Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Sections 2.6 and 2.7
More Problem Solving
Solving Percent Problems
Note: “Per cent” means “per 100”. For example, 17% = 17/100 = 0.17 .
A percent problem has three different parts:
amount = percent x base
Any one of the three quantities may be unknown.
1. When we do not know the amount:
n = 10% · 500
2. When we do not know the base:
50 = 10% · n
3. When we do not know the percent:
50 = n · 500
Solving a Percent Problem:
Amount Unknown
Example: What is 9% of 65?
n  9%  65
n  (0.09)(65)
n  5.85
5.85 is 9% of 65
Solving a Percent Problem:
Base Unknown
Example: 36 is 6% of what?
36  6%  n
36  0.06n
36
0.06n

0.06 0.06
600  n
36 is 6% of 600
Solving a Percent Problem:
Percent Unknown
Example: 24 is what percent of 144?
24  n 144
24  144n
24 144n

144 144
0.16  n
2
16 %  n
3
2
24 is 16 % of 144
3
Percent Problem Tip:
If you’re having trouble doing percent problems that
give you a new value after a certain percent increase
or decrease from an old value (such as sales tax
problems), try thinking about it this way:
Think about when you go shopping to buy, say, a TV.
Usually you know how much the TV costs, for
example $400, and the percent tax rate, for example
5.5%. Normally what you do (or the salesclerk’s
computer does) is calculate the TOTAL COST by
taking 5.5% of $400, then adding that amount back
onto the $400 price of the TV to get the total cost to
you.
The working equation is
PRICE + TAX = TOTAL COST.
In words, here’s what you did (after writing the 5.5% as
a decimal, 0.055):
PRICE + .055 times PRICE = TOTAL COST
Plugging in the numbers, we get
400 + .055 x 400 = 400 + 22 = 422.
Notice that you’ve multiplied the OLD VALUE (the price
before tax) by the .055.
The same basic format applies to anything
with a percent increase or decrease from an
original amount:
Old amount +/- % of old amount = new amount
(Remember to write the percent as a decimal.)
This equation works for raises in pay, population
increases or decreases, and many other percent
change problems, especially where you’re given
the new amount and the percent change and you
need to work backwards to find out the old
amount.
Example:
After a 6% pay raise, Nora’s 2013 salary is
$39,703. What was her salary in 2012? (Round
to the nearest dollar).
Solution: Recall the equation:
Old amount + % of old amount = new amount
The “old amount” is her 2012 salary, which is
unknown, so we’ll call it X.
This gives us the equation
X + 0.06X = 39703
Example (cont.)
After a 6% pay raise, Nora’s 2013 salary is
$39,703. What was her salary in 2012? (Round
to the nearest dollar).
1X + 0.06X = 39703
This simplifies to
1.06X = 39703
Divide both sides
X = 39703
by 1.06 to get X.
1.06
Answer: Her 2012 salary was $37,456
Now check your answer:
37456 + .06 x 37456 = 37456 + 2247 = 39703 
NOTE that this DOES NOT give you the same
answer as if you subtracted 6% of the new
salary (39703) from the new salary.
Try it and you’ll see that it doesn’t work.
(It’s not real far off, but enough to give you
the wrong answer, and the bigger the
percentage, the farther off you’ll be.)
On these kinds of problems, we won’t give
partial credit for those “close” answers on
tests.
Sample problem from today’s homework:
Answer: 30.23
Solving Mixture Problems
Example:
The owner of a candy store is mixing candy worth $6 per
pound with candy worth $8 per pound. She wants to obtain
144 pounds of candy worth $7.50 per pound. How much of
each type of candy should she use in the mixture?
Solution:
• Let n = the number of pounds of candy costing $6 per pound.
• Let 144 – n = candy costing $8 per pound.
Use a table to summarize the information.
Number of
Pounds
$6 candy
n
$8 candy
144  n
$7.50
144
candy
Price per
Pound
6
8
Value of Candy
7.50
144(7.50)
6n + 8(144  n) = 144(7.5)
# of
pounds
of $6
candy
# of
pounds of
$8 candy
# of
pounds of
$7.50
candy
6n
8(144  n)
6n + 8(144  n) = 144(7.5)
6n + 1152  8n = 1080
1152  2n = 1080
2n = 72
n = 36
She should use 36 pounds of the $6 per pound candy.
She should use 108 pounds of the $8 per pound candy.
(144  n) = 144  36 = 108
Check: Will using 36 pounds of the $6 per pound
candy and 108 pounds of the $8 per pound candy
yield 144 pounds of candy costing $7.50 per pound?
?
6(36) + 8(108) = 144(7.5)
?
216 + 864 = 1080
?
1080 = 1080 
Distance Problems:
When the amount in the formula is distance, we refer to the
formula as the distance formula.
distance = rate · time or d = r · t
The assignment on this material (HW 2.6/7) is due at
the start of the next class session.
Lab hours:
Mondays through Thursdays
8:00 a.m. to 6:30 p.m.