CHAPTER 5
ELEMENTARY ALGEBRA, FACTORING & RATIOS
One of the most important concepts in algebra may be stated as:
RULE #42 WHEN TWO ‘THINGS’ ARE EQUAL, YOU MAY CHANGE THE VALUE OF EACH,
BY THE SAME AMOUNT, AND RETAIN THE EQUALITY BETWEEN THEM.
‘CHANGING’ MAY INCLUDE ANY MATH OPERATION SUCH AS
‘ADDITION’, ‘MULTIPLICATION’, ETC.
EXAMPLE
BASIC EQUATION: 10 = 5 x 2
Amount to be changed = 5
Addition: 10 + 5 = (5 x 2) + 5
Multiplication: 10 x 5 = (5 x 2) x 5
In ‘every day’ terms, the concept may be a little easier to understand.
Tom has 10 pennies (cents). Jim has 2 nickels (5-cent pieces or 10 cents). They both have equal amounts.
If I give each of them 5 cents, they have more than before but they still have equal amounts !
What does ‘Y’ equal in the following equation ? : Y + N = 10 + N
‘Y’ must equal ‘10’ regardless of what value you may assign to ‘N’ ! Why ?
Because Rule #42 also works ‘backwards’. If two things: (Y + N) and (10 + N) are equal
after adding the same ‘thing’ (‘N’) to two different values (‘Y’ and ‘10’), then the two original ‘things’
must have been equal to start with !
Another reason is that you may subtract ‘N’ from both sides of the ‘=‘ sign (Tom and Jim each spend ‘N’).
--------------------------------------------------------------------------------------------------------
THE SUM OF 6 AND THREE(3) TIMES SOME NUMBER IS EQUAL TO 18. WHAT IS THAT NUMBER ?
6 + 3Y = 18 (let Y = unknown #. Equivalent equation from the words)
We start the isolation of ‘Y’ by ‘getting rid of’ the ‘6’ so that ‘3Y’ stands alone.
6 + 3Y - 6 = 18 - 6 (Subtract ‘6’ from both sides of the equation)
3Y = 12 3Y / 3 = 12 / 3 (Divide both sides by 3) Y = 4 (answer)
==================================================================================
Click for answer PROBLEMS
1. The length of a room is twice its width. The perimeter (total length of all 4 sides) is 66 feet.
How long is the length and width of the room?
2. The length of the ground shadow of a 24 foot high tree is 15 feet. If the shadow of a
nearby tree is 10 feet, how tall is the nearby tree ?
3. A right triangle's sides are in the ratio: 3: 4 : 5. What are the dimensions of a similar
triangle whose largest dimension is 25 ?
4. Two cars start out together from the same place traveling in the same direction. One
car travels at 67 miles per hour (MPH) while the other travels at 55 MPH. How far apart
will they be after 45 minutes ?
==========================================================================================
CHAPTER 6
COMMON FORMS OF EQUATIONS
The recognized difficulty with word problems is establishing the equation from the given words.
By establishing some common forms and relating them to specific examples, you may be able
to recognize then when you ‘see’ them.
FORM B: a x b x K = X
This form is similar to Form A except that you must include your own conversion ‘W/F’ of K
(not given in the problem).
EXAMPLE
A MOTOR SPINS AT A RATE OF 10 REVOLUTIONS PER SECOND.
HOW MANY REVOLUTIONS DOES IT MAKE IN TWO (2) MINUTES ?
a = 10 rev. /sec. b = 2 min. K = 60 sec. /min.
10 rev. x 2 min. x 60 sec. = ? revs. = 1200 rev. (ANSWER)
sec. min.
Note that we were able to cancel: sec. /sec. = 1 and min. / min. = 1 leaving only the
word 'rev' on the left side.
------------------------------------------------------------------------------------------------------
FORM E: Y(a ± b) = A
This form is one in which the answer (A) of the equation is known (including both the
answer name and its value). The unknown (‘Y’) is a factor of two different known values:
(‘a’ and ‘b’) which are first added or subtracted.
TWO CARS START OUT TOGETHER. ONE CAR IS TRAVELING AT
65 MILES PER HOUR (MPH) WHILE THE OTHER IS TRAVELING AT
55 MPH. HOW LONG WILL IT TAKE BEFORE THE FASTER CAR IS
300 MILES AHEAD OF THE SLOWER ONE ?
Y = ? hours a = 65 mi./hr. b = 55 mi./hr. A = 300 miles
Y hr. ( 65 mi/hr. - 50 mi./ hr) = 300 mi.
15 mi./hr. x Y hr. = 300 miles Y = 20 hours (ANSWER)
Note: using the common factor ‘Y’ eliminated the need for 2 multiplications; finding the distance each
car traveled was unnecessary.. difference only was asked for !
======================================================================================
Click for answers PROBLEMS
1. My father is 3 times as old as I. Five years ago, he was 4 times as old as I was. How old am I ?
2. An airplane is flying at 120 miles per hour (mph) above a road when it spots a car 10 miles ahead
(traveling in the same direction). If the car is traveling at 60 mph, how long will it take the airplane
to reach the car ?
3. A number is 9 less than twice another number. The second number is 3 less than twice the first.
What are the two numbers ?
4. A stock broker charges 1% commission on the first $2400 purchased plus 1/2% on any excess
amount. If I buy 100 shares of XYZ Co. at $57 per share, what is my commission cost ?
ELEMENTARY ALGEBRA, FACTORING & RATIOS
One of the most important concepts in algebra may be stated as:
RULE #42 WHEN TWO ‘THINGS’ ARE EQUAL, YOU MAY CHANGE THE VALUE OF EACH,
BY THE SAME AMOUNT, AND RETAIN THE EQUALITY BETWEEN THEM.
‘CHANGING’ MAY INCLUDE ANY MATH OPERATION SUCH AS
‘ADDITION’, ‘MULTIPLICATION’, ETC.
EXAMPLE
BASIC EQUATION: 10 = 5 x 2
Amount to be changed = 5
Addition: 10 + 5 = (5 x 2) + 5
Multiplication: 10 x 5 = (5 x 2) x 5
In ‘every day’ terms, the concept may be a little easier to understand.
Tom has 10 pennies (cents). Jim has 2 nickels (5-cent pieces or 10 cents). They both have equal amounts.
If I give each of them 5 cents, they have more than before but they still have equal amounts !
What does ‘Y’ equal in the following equation ? : Y + N = 10 + N
‘Y’ must equal ‘10’ regardless of what value you may assign to ‘N’ ! Why ?
Because Rule #42 also works ‘backwards’. If two things: (Y + N) and (10 + N) are equal
after adding the same ‘thing’ (‘N’) to two different values (‘Y’ and ‘10’), then the two original ‘things’
must have been equal to start with !
Another reason is that you may subtract ‘N’ from both sides of the ‘=‘ sign (Tom and Jim each spend ‘N’).
--------------------------------------------------------------------------------------------------------
THE SUM OF 6 AND THREE(3) TIMES SOME NUMBER IS EQUAL TO 18. WHAT IS THAT NUMBER ?
6 + 3Y = 18 (let Y = unknown #. Equivalent equation from the words)
We start the isolation of ‘Y’ by ‘getting rid of’ the ‘6’ so that ‘3Y’ stands alone.
6 + 3Y - 6 = 18 - 6 (Subtract ‘6’ from both sides of the equation)
3Y = 12 3Y / 3 = 12 / 3 (Divide both sides by 3) Y = 4 (answer)
==================================================================================
Click for answer PROBLEMS
1. The length of a room is twice its width. The perimeter (total length of all 4 sides) is 66 feet.
How long is the length and width of the room?
2. The length of the ground shadow of a 24 foot high tree is 15 feet. If the shadow of a
nearby tree is 10 feet, how tall is the nearby tree ?
3. A right triangle's sides are in the ratio: 3: 4 : 5. What are the dimensions of a similar
triangle whose largest dimension is 25 ?
4. Two cars start out together from the same place traveling in the same direction. One
car travels at 67 miles per hour (MPH) while the other travels at 55 MPH. How far apart
will they be after 45 minutes ?
==========================================================================================
CHAPTER 6
COMMON FORMS OF EQUATIONS
The recognized difficulty with word problems is establishing the equation from the given words.
By establishing some common forms and relating them to specific examples, you may be able
to recognize then when you ‘see’ them.
FORM B: a x b x K = X
This form is similar to Form A except that you must include your own conversion ‘W/F’ of K
(not given in the problem).
EXAMPLE
A MOTOR SPINS AT A RATE OF 10 REVOLUTIONS PER SECOND.
HOW MANY REVOLUTIONS DOES IT MAKE IN TWO (2) MINUTES ?
a = 10 rev. /sec. b = 2 min. K = 60 sec. /min.
10 rev. x 2 min. x 60 sec. = ? revs. = 1200 rev. (ANSWER)
sec. min.
Note that we were able to cancel: sec. /sec. = 1 and min. / min. = 1 leaving only the
word 'rev' on the left side.
------------------------------------------------------------------------------------------------------
FORM E: Y(a ± b) = A
This form is one in which the answer (A) of the equation is known (including both the
answer name and its value). The unknown (‘Y’) is a factor of two different known values:
(‘a’ and ‘b’) which are first added or subtracted.
TWO CARS START OUT TOGETHER. ONE CAR IS TRAVELING AT
65 MILES PER HOUR (MPH) WHILE THE OTHER IS TRAVELING AT
55 MPH. HOW LONG WILL IT TAKE BEFORE THE FASTER CAR IS
300 MILES AHEAD OF THE SLOWER ONE ?
Y = ? hours a = 65 mi./hr. b = 55 mi./hr. A = 300 miles
Y hr. ( 65 mi/hr. - 50 mi./ hr) = 300 mi.
15 mi./hr. x Y hr. = 300 miles Y = 20 hours (ANSWER)
Note: using the common factor ‘Y’ eliminated the need for 2 multiplications; finding the distance each
car traveled was unnecessary.. difference only was asked for !
======================================================================================
Click for answers PROBLEMS
1. My father is 3 times as old as I. Five years ago, he was 4 times as old as I was. How old am I ?
2. An airplane is flying at 120 miles per hour (mph) above a road when it spots a car 10 miles ahead
(traveling in the same direction). If the car is traveling at 60 mph, how long will it take the airplane
to reach the car ?
3. A number is 9 less than twice another number. The second number is 3 less than twice the first.
What are the two numbers ?
4. A stock broker charges 1% commission on the first $2400 purchased plus 1/2% on any excess
amount. If I buy 100 shares of XYZ Co. at $57 per share, what is my commission cost ?