7.1The Concept of Ratio
Definition of a Ratio
A ratio is an ordered pair of numbers used to show a comparison between like or unlike quantities, written x to y, x/y, x y, or x:y (y 0)
Definition of Equivalent Ratios
Two ratios are equivalent ratios if their respective fractions are equivalent or if the quotients of the respective terms are the same.
Using Ratios to Compare Like Quantities or Measures
A ratio involving two like quantities permits three types of comparisons:
Example: There are 7 boys and 12 girls in the class:
a)part to part
b)part to whole
c)whole to part
Ratios in Decimal Form
Example the cost factor is .85, which can be expressed as the ration .85:1, 85:100
Example:Given this ratio, what would the dealers cost be on a $15,000 auto?
7.2Proportional variation
Definition of Quantities Varying Proportionally
Two quantities vary proportionally if and only if, as their corresponding values increase or decrease, the ratios of the last two quantities are always equivalent.
Example: 1:3, 2:6, 3: 9 . . .
Multiplicative Property of Quantities That Vary Proportionally
When quantities a and b vary proportionally, a nonzero number k exists such that , or a = bk, for all corresponding values a and b.
:
Constant Change Property of Quantities That Vary Proportionally
When quantities a and b vary proportionally and the ratio of a to b is 1 to n, a unit change in a value of a always evokes a constant change of n in the corresponding value of b.
Example:Given the following ratios: what is the constant change?
1: 20, 2:40, 3: 60. . . .
Definition of a Proportion
A proportion is an equation stating that two ratios are equivalent.
Properties of Proportions
Reciprocal Property of Proportions
For nonzero integer a, b, c and d, if and only if
Cross-Product Property of Proportions
For integer a, b, c and d (bd 0), if and only if ad = bc.
Example:Solve
Applying the Method of Equating Numerators or Denominators
Example:A package of three grapefruits sells for $1.80. How many grapefruits could you buy for $9?
Example:
If there should be 3 tractors for every 4 farmers, how many tractors are needed for 44 farmers?
If a turtle travels 5in. every 10 seconds, how may feet does it travel in 50 second? (watch units!)
7.3Solving Percent Problems
Definition of Percent
A percent is a ratio with a denominator of 100.
To convert a percent to a decimal: simply divide it by 100 and drop the % sign.
Example:25%=
To convert a percent to a fraction: place the value over 100 and drop the % sign.
Example: 80%
To convert a fraction to a percent or decimal: write an equivalent fraction with denominator 100 or divide the numerator by the denominator.
Example:1/5 =
Solving Percent Problems
Finding a Percent of a number. What is 20% of 100? Multiply .20 100
Finding a Number when a Percent of it is known.
Find the ratio: For example: Barry determined that he had traveled 240 miles, which he estimated as being 40% of the trip. How long is the trip?
Finding the percent that one number is of another:
Example:Sherry learned that 72 of her 150-member class went to college. What percent of her class went to college?
Procedure for Finding the Percent Increase or Decrease
1)Determine the amount of increase or decrease
2)Divide this amount by the original amount.
3)Convert this fraction or decimal to a percent.
Example:Mary’s stock went form $2.50 a share to $3.75 what was the percent increase?
Simple Interest
I = prt
Compound Interest
Example:How much money would you have if you invested 1000 for 5 years at 8% compounded daily?
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Student Notes – Math 104