7th Grade Chapter 6
6-1 Ratios and Unit Rates
A ratio is a comparison of two quantities by division. A rate is a ratio that compares quantities in different units. A unit rate is a rate that has a denominator of 1 (and a unit), such as a temperature change of 2 degrees/h. Rates can be converted from one set of units to another. Dimensional analysis is a way to check whether a problem has been solved correctly by checking the units.
6-2 Proportions
A mathematical sentence that states that two ratios are equal is a proportion. Like any mathematical sentence, a proportion may be true, false, or open. Two ratios form a proportion if and only if their cross products are equal. To solve a proportion that contains a variable, find the value of the variable that makes the equation true.
6-3 Similar Figures and Scale Drawings
Not only are the corresponding sides of similar figures in proportion, but all other segments, such as medians and altitudes, are also in proportion. Similarly, the scale of a drawing - the ratio of a distance in the drawing to the corresponding actual distance - can be used to determine any drawing distance given an actual distance, and vice versa.
6-4 Fractions, Decimals and Percents
The word percent comes from the Latin phrase per centum, meaning by the hundred or per 100. "Per" suggests division. Since 28% means 28 per 100, you can also write 28% as 28/100 or 0.28.
6-5 Proportions
In general, a percent problem has a part over its whole equal to a part of 100 over 100. 35% means 35 equal parts out of 100 total parts. To find 35% of $623, set up a proportion as follows: 35 is to 100 as the part of 623 is to the whole amount of 623: x/623 = 35/100 . Then solve for x.
6-6 Percents and Equations
Some salespeople, such as car sales representatives, are often paid an amount based upon how much they sell. This is known as a commission. For example, a 3% commission c on a car sold for $17,000 can be calculated using a percent equation c = 0.03 * $17,000 = $510.00
6-7 Percentage of Change
A large percent of change may not necessarily mean a large amount of actual change. When 4 is increased to 6 by adding 2, the percent of change is 50%. However, when 400 is increased to 500 by adding 100, the percent of change is only 25%.
6-8 Markup and Discount
One example of percent of increase is the percent of markup that stores use to establish the selling price for merchandise. In Lesson 6-7, percent of change is defined as (amount of change)/(original amount). You can define percent of markup as (markup)/(original cost). An example of percent of decrease is the percent of discount when an item goes on sale. You can define percent of discount as (discount)/(original price).
6-9 Applications of Rational Numbers
Rational numbers in the form of fractions, decimals, or percents can be used to solve real-world, multistep problems involving distance and rate. A rate that is familiar to students is speed. Average speed is the total distance traveled divided by the amount of time during which that distance is traveled. For example, if Reza travels 117 miles in 2 hours, his average speed is 117mi/2h or 58.5 mph 6-10 Reasoning Strategy: Make a Table You can use a table to help solve problems that describe recurring relationships. Information can easily be compared when displayed in such an organized manner.
A ratio is a comparison of two quantities by division. A rate is a ratio that compares quantities in different units. A unit rate is a rate that has a denominator of 1 (and a unit), such as a temperature change of 2 degrees/h. Rates can be converted from one set of units to another. Dimensional analysis is a way to check whether a problem has been solved correctly by checking the units.
6-2 Proportions
A mathematical sentence that states that two ratios are equal is a proportion. Like any mathematical sentence, a proportion may be true, false, or open. Two ratios form a proportion if and only if their cross products are equal. To solve a proportion that contains a variable, find the value of the variable that makes the equation true.
6-3 Similar Figures and Scale Drawings
Not only are the corresponding sides of similar figures in proportion, but all other segments, such as medians and altitudes, are also in proportion. Similarly, the scale of a drawing - the ratio of a distance in the drawing to the corresponding actual distance - can be used to determine any drawing distance given an actual distance, and vice versa.
6-4 Fractions, Decimals and Percents
The word percent comes from the Latin phrase per centum, meaning by the hundred or per 100. "Per" suggests division. Since 28% means 28 per 100, you can also write 28% as 28/100 or 0.28.
6-5 Proportions
In general, a percent problem has a part over its whole equal to a part of 100 over 100. 35% means 35 equal parts out of 100 total parts. To find 35% of $623, set up a proportion as follows: 35 is to 100 as the part of 623 is to the whole amount of 623: x/623 = 35/100 . Then solve for x.
6-6 Percents and Equations
Some salespeople, such as car sales representatives, are often paid an amount based upon how much they sell. This is known as a commission. For example, a 3% commission c on a car sold for $17,000 can be calculated using a percent equation c = 0.03 * $17,000 = $510.00
6-7 Percentage of Change
A large percent of change may not necessarily mean a large amount of actual change. When 4 is increased to 6 by adding 2, the percent of change is 50%. However, when 400 is increased to 500 by adding 100, the percent of change is only 25%.
6-8 Markup and Discount
One example of percent of increase is the percent of markup that stores use to establish the selling price for merchandise. In Lesson 6-7, percent of change is defined as (amount of change)/(original amount). You can define percent of markup as (markup)/(original cost). An example of percent of decrease is the percent of discount when an item goes on sale. You can define percent of discount as (discount)/(original price).
6-9 Applications of Rational Numbers
Rational numbers in the form of fractions, decimals, or percents can be used to solve real-world, multistep problems involving distance and rate. A rate that is familiar to students is speed. Average speed is the total distance traveled divided by the amount of time during which that distance is traveled. For example, if Reza travels 117 miles in 2 hours, his average speed is 117mi/2h or 58.5 mph 6-10 Reasoning Strategy: Make a Table You can use a table to help solve problems that describe recurring relationships. Information can easily be compared when displayed in such an organized manner.