Proportions from Word Problems (using Ratios)

Ratios show how numbers are related.  When you put two ratios together in a proportion, you show that two sets of numbers are related in the same way.

Whenever you have a proportion (see Proportion lesson), or can create a proportion, you can solve by cross-multiplying, which is a great way to solve word problems using rates, scale, ratios (see Ratios lesson), and any other kinds of numbers that change proportionally. Hint, if you see the word "proportionally" in a word problem, you will definitely want to use a proportion to solve it!

Just as you have to be careful about order when setting up a ratio, when you set up a proportion, the trickiest part is setting it up.  Always write labels in your fractions.  Labels should match across the proportion (so labels on top should be the same and labels on the bottom should be the same).  If your labels are not the same, something is wrong!  Either you need to flip one of the ratios, OR, you need to do a calculation to get the proper item in the fraction.  

Example:

The teacher student ratio at school is 1:19.  If there are 29 teachers, how many students are there?

$$\eqalign{\dfrac{1 \text{ teacher}}{19 \text{ Students}}&=\dfrac{29 \text{ teachers}}{x \text{ students}}\qquad&&\text{Set up a proportion.  Make sure labels match}\\\dfrac{1}{19}\searrow\dfrac{29}{x} \text{ and }\dfrac{1}{19}\swarrow\dfrac{29}{x}&= 1\times x = 19 \times 29 \qquad && \text{Cross multiply for a new equation}\\x&=551\qquad&&\text{Solve the equation}}$$

There are 551 students in the school.

Sometimes, problems seem more complicated. The easiest way to start is to find your "base ratio" -- the one where you know both numbers. Then, just set it equal to another ratio, which will include your unknown value. 

Example:

A building has 3,500 square feet of surface that needs to be painted.  If 2 gallons of paint will cover 500 square feet, what is the least number of whole gallons that must be purchased to have enough paint to apply one coat to the surface?

$$\eqalign{\dfrac{2\text{gallons}}{500\text{ square feet}}& \qquad && \text{Find the fraction you can fill in the top and bottom for}\\\dfrac{2 \text{ gallons}}{500 \text{ square feet}}&=\dfrac{x \text{ gallons}}{3500 \text{ square feet}}\qquad&&\text{Set equal to fraction with variable. Match labels.}\\\dfrac{2}{500}\searrow\dfrac{x}{3500} \text{ and } \dfrac{2}{500}\swarrow\dfrac{x}{3500}&= 2\times 3500 = x \times 500 \qquad && \text{Cross multiply for a new equation}\\7000&=500x\qquad&&\text{Solve the equation by dividing each side by 500}\\\div 500 \quad & \quad \div 500 \\x&=7\qquad &&\text{You need 7 gallons of paint}}$$

Finally, although all of the problems that you need to set up a proportion are often included in the problem.  Sometimes they aren't. Sometimes you have to do some math just to set up your proportion (see Creating Ratios lesson for more).  Sometimes that math means finding a value that is not given (but is implied) in the problem.   Other times, it means converting to a different unit (feet to inches, seconds to minutes, etc.)

 Example:

Researchers worry about how much advertising students see when they are watching TV.  The average children's television show contains 21 minutes of program time and 9 minutes of commercial time. A new children's network argues that children should watch 4 hours of television a day.  If they do watch 4 hours of television a day, how many minutes of commercials will that be?

$$\eqalign{\dfrac{21\text{ minutes of program}}{9 \text{ minutes of commercial}}& \qquad && \text{Find the fraction you know the top and bottom for}\\\dfrac{21 \text{ minutes of program time}}{9\text{minutes of commercial time}}&=\dfrac{x \text{ minutes of commercial time}}{4 \text{ hours of television}}\qquad&&\text{Check: do the labels match?}}$$

There's a problem.  The labels don't match in a lot of ways!

First, "minutes of commercial" should match up, so let's flip the second fraction: $$\eqalign{\dfrac{21 \text{ minutes of program}}{9\text{minutes of commercial}}&=\dfrac{x \text{ minutes of commercial}}{4 \text{ hours of television}} \rightarrow \dfrac{21 \text{ minutes of program}}{9\text{minutes of commercial}}&=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}}}$$

Now, "minutes of commercial" is matched up on both fractions, but "program" is next to "television" and that won't work.  We know that we have 4 hours of total television time in our question, so let's leave that. Can we figure out the total time for our base ratio? There's 21 minutes of program time and 9 minutes of commercial time, so how many minutes of total television time are in our original ratio?

$21+9=30\text { minutes of total television}$

Let's rewrite our proportion: $$\eqalign{\dfrac{21 \text{ minutes of program}}{9\text{minutes of commercial}}&=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}} \rightarrow \dfrac{30 \text{ minutes of television}}{9\text{minutes of commercial}}&=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}}}$$

Now, we're close but there's still one issue.  Almost all of our labels are in minutes.  But we have 4 HOURS of television in our second ratio.  That won't work.  Let's convert that to minutes!

$4 \text{ hours} \times 60 \text { minutes} = 240 \text{ minutes}$

Plug that into the proportion and we're reading to cross multiply!

$$\eqalign{\dfrac{30 \text{ minutes of television}}{9\text{minutes of commercial}}=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}} \rightarrow \dfrac{30 \text{ minutes of television}}{9\text{minutes of commercial}}&=\dfrac{240 \text{ minutes of television}}{x \text{ minutes of commercial}}\\30 x&=2160\\x&=72}$$

In 4 hours of television there's 72 minutes of commercial time.

Overall, find your base ratio.  Then, match it up with a new ratio, that includes the number you need to find (as a variable).  Make sure labels match up (both the content and the metric).  Cross multiply and use algebra to solve.

Once you get used to using proportions, they're easy!  And you'll find so many uses for them!