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7 Ratio and Proportion

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Ratio7.1

• 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

• Express the ratio in lowest terms.

Ratio of two fractions

• 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 of unlike units

• 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.

Ratio

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Proportion7.2

• 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.

Proportion

• Determine whether or not the ratios and are equal.

• If 36 29 = 13 84, then .

• However, 36 29 = 1044 and 13 84 = 1092.

• Therefore,

Example 3

• 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.

Proportion

• 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

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.

Example cont’d

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1.13 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).

Percent

(a) This pitcher is 25% full.

(b) This pitcher is 83% full.

(c) This pitcher is 100% full.

• 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.”

Percent

Changing a Percent to a Decimal

• Change each percent to a fraction and then to a decimal.

• a.

• b.

• c.

• d.

Example 1

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 Percent to a Decimal

• Change to a fraction.

• So

Changing a Fractional % to a Simple Fraction

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

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%, Whole, and Part1.14

• 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.

%, Whole, and Part

• Given: 25% of $80 is $20. Identify %, W, and P.

• % is 25%.

• W is $80.

• P is $20.

Example 1

25 is the number with a percent sign. Remember tochange 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.

• 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.

Proportion

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)