3.1.7: Summary of Key Concepts

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Summary of Key Concepts

Denominate Numbers
Numbers that appear along with units are denominate numbers. The amounts 6 dollars and 4 pints are examples of denominate numbers.

Like and Unlike Denominate Numbers
Like denominate numbers are denominate numbers with like units. If the units are not the same, the numbers are unlike denominate numbers.

Pure Numbers
Numbers appearing without a unit are pure numbers.

Comparing Numbers by Subtraction and Division
Comparison of two numbers by subtraction indicates how much more one number is than another. Comparison by division indicates how many times larger or smaller one number is than another.

Comparing Pure or Like Denominate Numbers by Subtraction
Numbers can be compared by subtraction if and only if they are pure numbers or like denominate numbers.

Ratio Rate
A comparison, by division, of two like denominate numbers is a ratio. A comparison, by division, of two unlike denominate numbers is a rate.

Proportion
A proportion is a statement that two ratios or rates are equal.
\(\dfrac{\text{3 people}}{\text{2 jobs}} = \dfrac{\text{6 people}}{\text{4 jobs}}\) is a proportion.

Solving a Proportion
To solve a proportion that contains three known numbers and a letter that represents an unknown quantity, perform the cross multiplication, then divide the product of the two numbers by the number that multiplies the letter.

Proportions Involving Rates
When writing a proportion involving rates it is very important to write it so that the same type of units appears on the same side of either the equal sign or the fraction bar.
\(\dfrac{\text{unit type 1}}{\text{unit type 2}} = \dfrac{\text{unit type 1}}{\text{unit type 2}} \text{ or } \dfrac{\text{unit type 1}}{\text{unit type 1}} = \dfrac{\text{unit type 2}}{\text{unit type 2}}\)

Five-Step Method for Solving Proportions

By careful reading, determine what the unknown quantity is and represent it with some letter. There will be only one unknown in a problem.
Identify the three specified numbers.
Determine which comparisons are to be made and set up the proportion.
Solve the proportion.
Interpret and write a conclusion.

When solving applied problems, ALWAYS begin by determining the unknown quantity and representing it with a letter.

Percents
A ratio in which one number is compared to 100 is a percent. Percent means "for each hundred."

Conversion of Fractions, Decimals, and Percents
It is possible to convert decimals to percents, fractions to percents, percents to decimals, and percents to fractions.

Applications of Percents:
The three basic types of percent problems involve a base, a percentage, and a percent.

Base
The base is the number used for comparison.

Percentage
The percentage is the number being compared to the base.

Percent
By its definition, percent means part of.

Solving Problems
\(\text{Percentage} = \text{(percent)} \times \text{(base)}\)
\(\text{Percent} = \dfrac{\text{percentage}}{\text{base}}\)
\(\text{Base} = \dfrac{\text{percentage}}{\text{percent}}\)