Copyright © Cengage Learning. All rights reserved. 7 Ratio and Proportion Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 7.1 Ratio Copyright © Cengage Learning. All rights reserved.

Ratio The comparison of two numbers is a very important concept, and one of the most important of all comparisons is the ratio. The ratio of two numbers, a and b, is the first number divided by the second number. Ratios may be written in several different ways. For example, the ratio of 3 to 4 may be written as 3/4, 3 : 4, or 3  4. Each form is read “the ratio of 3 to 4.”

Ratio If the quantities to be compared include units, the units should be the same whenever possible. To find the ratio of 1 ft to 15 in., first express both quantities in inches and then find the ratio: Ratios are usually given in lowest terms.

Ratio of two fractions Express the ratio in lowest terms.

Ratio of unlike units Ratios can compare unlike units as well. Suppose you drive 75 miles and use 3 gallons of gasoline. Your mileage would be found as follows:

Ratio We say that your mileage is 25 miles per gallon. Note that each of these two fractions compares unlike quantities: miles and gallons. A rate is the comparison of two unlike quantities whose units do not cancel.

Copyright © Cengage Learning. All rights reserved. 7.2 Proportion Copyright © Cengage Learning. All rights reserved.

Proportion A proportion states that two ratios or two rates are equal. Thus, and are proportions. A proportion has four terms. In the proportion the first term is 2, the second term is 5, the third term is 4, and the fourth term is 10.

Proportion Proportion In any proportion, the product of the means equals the product of the extremes. That is, if , then bc = ad. To determine whether two ratios are equal, put the two ratios in the form of a proportion. If the product of the means equals the product of the extremes, the ratios are equal. We normally call this cross multiplying.

Example 3 Determine whether or not the ratios and are equal. If 36  29 = 13  84, then . However, 36  29 = 1044 and 13  84 = 1092. Therefore,

Proportion To solve a proportion means to find the missing term. To do this set up your ratios using labels to help you place numbers in correct spots Cross multiply and then divide to solve the resulting equation.

Example A nurse needs to give an IV drip at the following rate: 50ml every 5 seconds. If a patient needs 5000 ml how long will she have until she needs to check on the IV to see if it is empty? 50 ml = 5000 ml 5 sec ? Cross multiply so 50 (?) = 5000 (5) or 50(?) = 25000 Then divide by 50 to find the ? 25000/50 = 500 seconds She has 500 seconds before it will run out.

Example cont’d Alex’s soccer team had a record last year of 10 wins to 5 losses. If his team is playing 30 games this season, how many losses can they expect if they have the same rate? 5 losses = ? 15 games 30 games Cross multiply so 15(?)=5(30) or 15(?)=150 Divide by 15 to find ? 150/15 = 10 losses.

Copyright © Cengage Learning. All rights reserved. 1.13 Percent Copyright © Cengage Learning. All rights reserved.

Percent Percent is the comparison of any number of parts to 100 parts. The word percent means “per hundred.” The symbol for percent is %. You wish to put milk in a pitcher so that it is 25% “full” (Figure 1.34a). (a) This pitcher is 25% full. (b) This pitcher is 83% full. (c) This pitcher is 100% full.

Percent First, imagine a line drawn down the side of the pitcher. Then imagine the line divided into 100 equal parts. Each mark shows 1%: that is, each mark shows one out of 100 parts. Finally, count 25 marks from the bottom. The amount of milk below the line is 25% of what the pitcher will hold. Note that 100% is a full, or one whole, pitcher of milk.

Percent One dollar equals 100 cents or 100 pennies. Then, 36% of one dollar equals 36 of 100 parts, or 36 cents or 36 pennies.

Percent A car’s radiator holds a mixture that is 25% antifreeze. That is, in each hundred parts of mixture, there are 25 parts of pure antifreeze. A state charges a 5% sales tax. That is, for each $100 of goods that you buy, a tax of $5 is added to your bill. The $5, a 5% tax, is then paid to the state. Just remember: percent means “per hundred.”

Changing a Percent to a Decimal

Example 1 Change each percent to a fraction and then to a decimal. a. b. c. d. 75 hundredths 45 hundredths 16 hundredths 7 hundredths

Changing a Percent to a Decimal To change a percent to a decimal, move the decimal point two places to the left (divide by 100). Then remove the percent sign (%). If the percent contains a fraction, write the fraction as a decimal. Ex. 33 1/3 % = 33.3 % = .333 For problems involving percents, we must use the decimal form of the percent, or its equivalent fractional form.

Changing a Fractional % to a Simple Fraction Change to a fraction. So First, change the mixed number to an improper fraction. Since 100/3 is a % it is over 100 which would mean to divide by 100/1 or to more simply to multiply by 1/100

Copyright © Cengage Learning. All rights reserved. 1.14 %, Whole, and Part Copyright © Cengage Learning. All rights reserved.

%, Whole, and Part Any percent problem calls for finding one of three things: 1. the rate (percent), 2. the Whole amount the % is being compared to 3. the part, or the piece that we get when the % is compared to the whole Such problems are solved using one of three percent formulas. In these formulas, we let %= the rate (percent) W = the whole P = the part or amount (sometimes called the percentage)

%, Whole, and Part The following may help you identify which letter stands for each given number and the unknown in a problem: 1. The rate, %, usually has either a percent sign (%) or the word percent with it. 2. The whole, W, is the whole (or entire) amount that the % is being compared to or taken out of and often follows the word of. 3. The part, P, is the number that you get as answer when you apply the % to the whole and often follows the word is or the = sign.

Example 1 Given: 25% of $80 is $20. Identify %, W, and P. % is 25%. W is $80. P is $20. 25 is the number with a percent sign. Remember to change 25% to the decimal 0.25 for use in a formula. $80 is the whole amount. It also follows the word of. $20 is the part. It is also the number that is not R or B.

Proportion Knowing this formula and knowing the fact that percent means “per hundred,” we can write the proportion % = P 100 W If we then can identify and fill in the pieces given, we will have one spot empty and we can cross multiply and divide to find the missing number.

The % Proportion Remember: we usually find the part near the word “is” or an = sign we find the whole near the word “of” often. The rate is always expressed as a %.

Solving for the Part Find 75% of 180. Set up the proportion: 75/100 = ?/ 180 (note 180 is the whole and the part is missing) Cross multiply so: 75(180) = 100(?) Or 13500= 100 (?) Then divide by 100 to find the par so 13500/100= 135 Hence 75% of 180 = 135

Solving for the Whole Aluminum is 12% of the mass of a car. If a car has 186 kg of aluminum in it, what is the total mass of the car? Set up the proportion: 12/100 = 186/? Cross multiply: 12(?) = 100(186) Or 12(?) = 18600 Divide: 18600/12 = 1550 Hence the total mass of the car is 1550 kg

Solving for the Rate (%) What % of 20 meters is 5 meters? Set up the proportion: ?/100 = 5/20 (note the % or rate is missing and the 20 is near the “of” and the 5 is near the “is”) Cross multiply: ?(20) = 5(100) so ?(20) = 500 Divide: 500/20 = 25 So 5 out of 20 meters is 25%

Practice Problems Page 282 7-12 , 14, 17 Page 288 13-15, 25, 37, 48-50 Page 80 question 81 Page 84 15, 16, 19, 21, 28 Hints: use 942in for 78ft 6in and 146 in for 12ft 2in (why?) then find that area, next find 20% of it then divide by the area of one window (24in*72in)