Algebra Honors ( Periods 6 & 7)

Percents 7-5

You have been doing THIS since 6^th grade ( or even  before!)

The word percent means per cent or hundredths or divided by 100.

29 percent = 29%= 29/100 = 0.29

2.6 percent = 2.6% = 2.6/100 = 26/1000 = 0.026

637 percent = 637% = 637/100 = 6  37/100 = 6.37

0.02 percent = 0.02% = 0.02/100 =  2/10,000 = 0.0002

¼ percent = ¼% = 0.25% = 0.25/100 = 25/10,000 = 0.0025

33 ⅓ percent = 33 ⅓% = 100/3% = 100/3 ÷ 100 = 1/3

It is easy to translate word problem questions about percents if you use the methods reviewed in class:

Translating into AN EQUATION

Remember  when translating into an equation “is” means =  and “of” means multiplication
Change the % into a decimal ( or fraction)

Example 1

15%   of   180    is    what number?

0.15 (180)   =   x

or as a fraction

(15/100) (180) = x

In both cases you ge

x = 27

So 15% of 180 is 27.

Example 2

23 is 25% of what number?

23 = .25(x)

or as a fraction

23 = (25/100)(x)

Again in both cases you will arrive at

x = 92

23 is 25% of 92

Example 3

What percent  of 64  is 48?

x(64) = 48

This time you need to remember you found

x = 48/64 which simplifies to ¾

You need to remember to change this into a % 

 ¾ = 75%

This is more easily done using a fraction

(x/100)(64) = 48

x = 75

You just need to remember to add the percent symbol!

Let’s use those three examples but using the proportion method;

Translating into A PROPORTION

Remember  when translating into a proportion “is” means part  and “of” means whole
Change the % into a fraction  with 100 as the denominator. You will have 3 out of 4 numbers. The one missing is the one you are solving for—in each problem

  

That’s all you need to remember

Example 1

15%   of   180    is    what number?

You definitely want to simplify before using cross products

but you will get

x = 27

Example 2

23 is 25% of what number?

You definitely want to simplify before using cross products

but you will get

x = 92

Example 3

What percent  of 64  is 48?

You definitely want to simplify before using cross products

but you will get

x = 75

so it is 75%

Always remember to check what the question is asking for—so you label it appropriately.

When you solve an equation with decimal coefficients, you can multiply both sides of the equation by a power of 10 (10, 100, and so on)  to get an equivalent equation with integral coefficients.

Solve    1.2x = 36 + 0.4x

Multiply both sides by 10 when the coefficients are tenths

10(1.2x)  = 10(36 + 0.4x)

12x = 360 + 4x
8x = 360
x= 45
{45}

Solve     94 = 0.15x + 0.08(1000 - x)

Multiply BOTH SIDES by 100 when the coefficients are hundredths

100(94) = 100[0.15x + 0.08(1000 - x)]

9400 = 15x +8(1000 –x)
9400 = 15x + 8000 -8x
1400 = 7x
200 = x
{200}

Word Problem Example:
During a sale, a sporting goods store gave a 40% discount on sleeping bags. How much did Ross pay for a sleeping bag with an original price of $75?

Two different methods to find the same solution:
Method 1
Find 40% of $75? 0.40 × 75 = 30
Subtract the amount of discount from the original price
75-30 = $45
Ross paid $45 for the sleeping bag

Method 2
If the sleeping bag was discounted 40%, it then cost 100% -40% or 60% of its original price
Find 60% of 75 0.60 × 75 = 45
Ross paid $45 for the sleeping bag.

Algebra Honors ( Periods 5 & 6)

 Percents 7.5
You have been doing this since at least 6th grade so this portion should really be review.
The word percent means hundredths or divided by 100

Some examples:
29 percent = 29% = 29/100 = 0.29
2.6 percent = 2.6 % = 2.6/100 = 26/1000 = 0.026
637 percent = 637% = 637/100 = 6 37/100 = 6.37
0.02 percent = 0.02% = 0.02/100 = 2/10000 = 0.0002
1/4 percent = 1/2% = 0.25 % = .25/100 = 25/10000 = 0.0025
33 1/3 percent = 33 1/3 % = 100/3% = 100/3 ÷ 100 = 1/3

Most calculators have a % key which will enable you to check your work!!
Remember in percent problems, the word, OF means Multiply and the word IS means EQUALS.

You can set up your problems either with an equation or with a proportion
Example 1:
15% of 180 is what number?
15/100 × 180 = x or
.15 × 180 = x

or set up a proportion
%/100 = 'is'/'of'
15/100 = x/180

In both cases x = 27

Example 2:
23 is 25% of what number?
23 = .25×n

or as a proportion
25/100 = 23/x

Again in both cases
x = 92

Example 3:

What Percent of 64 is 48?
x% × 64 = 48
or
x/100 × 64 = 48
64x/100 = 48

Set up as a proportion
x/100 = 48/64

x = 75
Remember to look at the question to make sure you have answered it. In this case you need to make sure you answer with 75%

When you solve with decimal coefficients, you can multiply both sides of the equation by a power of 10 (10, 100, 100 and so on) to get an equivalent equation with integral coefficients.

Example 4:
1.2x = 36 + 0.4x
Multiply both sides by 10 because the coefficients are tenths
12x = 360 + 4x
8x = 360
x = 45
{45}

Example 5:

94 = 0.15x + 0.08(1000 - x)
Multiply both sides by 100 because the coefficients are hundredths

9400 = 15x + 8(1000 - x)
9400 = 15x + 8000 - 8x
1400 = 7x
200 = x
{200}

Word Problem Example:
During a sale, a sporting goods store gave a 40% discount on sleeping bags. How much did Ross pay for a sleeping bag with an original price of $75?

Two different methods to find the same solution:
Method 1
Find 40% of $75? 0.40 × 75 = 30
Subtract the amount of discount from the original price
75-30 = $45
Ross paid $45 for the sleeping bag

Method 2
If the sleeping bag was discounted 40%, it then cost 100% -40% or 60% of its original price
Find 60% of 75 0.60 × 75 = 45
Ross paid $45 for the sleeping bag.