~More on Proportions & Percents~

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Review of basics:

~A Ratio of a to b is a division, a/b. It is a way of comparing two quantities or categories.
Also written a:b

~Note: A ratio does not necessarily give you the exact amounts of each category. It may or may not.

Ex: If there are 17 girls and 11 boys in a class, the ratio of girls to boys would be 17/11. Since this can not be reduced, the exact numbers appear. However, if a class consists of 18 girls and 12 boys, the ratio would be 18/12 and could be given in reduced form as 3/2, obviously, this form does not give the exact numbers of girls & boys in the class.

~Use the following set-up for Ratio Word Problems: If the ratio of two categories is a to b, let the first be represented by ax and the second by bx. (since a & b are usually not the exact numbers of each). Then, in basic problems, take their sum and equate this to the total which is usually given.

Ex:  The ratio of people 21 or older to those under 21 at a given event is  7 to 5.  There were 228 people attending this event. Find the number of each that attended.

Solution:  Let  the # of people 21 or older = 7x & the # of those under 21= 5x. Since the total attendence was 228, we get 7x+5x=228 or 12x=228 or x=19. So, the # of people 21 or older would be 7x=7(19)=133 and those under 21 would be 5x=5(19)=95.

~A Proportion is established when two ratios are equal.
The form will look like this: a/b = c/d  or a:b = c:d.

~The Means in a proportion are the elements b & c (the inside elements in the second form)
~The Extremes in a proportion are the elements a & d (the outside elements in the second form).

~Properties of a Proportion:

1) In any proportion, the product of the Means = the product of the Extremes. This is known in common terms as  Cross-Multiplication. If two ratios are not equal, then there is no proportion & these products would not be equal. This is a good way of checking whether or not you have a proportion.
(or if two fractions have equal value).

2) You may interchange the elements of the means and/or the elements of the extremes at any time. This can save you time in solving some equations involving proportions.

Ex:  a/b = c/d,  a/c = b/d, d/b = c/a

3) You may take the reciprocal of both sides of a proportion. Note that this changes the means & extremes.
Ex: a/b = c/d,  b/a = d/c

4) You may take any element & multiply the other element along the means or the extremes.
Ex: a/b = c/d,  a = bc/d  or c =ad/b or b = ad/c or d = bc/a

~If two quantities are related LINEARLY, proportions are useful for finding many solutions.

Ex: Under normal conditions, an individual can walk 2 miles in 22 minutes (my estimation). How far can this individual walk in an hour & a half? 

Ex: A 185 lb diabetic needs 22 cc’s of insulin daily. Under normal conditions, how much does a 135 lb diabetic need?

Solution:   x/135 lb = 22 cc/185 lb.  The label of lb will go & leave you with cc's.
We get x = 135(22)/185 cc's = 16.05 cc's.

~Percents: 

1) percent to decimal (move the decimal two places to the left & drop the % symbol)

2) decimal to percent (move the decimal two places to the right & introduce the % symbol)

3) fraction to percent (divide to get a decimal & proceed as in 2 above)

4) percent to fraction (place the percentage over 100 & drop the % symbol)(remove all decimals from top & bottom by multiplying top & bottom by the appropiate power of 10)(reduced, if possible)

~Solving problems involving percentages:  Translate the problem into an equation & solve.

1) Use the decimal equivalent of the percent in the equation.
2) The word “of” translates “to multiply”.
3) The word “is” translates to “equality”.

Ex:  In a given town, 47.3% of the residents live in apartments. The total number of residents of the town is 3,755.  (give answer to the nearest whole resident).

Solution:  x = (.473)(3755)= 1776 (nearest whole resident)

Ex: 56.8% of students at a local college drink Pepsi. This number is estimated to be 2,766. How many students attend this local college?

Solution:   .568 x = 2766  gives x = 2766/.568 = 4870 (nearest integer)

~Review Percent Increase & Percent Decrease.  (missed often by many) (Important in business)

~To find the percentage increase or decrease, find the amount of increase or decrease & place it over the STARTING POINT. 

Ex: You purchased a penny stock for  $0.24. In one day session it increased to $1.13. What is the percentage increase?

Solution:    % increase = (1.13-0.24)/0.24 = 3.71=371% (nearest %)

Ex: You don’t sell this stock (bad decision). The next day, it goes back down & closes at the original price you paid, at $0.24. What was the percentage decrease in this stock that day?

Solution:    % decrease = (1.13-0.24)/1.13 = 0.79 = 79% (nearest %)

~Note:  The bottoms in the above problems are different since their starting points differ.

~Compound Interest:   P = Po (1 + r/n)^nt . Po are deposited at an APR (annual percent rate) of r %, compounded n times a year, the amount, at the end of t years, will be P.

Ex: How much must be invested at the annual rate of 12%, compounded monthly, in order for the value of the investment to be $20,000 at the end of 5 years?  (P=20,000, r=.12, n=12, t=5, find Po)

20000 = x(1 + .12/12)^(12)(5) = x (1.01)⁶⁰, then x = 20000/(1.01)⁶⁰

 (do 20000 divided by 1.01 y^x 60 enter)

x = $11,009 (nearest dollar).