Math 6A (periods 2 & 4)

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

Algebra Honors (Periods 5 & 6)

Scientific Notation 7-10 Scientific Notation 7-10
Scientific notation makes it easier to work with either very large numbers or very small numbers. To write a positive number in scientific notation, you express it as the product of a number greater than or equal to 1 BUT less than 10 AND an integral power of 10.
When a positive number greater than or equal to 10 is written in scientific notation, the power of 10 is positive. When the number is less than 1, the power of 10 will be negative.

Scientific notation consists of two parts
1≤ x< 10 × POWER¹⁰
for example
58,120,000,000
Move the decimal point left 10 spaces to between the 5 and the 8 to get a number BETWEEN 1 and 10
5.8
so 5.8 × 10¹⁰
0.00000072
Move the decimal point 7 places to the right to get a number between 1 and 10
7.2
7.2 × 10^-7

Numbers written in scientific notation can be multiplied and divided easily by using the rules of exponents
Simplify. Keep your answers in scientific notation
3.2 × 10⁷/2.0 × 10⁴
= 3.2/2.0 ×10⁷/10⁴
=1.6 × 10^7-4
=1.6 × 10³

(2.5 × 10³)(6.0 × 10²)
=(2.5 ×6.0) × (10³ × 10²) Add exponents
= (15)(10³⁺²)
=15 × 10⁵
But wait 15 is not between 1 and 10
15 itself is 1.5 × 10¹
so it becomes
1.5 ×10¹× 10⁵
1.5 × 10¹⁺⁵
=1.5 × 10⁶
The distance from the sun to Mercury is approximately 6 × 10⁸ km.
The distance from the sun to Pluto is approximately 5.9 × 10⁹ km
Find the ratio of the first distance to the second distance
Distance from sun to Mercury
Distance from sun to Pluto
6 × 10⁸
5.9 × 10⁹
= 6/5.9 × (10⁸)/10⁹

= 6/5.9 ×10^8-9
6/5.9 ×10^-1
=6/5.9 ×1/10
= 6/59

You learned expanded notation in 6th grade... but a review of writing numbers in expanded notation using powers of 10 follows:
8572 = 8000 + 500 + 70 + 2
=8(10³) + 5(10²) + 7(10¹) + 2(10⁰)

0.3946 = 0.3 + 0.09 + 0.004 + 0.0006
= 3(10^-1) + 9(10^-2) +4(10^-3) + 6(10^-4)

25.03 = 2(10¹) +5(10⁰) + 0(10^-1) + 3(10^-2)

The metric system is also based on powers of TEN.
To change from one metric unit to another, you simply multiply by a power of 10.
Remember:
King Henry Died By Drinking Chocolate Milk
1 km = 10³m = 1000m
1mL =10^-3 L = 1/1000 L

Algebra Honors (Periods 5 & 6)

 Negative Exponents 7-9
If a is a nonzero real number and n is a positive integer
a^-n = 1/aⁿ
so
10^-3 = 1/10³ = 1/1000
5^-4= 1/5⁴ = 1/625

16^-1 = 1/16

The rule of exponents for division (page 190) will help you understand why
a^-n = 1/aⁿ
recall that for m > n a^m/aⁿ= a^m-n
For example
a⁷/a³ = a^7-3= a⁴
you can also apply this rule when m < n that is when m - n becomes a negative number. For example a³/a⁷ = a^3-7 = a^-4
since
a⁷/a³ and a³/a⁷ are reciprocals then
a⁴ and a^-4 must also be reciprocals.
Thus
a^-4= 1/a⁴
a⁵/ a⁵ = a^5-5 = a⁰
But you already know that a⁵/a⁵ = 1
SO, definition of a⁰
a⁰ = 1
However, the expression 0⁰ has no meaning

All the rules for positive exponents also hold for zero and negative exponents.

Summary of Rules for Exponents
Let m and n be any integers
Let a and b be any non zero integers
Review—>But you should really know these because of our Powers Project
Products of Powers
b^mbⁿ = b^m+n
Example with negative exponents
2³⋅2^-5 = 2^3+(-5) = 2^-2 = 1/2² = 1/4
Quotient of Powers
b^m ÷ bⁿ = b^m-n
Example with negative exponents
6³÷6⁷= 6^3-7= 6^-4= 1/6⁴= 1/1296
Power of Powers
(b^m)ⁿ = b^mn
Example with negative exponents
(2³)^-2 = 2^-6 = 1/2⁶ = 1/64
Power of a Product
(ab)^m= a^mb^m
Example with negative exponents
(3x)^-2 = 3^-2 ⋅x^-2 = 1/3²⋅1/x² = 1/9x²
Power of a Quotient
(a/b)^m= a^m/b^m
Example with negative exponents
(3/5)^-2= 3^-2/5^-2= (1/3²)/ (1/5²)= 1/3² ÷ 1/5² which means
1/3² ⋅5²/1= 5²/3²= 25/9

Algebra Honors (Periods 5 & 6)

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

Math 6A (Periods 2 & 4)

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%

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%