A proportion is a set of 2 fractions that equal each other.
Real World Uses of Proportions
- Modifying a budget for a restaurant chain that is expanding from 3 locations to 20 locations
- Creating a skyscraper from blueprints
- Calculating tips, commissions, and sales tax
Modifying a Recipe
On Monday, you are cooking enough white rice to serve exactly 3 people. The recipe calls for 2 cups of water and 1 cup of dry rice. On Sunday, you are going to serve rice to 12 people. How would the recipe change? If you’ve ever made rice, you know that this ratio — 1 part dry rice and 2 parts water — is important. Mess it up, and you’ll be scooping a gummy, hot mess on top of your guests' crawfish étouffée.
Because you are quadrupling your guest list (3 people * 4 = 12 people), you must quadruple your recipe. Cook 8 cups of water and 4 cups of dry rice. These shifts in a recipe demonstrate the heart of proportions: use a ratio to accommodate life's greater and smaller changes.
Algebra and Proportions 1
Sure, with the right numbers, you can forgo setting up an algebraic equation to determine the amounts of dry rice and water. What happens when the numbers are not so friendly? On Thanksgiving, you'll be serving rice to 25 people. How much water do you need?
Because the ratio of 2 parts water and 1 part dry rice applies to cooking 25 servings of rice, use a proportion to determine the quantity of ingredients.
Note: Translating a word problem into an equation is super important. Yes, you can solve an incorrectly set up equation and find an answer. You can also mix rice and water together to create "food" to serve at Thanksgiving. Whether the answer or food is palatable depends on the equation.
Think about what you know:
- 3 servings of cooked rice = 2 cups of water; 1 cup of dry rice
25 servings of cooked rice = ? cups of water; ? cup of dry rice - 3 servings of cooked rice/25 servings of cooked rice = 2 cups of water/x cups of water
- 3/25 = 2/x
Cross multiply. Hint: Write these fractions vertically to get the full understanding of cross multiplying. To cross multiply, take the first fraction's numerator and multiply it by the second fraction's denominator. Then take the second fraction's numerator and multiply it by the first fraction's denominator.
3 * x = 2 * 25
3x = 50
Divide both sides of the equation by 3 to solve for x.
3x/3 = 50/3
x = 16.6667 cups of water
Freeze- verify that the answer is correct.
Is 3/25 = 2/16.6667?
3/25 = .12
2/16.6667= .12
Whoo hoo! The first proportion is right.
Algebra and Proportions 2
Remember that x will not always be in the numerator. Sometimes the variable is in the denominator, but the process is the same.
Solve the following for x.
36/x = 108/12
Cross multiply:
36 * 12 = 108 * x
432 = 108x
Divide both sides by 108 to solve for x.
432/108 = 108x/108
4 = x
Check and make sure the answer is right. Remember, a proportion is defined as 2 equivalent fractions:
Does 36/4 = 108/12?
36/4 = 9
108/12 = 9
It’s right!
Answers and Explanations
1. A nurse discovers that his patient’s heart rate is 19 beats per 15 seconds. How many times does the patient’s heart beat in 60 seconds?
What do you know?
19 beats = 15 seconds
? beats = 60 seconds
19 beats/? beats = 15 seconds/60 seconds
19/x = 15/60
Cross Multiply.
19 * 60 = 15 * x
1140 = 15x
Divide both sides of the equation by 15 to solve for x.
1140/15 = 15x/15
76 = x
Use common sense to verify the answer.
Every 15 seconds, the patient’s heart beats 19 times. So by quadrupling the time to 60 seconds, that means that the number of heart beats should quadruple. 19 * 4 = 76
2. Chad swims seven laps in 2 minutes. At this rate, how many laps can he swim in 11 minutes?
What do you know?
7 laps = 2 minutes
? laps = 11 minutes
7 laps/? laps = 2 minutes/11 minutes
7/x = 2/11
Cross Multiply.
7 * 11 = 2 * x
77 = 2x
Divide both sides by 2 to solve for x.
77/2 = 2x/2
38.5 = x
Use Algebra to verify the answer.
Does 7/38.5 = 2/11?
7/38.5 = .181818…
2/11 = .181818…
3. Renee’s photo album, Summer Vacation, contains 11 photos and occupies 18.8 megabytes. She has another photo album, Old Skool, that contains 275 pictures. How many megabytes does Old Skool occupy?
What do you know?
11 photos = 18.8 megabytes
275 photos = ? megabytes
11photos/275 photos
18.8 megabytes = ? megabytes
11/275 = 18.8/x
Cross Multiply.
11 * x = 18.8 * 275
11x = 5170
Divide by sides by 11 to solve for x.
11x/11 = 5170/11
x = 470
Use Algebra to verify the answer.
Does 11/275 = 18.8/470?
11/275 = .04
18.8/470 = .04