Converting between fraction, decimal, and percent

Fraction → Decimal → Percent
Given a fraction a/b, do the division to get the decimal equivalent, then move the decimal point two places to the right (which is multiplying by 100%, which is 1) to get the percent.
5/8 = 0.625 = 62.5%
9/16 = 0.5625 = 56.25%
3/10 = 0.3 = 30%
3/100 = 0.03 = 3%
3/1000 = 0.003 = 0.3%

Decimal → Fraction
Given a decimal number .abc etc., make those digits the numerator and make the denominator 1 followed by digits number of 0's. Simplify the fraction.
.abc = abc/1000
.abcd = abcd/10000
0.5625 = 5625/10000 = 9/16
.3 = 3/10
0.625 = 625/1000 = 5/8
0.0625 = 0625/10000 = 1/16

n/d : numerator is n÷d·100% of denominator.
3/4 = 3÷4 = .75 = 75%
   3 is 75% of 4
5/8 = 5÷8 = .625 = 62.5%
   5 is 62.5% of 8

Basic percent equation:

A = PB    Amount is Percentage of Base    Ex. 20 = 25% of 80
"of" means multiply: 25% of 80 = .25 * 80 = 20

Know the values of any two of the variables, can determine the value of the third.
What (amount) is 25% of 80?    A = 25% of 80    A = .25*80 = 20
20 is what (percentage) of 80?    20 = P of 80    20 = P*80    Solve for P: P = 20/80 = .25=25%
20 is 25% of what (base)?    20 = 25% of B    20 = .25*B    Solve for B: B = 20/.25 = 80

Percent increase and decrease are not "symmetric":
               

Percent increase/decrease:

Time:    start → end    old → new    then → now    previous → current    current → future
Place: here ⇄ there   
me ⇄ you    it ⇄ that    etc.

Percent change = (end-start)/start = (New-Old)/Old = NO/O

In the beginning there were 5000 things, then at the end there were 8000.
   What is the percent change?    (8000-5000)/5000 = .6=60% (positive, so is an increase)

In the beginning there were 8000 things, then at the end there were 5000.
   What is the percent change?    (5000-8000)/8000 = -.375=-37.5% (negative, so is a decrease)

x → y    %change = (y-x) / x
y → x    %change = (x-y) / y
The %'s will only be the same when x=y, i.e. there is no change. 0%.

Markup:

S = B + PB     Selling,retail price is wholesale price + markup,profit amount
120 = 100 + 20% of 100

A thing with wholesale price of 60 is marked up 25%.
   What is the selling price?    S=60+25% 60    S=75

A thing's selling price is 75. It was marked up 25%.
   What is its wholesale price?    75=B+25% B    B=60

A thing with wholesale price of 60 is sold for 75.
   What is the markup percentage?    75=60+P 60    P=.25=25%

Discount:

S = B - PB     Selling,discount price is regular price - discount amount
80 = 100 - 20% of 100

A thing with regular price of 100 is discounted by 20%.
   What is the selling price?    S=100-20%100    S=80

A thing's sale price is 80. It was discounted 20%.
   What is the regular price?    80=B-20%B    B=100

A thing with regular price of 100 is sold for 80.
   What is the discount percentage?    80=100-P 100    P=.20=20%

A thing's sale price is 80. This is 63% of the original price.
   What is the regular price? S = PB    80=63%B    B=126.98

Percent mixture:

Quantity1 of p1% 'stuff' mixed with (+) Quantity2 of p2% 'stuff' = amount of 'stuff' in the mixture
   = Q1*p1% + Q2*p2%
Total mixture quantity = Q1+Q2.
Percent of the total mixture that is 'stuff' is (p1% of Q1 mixed with (+) p2% of Q2) / (total mixture)
   = (Q1*p1% + Q2*p2%) / (Q1+Q2)

Gold (Au) alloy measured in karat K: 24K=100%, 20K=20/24=83.33%, 18K=18/24=75%, 12K=12/24=50%
3 oz of 20K gold mixed with 4 oz of 18K gold.
Amount of gold in the 7 oz mixture = 3oz*83.3% + 4oz*75% = 5.5oz
Percent of the mixture that is gold = 5.5oz / 7 oz = 78.57%

Two-stage percent increases or decreases


.25 + .10 + (.25)(.10) = 0.375     OR:   (1.25)(1.10) = 1.375
-.25 + -.10 + (-.25)(-.10) = -0.325     OR:   (-0.75)(-0.90) = 0.675

Increase then decrease, or vice versa:
Increase x% then decrease y% == Decrease y% then increase x%
Increase 30% then decrease 20%: (1.30)(0.80) = 1.04, ie. 4% increase.
Decrease 20% then increase 30%: (0.80)(1.30) = 1.04, ie. 4% increase.
Increase 20% then decrease 30%: (1.20)(0.70) = 0.84, ie. 16% decrease.
Decrease 30% then increase 20%: (0.70)(1.20) = 0.84, ie. 16% decrease.

Repeated same percent increase or decrease: exponential.

x decreases by 10% = 90% of x
90% of something, then 90% of that = 81% of the original something.
90% of 90% of x = 81% of x
(90%)(90%)x = (100%-10%)(100%-10%)x = (1-0.1)(1-0.1)x = (0.9)(0.9)x = (0.9)2x = .81x

10% decrease each of three years: 90% of 90% of 90% of x = .93x = 0.729x =72.9% of x
10% increase each of three years: 110% of 110% of 110% of x = 1.13x = 1.331x =133.1% of x

Q(1+r)t = Q*(100%+R%)t      (Q is starting quantity)

p% of q = q% of p

10% of 20 = 20% of 10