our psb modle 1

1. Equations
Solve equations using multiplication or division.
Quantitative Skills
Solve equations using addition or subtraction.
Topic 1-C
Solve equations using more than one operation.
Equations Solve equations containing multiple unknown terms.
Solve equations containing parentheses.
Solve equations that are proportions.
Solve Equations Using
Multiplication or Division Key Terms
The letters (x,y,z) represent unknown amounts and
An equation is a mathematical statement in which two are called unknowns or variables.
quantities are equal.
The numbers are called known or given amounts.
Solving an equation means finding the value of an
unknown.
For example: 8x = 24 4x = 16
To solve this equation, the value of x must be
discovered.
Division is used to solve this equation.
How to solve an equation with
Remember! multiplication and division
8x = 24
Any operation performed on one side of the equation
must be performed on the other side of the equation Step one: Isolate the unknown value.
as well. Determine if multiplication or
If you “multiply by 2” on one side, you must “multiply division is needed.
by 2” on the other side.
Step two: Use division to divide both
If you “divide by 3” on one side, you must “divide by sides by “8.”
3” on the other side and so on.
Step three: Simplify: x = 3

2. Find the value of an unknown
using multiplication Do this example
Solve the following:
Find the value of “a” in the following equation. 2b = 40
a/3 = 6
1. Determine which operation is needed.
Multiply both sides by 3 to isolate “a.” Division
The left side becomes 1a or “a.” 2. Perform the same operation to both sides.
Divide both sides by “2.”
The right side becomes the product of 3. Isolate the variable and solve the equation
6 x 3 or “18.”
b = 40/2 = 20
a = 18
Solve an Equation with Addition or
Subtraction
Don’t forget!
Adding or subtracting any number from one side must
4 + x = 10 be carried out on the other side as well.
Step one: Isolate the unknown value.
Subtract “the given amount” from both sides.
Determine if addition or
subtraction is needed. Would solving 4 + x = 16 require addition or
subtraction of “4” from each side?
Step two: Use subtraction to isolate “x.” Subtraction
Step three: Simplify: x = 6
Solve Equations Using More Than
Do this example One Operation
Solve the following: Isolate the unknown value.
b - 12 = 8 Add or subtract as necessary first.
1. Determine which operation is needed.
Multiply or divide as necessary second.
Addition
Identify the solution: the number on the side
2. Perform the same operation to both sides. opposite the unknown.
Add “12” to both sides.
Check the solution by “plugging in” the number
3. Isolate the variable and solve the equation. using the original equation.
b = 8 + 12 = 20

3. Order of Operations “Undo the operations”
When two or more calculations are written To solve an equation, we undo the operations, so
symbolically, it is agreed to perform the operations we work in reverse order.
according to a specified order of operations.
1. Undo the addition or subtraction.
Perform multiplication and division as they appear
from left to right. 2. Undo multiplication or division.
Perform addition and subtraction as they appear
from left to right. 7x + 4 = 39
Try this example Equations Containing Multiple
Unknown Terms
7x + 4 = 39
First, undo the addition by subtracting 4 from each side. In some equations, the unknown value may occur more than
once.
And that becomes 7x = 35
The simplest instance is when the unknown value occurs in
Next, divide each side by 7. two addends.
And that becomes x = 35/7 = 5 For example: 3a + 2a = 25
Verify the result by “plugging 5 in” the place of “x.” Add the numbers in each addend (2+3).
7 (5) + 4 = 39 Multiply the sum by the unknown (5a = 25).
35 + 4 = 39 Solve for “a.” (a = 5)
Solve Equations Containing
Try this example Parentheses
Find a if: a +4a – 5 = 30
1. Eliminate the parentheses:
Combine the unknown value addends.
a. Multiply the number just outside the parentheses
a + 4a = 5a 5a – 5 = 30 by each addend inside the parentheses.
“Undo” the subtraction. 5a = 35 b. Show the resulting products as addition or
subtraction as indicated.
“Undo” the multiplication. a=7
2. Solve the resulting equation.
Check by replacing “a” with “7.
It is correct.

4. Look at this example Tip!
Solve the equation: Remove the parentheses first.
6(A + 2) = 24 5 (x - 2) = 45
Multiply “6” by each addend. Do me first !
6 multiplied by A + 6 multiplied by 2
Show the resulting products.
6A + 12 = 24
5x -10 = 45
Solve the equation.
6A = 12 5x = 55
A=2 x = 11
Solve Equations That
Cross products
are Proportions
An important property of proportions is that the cross
A proportion is based on two pairs of related products are equal.
quantities.
A cross product is the product of the numerator of one
The most common way to write proportions is to use fraction times the denominator of another fraction.
fraction notation.
A number written in fraction notation is also called a Example: 4/6 = 6/9
ratio. Multiply 4 x 9 = 6 x 6 36 = 36
When two ratios are equal, they form a proportion.
Verify that two fractions Using Equations to
form a proportion Solve Problems
Do 4/12 and 6/18 form a proportion? There is a list of key words and what operations they
imply in your textbook. Please refer to it.
1. Multiply the numerator from the first fraction by the
denominator of the second fraction.
These words help you interpret the information and
4 x 18 = 72
begin to set up the equation to solve the problem.
2. Multiply the denominator of the first fraction by the
numerator of the second fraction.
Example: “of” often implies multiplication.
6 x 12 = 72
“¼ of her salary” means “multiply her salary
3. Are they equal?
by ¼”
Yes, they form a proportion.

5. Five-step problem solving
approach for equations Use the solution plan
What you know.
Known or given facts. Full time employees work more hours than part-time
employees. If the difference is four per day, and
What you are looking for?
part-time employees work six hours per day, how
Unknown or missing amounts. many hours per day do full-timers work?
Solution Plan
What are we looking for?
Equation or relationship among known / unknown facts. Number of hours that FT work
Solution
What do we know?
Solve the equation.
PT work 6 hours;
Conclusion
The difference between FT and PT is 4 hours.
Solution interpreted within context of problem.
Use the solution plan Try this example
We also know that “difference” implies subtraction. Jill has three times as many trading cards as Matt. If
Set up a solution plan. the total number that both have is 200, how many
cards does Jill have?
FT – PT = 4
FT = N [unknown] PT = 6 hours Use the five-step solution plan to solve this problem:
1. What are you looking for?
N–6=4
2. What do you know?
Solution plan: N = 4 + 6 = 10 3. Set up a solution plan.
4. Solve it.
Conclusion: Full time employees work 10 hours.
5. Draw the conclusion.
Solving a word problem with a
Solution Plan total of two types of items
What are you looking for?
The number of cards that Jill has.
What do you know? Diane’s Card Shop spent a total of $950 ordering 600
The relationship in the number of cards is 3:1; total is 200. cards from Wit’s End Co., whose humorous cards cost
Solution plan $1.75 each and whose nature cards cost $1.50 each.
x (Matt’s) + 3x (Jill’s) = 200 How many of each style of card did the card shop
Solve order?
x + 3x = 200
4x = 200; x = 50 Use the solution plan to solve this problem.
Conclusion
Jill has “3x” or 150 cards.

6. What are you looking for? Organize the information
How many humorous cards were ordered and how A total of $950 was
What do you
many nature cards were ordered.
know? spent.
The total of H + N = 600
Two types of cards were
Another way to look at this is: ordered.
N = 600 – H
The total number of cards
If we let “H” represent the humorous cards, Nature ordered was 600.
cards will be 600- H.
The humorous cards cost
This will simplify the solution process by using only one $1.75 each/nature cards
unknown: “H.” cost $1.50 each.
Solution plan Solve the equation
$1.75H + $1.50(600-H) = $950.00
Set up the equation by multiplying the unit price of
each by the volume, represented by the unknowns $1.75H + $900.00 - $1.50H = $950.00
equaling the total amount spent.
$0.25H + $900.00 = $950.00
$1.75(H) + $1.50 (600 – H) = $950.00
$0.25H = $50.00
Unit
prices
Volume Total
spent
H = 200
“unknowns”
Conclusion Try this problem
H = 200
Denise ordered 75 dinners for the awards
banquet. Fish dinners cost $11.75 and chicken
The number of humorous cards ordered is 200.
dinners cost $9.25 each. If she spent a total of
$756.25, how many of each type of dinner did
Since nature cards are 600 – H, we can conclude that
she order?
400 nature cards were ordered.
Using “200” and “400” in the original equation Use the solution plan to organize the information
proves that the volume amounts are correct. and solve the problem.

7. Denise’s order Proportions
The relationship between two factors is often
$11.75(F) + $9.25(75-F) = $756.25 described in proportions. You can use proportions to
solve for unknowns.
$11.75 F + $693.75 - $9.25F = $756.25
$2.50F + $693.75 = $756.25 Example: The label on a container of weed killer
gives directions to mix three ounces of weed killer
$2.50F = $62.50 with every two gallons of water. For five gallons of
water, how many ounces of weed killer should you
F = 25 use?
Conclusion: 25 fish dinners and 50 chicken dinners were
ordered.
Use the solution plan Proportions
What are you looking for? Your car gets 23 miles to the gallon. How far can you
The number of ounces of weed killer needed for 5 gallons go on 16 gallons of gas?
of water.
What do you know? 1 gallon/23 miles = 16 gallons/ x miles
For every 2 gallons of water, you need 3 oz. of weed killer.
Cross multiply: 1x = 368 miles
Set up solution plan.
2/3 = 5/x Conclusion: You can travel 368 miles on 16 gallons of
Solve the equation. gas.
Cross multiply.
2x = 15; x = 7.5
Conclude
You need 7.5 ounces of weed killer for 5 gallons of water.
Direct Proportions Formulas
Many business-related problems that involve pairs of
numbers that are proportional involve direct Evaluate a formula.
proportions. Find a variation of a formula by rearranging the
An increase (or decrease) in one amount causes an formula.
increase (or decrease) in the number that pairs with it.
In the previous example, an increase in the amount of
gas would directly and proportionately increase the
mileage yielded.

8. How to evaluate a formula Try this problem
Write the formula. A plasma TV that costs $2,145 is marked up
Rewrite the formula substituting known values $854. What is the selling price of the TV?
for the letters of the formula. Use the formula S = C + M where S is the
selling price, C is the cost, and M is Markup.
Solve the equation for the unknown letter or
perform the indicated operations, applying the S = $2,145 + $854
order of operations. S or Selling Price = $2,999
Interpret the solution within the context of the
formula.
Find a variation of a formula by
rearranging the formula Try this problem
Determine which variable of the formula is to be isolated The formula for Square Footage = Length x
(solved for). Width or S = L x W. Solve the formula for W or
Highlight or mentally locate all instances of the variable to width.
be isolated.
Isolate W by dividing both sides by L
Treat all other variables of the formula as you would treat
numbers in an equation, and perform normal steps for The new formula is then: S/L = W
solving an equation.
If the isolated variable is on the right side of the equation,
interchange the sides so that it appears on the left side.
Simultaneous Equations Simultaneous Equations
Exercise 1: Exercise 2:
Solve for x & y: x 2y 6 Solve for x & y: 2 x 3 y 12
x 3y 4 5x 4 y 23

9. Simultaneous Equations Simultaneous Equations
Exercise 3: Exercise 4:
A cup and a saucer cost $5.25 together. A cup and A shop sells bread rolls. If five brown rolls and six
two saucers cost $7.50. Find the cost of a cup. white rolls cost $2.94 and three brown rolls and
four white rolls cost $1.86 find the cost of each
type of roll.