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.
Monday, May 19, 2014
Tuesday, May 13, 2014
Math 6A (Periods 1 & 2)
Working With Percents
Review: When changing a percent to a fraction we use the following
proportion:
3.5% = 3.5/100 = 35/1000 = 7/200
Looking at
What percent of 225 is 90?
When setting up an equation we get
225n = 90
However we can use a similar proportion to
to solve any percent
problem! The fraction a/b really represents the part/ whole
Looking at the above problem we have
Of course we simplify as much as possible before we cross
multiply
we get n = 10(4) = 40
so the solution is 40%
EXAMPLE:
45% of 600 is what number?
Now this simplifies really easily
and we just have
n = 6(45)
n = 270
The number is 270.
EXAMPLE:
96% of 85 is what number?
Again, this simplifies to
With cross products we get
5n = 24(17)
5n = 408
Divide both sides by 5
5n/5 = 408/5
n = 81.6
81.6 is our solutions
EXAMPLE:
What number is 76% of 350?
x = 266
EXAMPLE:
56 is 4% of what numbers?
Definitely simplify and you get
x = 25(56)
x = 1400
Thursday, May 8, 2014
Math 6A ( Periods 1 & 2)
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.
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.
Wednesday, May 7, 2014
Math 6A (Periods 1 & 2)
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
SET UP the LADYBUG
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 X 7800 = $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?
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
SET UP the LADYBUG
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 X 7800 = $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?
Monday, May 5, 2014
Math 6A (Periods 1 & 2)
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?
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?
Friday, May 2, 2014
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 = (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?
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
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
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)
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
0.65= 25r
Solve this one
step equation
Divide both sides by 25
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%
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%
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
Thursday, May 1, 2014
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
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
Wednesday, April 30, 2014
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
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
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?”
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%
Tuesday, April 29, 2014
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%
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%
Monday, April 28, 2014
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%
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!!
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!!
Tuesday, April 15, 2014
Math 6A (Periods 1 & 2)
Graphs of Equations 11-9
An equation in two variables y = x + 2 produces an infinite number of ordered pairs
If we give x the value of 1, a corresponding value of y is determined
y = (1) + 2 = 3
The ordered pair is (1, 3)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -1
y = (-1) + 2 = 1 ( -1, 1)
I like to remember ordered pairs---> ( ordered, pairs)
We graphed the line on a mini graph stickie.
The line is graphed using a ruler and connecting all the points we plotted. Put arrows at each end (since a line continues with out end) and write the line's equation right above the line.
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
y = x + 2
in the future you will see it written as
f(x) = x + 2
so if x = 2
f(2) = (2) + 2 = 4
if x = 5
f(5) = (5) +4 = 9
We used a three column chart to compute our ordered pairs.
An equation in two variables y = x + 2 produces an infinite number of ordered pairs
If we give x the value of 1, a corresponding value of y is determined
y = (1) + 2 = 3
The ordered pair is (1, 3)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -1
y = (-1) + 2 = 1 ( -1, 1)
I like to remember ordered pairs---> ( ordered, pairs)
We graphed the line on a mini graph stickie.
The line is graphed using a ruler and connecting all the points we plotted. Put arrows at each end (since a line continues with out end) and write the line's equation right above the line.
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
y = x + 2
in the future you will see it written as
f(x) = x + 2
so if x = 2
f(2) = (2) + 2 = 4
if x = 5
f(5) = (5) +4 = 9
We used a three column chart to compute our ordered pairs.
Subscribe to:
Posts (Atom)
