Ratio, Proportion, and Percent - I
Ratios and Proportions
In this module we extend the idea of equivalent fractions, converting fractions to decimals, andequations. Recall that two fractions are equivalent (or equal) if they both have the same decimalform. For example, the fractionsandare equivalent, since they both have a decimal formof 0.75. Note that they are also equivalent sincecan be reduced to, but this connection isnot as relevant to our discussions in this module. There are other quantities which are equivalentto 0.75, such asand, though they generally do not fit the form of a simplified fraction.A ratio can be written in three ways:
Using the word to 3 to 4
Using a colon 3:4 These all read 3 to 4.
Writing a fraction
They do fit the form of a ratio, which refers to any quotient of two quantities. Ratios are oftenassociated with units to give the quantities meaning. When a ratio is used to compare two different kinds of quantities, such as miles to gallons, it is called a rate. Below are examples of these typesof ratios:
speed of a car: = 65 miles per hour
unit cost of food: = 20 cents per ounce
cost of lumber: = 2.125 dollars per foot
Note the use of the word “per”, which is frequently used in expressing ratios.
Example 1 Write the following mathematical quantities as a ratio, and simplify.
a. 214 miles traveled in 3 hours
71 miles per hour
b. $17.10 paid for 3.6 pounds of steak
$4.75 per pound
c. $74.50 paid for 24 pieces of 2×4 lumber
$3.1 per piece
d. $195,000 paid for 15.6 acres of land
$12,500 per acre
Practice 1 Compare the two given quantities using unit value comparisons.
a. Which is cheaper b. Which is faster
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Price per Earnings (PE ratio) of Stock
$94.71 for $3.85 of company A’searnings
$48.26 for $1.86 of company B’searnings
Speed of a Car
170 miles in 2.5 hours for car A
225 miles in 3.2 hours for car B
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24.60 dollars per dollar earnings of company A 68 miles per hour for car A
25.95 dollars per dollar earnings of company B 70.3 miles per hour for car B
Practice 2You can buy 18 ounces of shampoo for $3.99 or 24 ounces for $5.09. Which is the better buy?
$3.99/18 = $0.22 per ounce
$5.09/24 = $0.21 per ounce – the better choice
Ratios are useful in comparing quantities, especially prices, as seen above and recipes (practice 4).
Example 2 To make short crust pastry, one recipe book says ‘use one part of fat to two parts of flour’; another recipe says‘use fat and flour in the ratio of one to two’; and yet another says ‘use half as much fat as flour’. These aredifferent ways of expressing the same ratio. Ratios are often expressed as fractions. So in this case:
=
Since you can multiply top and bottom of a fraction by the same number and get an equivalent fraction, you canuse the ratio in a number of ways. If you have 100 grams of fat then
==
So you need 200 grams of flour to 100 grams of fat. There are many ways to arrive at this answer. Theimportant point is that a ratio of 100 to 200 is equivalent to 1 to 2.
To make concrete, the instructions are ‘use sand and cement in the ratio three to one’. This means
===
If you have 30 kg of cement, then you need 90 kg of sand.
Practice 3If the ratio of distance measured in miles to the same distance measured in kilometers is 5 to 8. What is the speed of 70 miles per hour in kilometers per hour?
Ans: = x=112
Up to this point we have constructed ratios for the purpose of comparing them. Suppose,however, that we know two ratios are equal. Any time two ratios are equal we call the resultingequation a proportion. That is, a proportion is a statement of the form:
=
To solve a proportion, we multiply by the Least Common Multiplier(LCM) = bd:
bd •=bd •
ad = bc This can also be obtained by using cross multiplication, shown at right.
Means and Extremes
When you set up a proportion in the form A: B = C: D, the values A and D are the extremes.
The values B and C are the means. If you have trouble remembering which is which, think,
“Extremes are on the ends. Means are in the middle.” This is essentially another form of cross product of proportion shown above.
In any true proportion, the product of the means always equals the product of the extremes.
Example 3. Suppose we are given the proportion:=. Find x.
x= 9/5
Practice 4Solve each proportion.
a. =x = - 5/3 b. =y = 7.75
Practice 5 In one state, property is taxed at a rate of $1.04 for every $120 in property value.
If a home has a property value of $180,000, how much will be the property tax?
1.04:120 = x:180,000
Ans. The property tax will be $1560.
Practice 6 A map uses a scale of inch = 30 miles. On a map, two cities are 3inches apart. How many miles apart are the two cities?
¼:30 = 3.125:x
Ans. The two cities are 375 miles apart.
Practice 7Compare the two given quantities using unit value comparisons. Circle the one that is cheaper.
A. Orange Juice
12 oz for $1.36
17 oz for $1.95
B. Toilet Paper
8 rolls for $2.88
15 rolls for $5.40 They are the same.
C. Speed of a Car
235 miles in 4.2 hours
295 miles in 5.6 hours
Practice 8Solve each proportion.
a. =z = - 14/3 b. =y = 75/4
c. –=t = 24d. =x = 9/2
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