Introduction to Exploring Ratios and Proportions
Mathematics is a language.
—JOSIAH WILLARD GIBBS, theoretical physicist (1839–1903)
This lesson begins by exploring ratios, using familiar examples to explain the mathematics behind the ratio concept. It concludes with the related notion of proportions, again illustrating the math with everyday examples.
Ratios
A ratio is a comparison of two numbers. For example, let's say that there are 3 men for every 5 women in a particular club. That means that the ratio of men to women is 3 to 5. It doesn't necessarily mean that there are exactly 3 men and 5 women in the club, but it does mean that for every group of 3 men, there is a corresponding group of 5 women. The table below shows some of the possible sizes of this club.
In other words, the number of men is 3 times the number of groups, and the number of women is 5 times that same number of groups.
A ratio can be expressed in several ways:
 using "to" (3 to 5)
 using "out of" (3 out of 5)
 using a colon (3:5)
 as a fraction
 as a decimal (0.6)
Like a fraction, a ratio should always be reduced to lowest terms. For example, the ratio of 6 to 10 should be reduced to 3 to 5 (because the fraction reduces to ).
Here are some examples of ratios in familiar contexts:

Last year, it snowed 13 out of 52 weekends in New York City. The ratio 13 out of 52 can be reduced to lowest terms (1 out of 4) and expressed as any of the following: 

Reducing to lowest terms tells you that it snowed 1 out of 4 weekends, ( of the weekends or 25% of the weekends). 

Lloyd drove 140 miles on 3.5 gallons of gas, for a ratio (or gas consumption rate) of 40 miles per gallon: miles per gallon 

The studentteacher ratio at Clarksdale High School is 7 to 1. That means for every 7 students in the school, there is 1 teacher. For example, if Clarksdale has 140 students, then it has 20 teachers. (There are 20 groups, each with 7 students and 1 teacher.) 

Pearl's Pub has 5 chairs for every table. If it has 100 chairs, then it has 20 tables. 

The Pirates won 27 games and lost 18, for a ratio of 3 wins to 2 losses. Their win rate was 60%, because they won 60% of the games they played. 
In word problems, the word per translates to division. For example, 30 miles per hour is equivalent to . Phrases with the word per are ratios with a bottom number of 1, like these:
24 miles per gallon 
$12 per hour 
3 meals per day 
4 cups per quart 
Tip
Ratios can also be used to relate more than two items, but then they are not written as fractions. Example: If the ratio of infants to teens to adults at a school event is 2 to 7 to 5, it is written as 2:7:5.

Ratios and Totals
A ratio usually tells you something about the total number of things being compared. In our first ratio example of a club with 3 men for every 5 women, the club's total membership is a multiple of 8 because each group contains 3 men and 5 women. The following example illustrates some of the total questions you could be asked about a particular ratio:
Example: Wyatt bought a total of 12 books, purchasing two $5 books for every $8 book.
 How many $5 books did he buy?
 How many $8 books did he buy?
 How much money did he spend in total?
Solution: The total number of books Wyatt bought is a multiple of 3 (each group of books contained two $5 books plus one $8 book). Since he bought a total of 12 books. he bought 4 groups of books (4 groups X 3 books = 12 books in total).
Total books: 8 $5 books + 4 $8 books = 12 books Total cost: $40 + $32 = $72
Tip
When a ratio involves three different things, the sum of the three becomes your multiple. For example, if there are 72 cars and the ratio of red to blue to white cars is 5:1:3, then the cars can be broken into multiples of 8.

Proportions
A proportion states that two ratios are equal to each other. For example, have you ever heard someone say something like this?
Nine out of ten professional athletes suffer at least one injury each season.
The words nine out of ten are a ratio. They tell you that of professional athletes suffer at least one injury each season. But there are more than 10 professional athletes. Suppose that there are 100 professional athletes. Then of the 100 athletes, or 90 out of 100 professional athletes, suffer at least one injury per season. The two ratios are equivalent and form a proportion:
Here are some other proportions:
Notice that a proportion reflects equivalent fractions: Each fraction reduces to the same value.
Cross Products
As with fractions, the cross products of a proportion are equal.
3 × 10 = 5 × 6
Many proportion word problems are easily solved with fractions and cross products. In each fraction, the units must be written in the same order. For example, let's say we have two ratios (ratio #1 and ratio #2) that compare red marbles to white marbles. When you set up the proportion, both fractions must be set up the same way— with the red marbles on top and the corresponding white marbles on bottom, or with the white marbles on top and the corresponding red marbles on bottom:
Alternatively, one fraction may compare the red marbles while the other fraction compares the white marbles, with both comparisons in the same order:
Here's a variation of an example used earlier. The story is the same, but the questions are different.
Example: 
Wyatt bought two $5 books for every $8 book. If he bought eight $5 books, how many $8 books did Wyatt buy? How many books did Wyatt buy in total? How much money did Wyatt spend in total? 
Solution: 
The ratio of books Wyatt bought is 2:1, or . For the second ratio, the eight $5 books goes on top of the fraction to correspond with the top of the first fraction, the number of $5 books. Therefore, the unknown b (the number of $8 books) goes on the bottom of the second fraction. Here's the proportion: 
Solve it using cross products:
2 × b × 1 = 8
2 × 4 = 800
Thus, Wyatt bought 4 $8 books and 8 $5 books, for a total of 12 books, spending $72 in total:
(8 × $5) + (4 × $8) = $72.
Check: 
Reduce , the ratio of $5 books to $8 books that Wyatt bought, to ensure getting back , the original ratio. 
Shortcut: 
On a multiplechoice test question that asks for the total number of books, you can automatically eliminate all the answer choices that aren't multiples of 3, perhaps solving the problem without any real work! 
Tip
Go to a grocery store and look closely at the prices listed on the shelves. Pick out a type of food that you would like to buy, such as cold cereal, pickles, or ice cream. To determine which brand has the cheapest price, you need to figure out each item's unit price. The unit price is a ratio that gives you the price per unit of measurement for an item. Without looking at the tags that give you this figure, calculate the unit price for three products, using the price and size of each item. Then, check your answers by looking at each item's price label that specifies its unit price.

Exploring Ratios and Proportions Sample Questions
 Every day, Bob's Bakery makes fresh cakes, pies, and muffins in the ratio of 3:2:5. If a total of 300 cakes, pies, and muffins is baked on Tuesdays, how many of each item is baked?
 The ratio of men to women at a certain meeting is 3 to 5. If there are 18 men at the meeting, how many people are at the meeting?
Solutions to Sample Questions
Question 1
1. 
The total number of items baked is a multiple of 10: 
3 + 2 + 5 = 10 
2. 
Divide 10 into the total of 300 to find out how many groups 

of 3 cakes, 2 pies, and 5 muffins are baked: 
300 ÷ 10 = 30 
3. 
Since there are 30 groups, multiply the ratio 3:2:5 by 30 

to determine the number of cakes, pies, and muffins baked: 
30 × 3 = 90 cakes 


30 × 2 = 60 pies 


30 × 5 = 150 muffins 
Check:


Add up the number of cakes, pies, and muffins: 
90 + 60 + 150 = 300 


Since the total is 300, the answer is correct. 
Question 2
On a multiplechoice question, you can eliminate any answer that's not a multiple of 8 (3 + 5 = 8). If more than one answer is a multiple of 8 or if this isn't a multiplechoice question, then you'll have to do some work. The first step of the solution is finding a fraction equivalent to with 18 as its top number (because both top numbers must reflect the same thing—in this case, the number of men). Since we don't know the number of women at the meeting, we'll use the unknown w to represent them. Here's the mathematical setup and solution:
3 × w =5 × 18
3 × w = 90
3 × 30 = 90
Since there are 30 women and 18 men, a total of 48 people are at the meeting.
Check:
Reduce Since you get .(the original ratio), the answer is correct.
Find practice problems and solutions for these concepts at Exploring Ratios and Proportions Practice Questions.
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From Practical Math Success in 20 Minutes A Day. Copyright © 2009 by LearningExpress, LLC. All Rights Reserved.