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7 Ratio and Proportion
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Ratio7.1
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• 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
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• 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
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• Express the ratio in lowest terms.
Ratio of two fractions
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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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
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• 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.
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• 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
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• 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
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• 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
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Changing a Percent to a Decimal
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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.
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• 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
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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 %.
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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
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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
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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%
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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)