# Ratios

As always, remember to read carefully, mark the bottom line, assess your options (by asking "What could I do?" and "What should I do?"), attack the problem, and loop back to ensure that you actually answered the question.

The odometer of a new automobile functions improperly and registers only 2 miles for every 3 miles driven. If the odometer indicates 48 miles, how many miles has the automobile actually been driven?

Now that you have read the problem--I hope you read carefully!--note the bottom line at the top of your scratch work. In this problem, you are trying to figure out how many miles the car has traveled, so the best variable to use is m. At the top of your page, write

m=?

Next, assess your options. What has the problem told you? Are there any hints as to what strategy you could use? You could simply count out to 48, but that would be time-consuming, and on timed tests the long way is the wrong way. In fact the phrase "for every" is a clue; it tells you that the problem has to do with relationships between numbers, and that often means that you will need to use ratios.

When dealing with ratios and proportions, the first step is to assign labels to the various parts of your scratch work. Put the variable you need to find in the numerator and the variable you know the most about in the denominator. Equal to that, put the ratio provided in the problem. Even though this will sometimes give you an improper fraction, is is the fastest way to find the answer.

$\frac{actual}{odometer}=\frac{3}{2}$

Next, add another ratio to create a proportion. Make sure that the numbers you add match the labels on the left. This time, the variable should go in the numerator and the denominator should have a number relating to the label. The problem says that the odometer indicates 48 miles, so write

$\frac{actual}{odometer}=\frac{3}{2}=\frac{m}{48}$

Now you can solve for x. Note that since the variable is in the numerator, you do not need to cross-multiply. Instead, simply multiply both sides by 48.

$\frac{3\ast 48}{2}=\frac{48m}{48}$

Then simplify. Remember to find the fastest and easiest way to simplify; here, a good way to save yourself effort is to divide 48 by 2 instead of multiplying it by 3.

${3\ast 24}=m$

$72=m$

Loop back to make sure you found the correct value. The problem needed to know the total number of miles the car had actually traveled. Based on the ratio, you found exactly that. Now check the answer choices.

A) 144

B) 72

C) 64

D) 32

E) 24