A ratio is a comparison of two quantities by division. The ratio of a to b can be written a : b or a/b, where b ≠ 0.
Ratios that name the same comparison are said to be equivalent. A statement that two ratios are equivalent, such as 1/12 = 2/24 , is called a proportion.
Example 1 :
The ratio of faculty members to students at a college is 1:15. There are 750 students. How many faculty members are there?
Solution :
Write a ratio comparing faculty to students.
Faculty ----> 1
Students ----> 15
Write a proportion. Let x be the number of faculty members.
x/750 = 1/15
Because x is divided by 750, multiply each side of the equation by 750.
750 ⋅ (x/750) = (1/15) ⋅ 750
x = 50
There are 45 faculty members.
A rate is a ratio of two quantities with different units, such as 48 mi/3 gal. Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as 16 mi/1 gal, or 16 mi/gal. You can convert any rate to a unit rate.
Example 2 :
David ate 58.5 hot dogs in 13 minutes to win a contest. Find the unit rate in hot dogs per minute. Round to the nearest hundredth.
Solution :
Write a proportion to find an equivalent ratio with a second quantity of 1.
58.5/13 = x/1
4.5 = x
The unit rate is approximately 4.46 hot dogs per minute.
Dimensional analysis is a process that uses rates to convert measurements from one unit to another. A rate such as 12 in./1 ft, in which the two quantities are equal but use different units, is called a conversion factor . To convert from one set of units to another, multiply by a conversion factor.
Example 3 :
A large adult male human has about 12 pints of blood. Use dimensional analysis to convert this quantity to gallons.
Solution :
Step 1 :
Convert pints to quarts.
Multiply by a conversion factor whose first quantity is quarts and whose second quantity is pints.
12 pt ⋅ (1 qt/2 pt) = 6 qt
12 pints is 6 quarts.
Step 2 :
Convert quarts to gallons.
Multiply by a conversion factor whose first quantity is gallons and whose second quantity is quarts.
6 qt ⋅ (1 gal/4 qt) = 6/4 gal = 1½ gal
A large adult male human has about 1½ gallons of blood.
In the proportion a/b = c/d, the products a · d and b · c are called cross products.
You can solve a proportion for a missing value by using the Cross Products Property.
Words :
In a proportion, cross products are equal.
Numbers :
Algebra :
Solve each proportion.
Example 4 :
7/9 = 3/x
Solution :
7/9 = 3/x
Use cross products.
7x = 3(9)
7x = 27
Divide each side by 7.
x = 27/7
Example 5 :
5/9 = 10/(x + 2)
Solution :
5/9 = 10/(x + 2)
Use cross products.
5(x + 2) = 9(10)
5x + 10 = 90
Subtract 10 from each side.
5x = 80
Divide each side by 5.
x = 16
A scale is a ratio between two sets of measurements, such as 1 in : 7 mi. A scale drawing or scale model uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing.
Example 6 :
On a map with a scale of 1 in = 18 mi, the distance from Chicago to Evanston is 0.63 in. What is the actual distance ?
Solution :
Write the scale as a fraction.
map/actual = 1 in/18 mi
Let x be the actual distance.
1/18 = 0.63/x
Use cross products to solve.
1(x) = 18(0.63)
x = 11.34 miles
The actual distance is 11.34 miles.
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