both equations.
Therefore, the system has NO SOLUTION.

Solve the system by graphing
2x + 3y = 6
4x + 6y = 12

When you graph the equations in the same coordinate plane, you see that the graphs coincide. The equations are equivalent. Every point on the line represents a solution of BOTH equations.
Therefore, the system has infinitely many solutions.

The Graphing Method in review:
To solve a system of linear equations in two variables, draw the graph of each linear equation in the same coordinate plane...
--> if the lines interset there is only one solutions, namely the intersection point.
--> if the lines are parallel, there is no solution
--> if the lines coincide, there are infinitely many solutions.

The Substitution Method 9-2

There are several ways to solve a system of equations, In the substitution method we use either equation to solve for one variable in terms of the other.
Solve
x + y = 15
4x + 3y = 38

Solve the first equation for y
x + y = 15
becomes
y = -x + 15
Substitute this expression for y in the other equation, and solve for x
4x + 3y = 38
4x + 3(-x+15) = 38
4x -3x + 45 = 38
x + 45 = 38
x = -7

Substitute the value of x in the equation in your first step and solve for y
y = -x + 15
y = -(-7) + 15
y = +7 +15
y = 22
CHeck x = -7 and y = 22 on BOTH equations
x + y = 15
(Here let ?=? represent having a ? above the equals sign)

-7 + 22 ?=? 15
15 = 15
and
4x + 3y = 38
4(-7) + 3(22) ?=? 38
-28 + 66 ?=? 38
38 = 38

It checks for both equations so the solution is (-7, 22)

Solve
2x - 3y = 4
x + 4y = -9

Using the 2nd equation is easier to manipulate so solve for x since x has a coefficient of 1
x = -4y - 9
substitute this expression for x in the other equation and solve for y
2x - 3y = 4
2(-4y-9) - 3y = 4
-8y -18 -3y = 4
-11y = 22
y = -2
Substitute the value of y in the equation in step 1 and solve for x
x = -4y -9
x = -4(-2) -9
x = 8 -9 = -1
Check both equations... and you discover that the solution is ( -1, -2)

The substitution method is most convenient to use when the coefficient of one of the variables is 1 or -1.

The Substitution Method in review:
To solve a system of linear equations in two variables:
--> Solve one equation for one of the variables
--> Substitute this expression in the other equation and solve fore the other variable.
--> Substitute this value n the equation in step 1 and solve
--> Check the alues in BOTH equations.


Solve by the substitution method
2x -8y = 6
x - 4y = 8

x = 4y + 8

2x-8y = 6
2(4y+8) - 8y = 6
8y + 16 -8y = 6
16= 6 WAIT that's FALSE

The false statement indicates that there is NO ordered pair (x, y) that satisfies BOTH equations. If you had graphed the equations you would see that these lines are actually parallel.

Solve by substitution method
y/2 = 2 -x
6x + 3y = 12

The first equation is easy to change to y = 4 - 2x by multiplying both sides by 2 to solve for y

6x + 3y = 12
6x + 3(4-2x) = 12
6x + 12 - 6x = 12
12 = 12 WAIT THat's TRUE... always
Every ordered pair (x, y) that satisfies one of the equations aso satisfies the other. IF you graph these two equations you will see that the lines coincide

Therefore, the system has infinitely many solutions.

Math 6A ( Periods 2 & 4)

 Percent of Increase or Decrease 9-4

Let's say we have an iPod that originally sold for $260. It is on sale for $208. What is the amount of change? "How much did you save?"
Just subtract
260-208 = 52
$52.
What is the percent of change?
The percent of change = amount of change/original
52/260 - x/100
or just divide 52 by 260 = .2
which is 20%
REMEMBER: The denominator in the formula is ALWAYS the ORIGINAL AMOUNT.
Amount of change = percent of change X the original amount.
Find the new number when 75 is decreased by 26%
Amount of change - 26% (75
= .26(75)
=19.5
Now take the difference (the amount of change) and subtract THAT from 75
75- 19.5 = 55.5

Remember the circle with the various parts of this formula?

Difference or amount of change
% of change X original amount

Difficult to show here so if you missed these notes make sure to ask a fellow student to see this!! IT is a great way to remember what to do!!

State the increase or decrease. Tell what the amount of change is and the percent of change.
from:
10 to 12
increase
amount of increase: 2
% of change : 20%

4 to 3
decrease
amount of decrease:
% of change : 25%

2 to 5
increase
amount of increase: 3
% of change : 3/2 = 1.5 = 150%

12 to 6
decrease
amount of decrease: 6
% of change : 50%

6 to 12
increase
amount of increase: 6
% of change : 6/6 = 1 = 100%

Find the new number produced when the given number is increased or decrease by the given percent.

120; 20% decrease
120(.20) = 24 120 -24 = 96

30; 10% decrease
30(.10) = 3 30 -3 = 27

48: 50% increase
48(.50) = 24 48 + 24 = 72

128: decrease by 25%, then increased by 25%
What... why multiply by .25 if you can use a fraction and work smarter?
128(1/4) = 32
128 - 32 = 96
then 96 ( 1/4) = 24
96 + 24 = 120

Did you think it would be the starting number? Why wasnn't it?