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When a Proportion is not a Proportion – GMAT Algebra

There’s an old adage that if all you have is a hammer, everything looks like a nail. The meaning is clear: If you only have one tool in your tool belt, you end up treating all problems the same way.

For many GMAT students, that tendency to try to use the same tool for all problems presents itself on certain GMAT algebra questions — specifically, ratio variation questions. Their default reaction when trying to “solve for x” on these types of questions is to set up a proportion and cross multiply to find the missing element.

But sometimes, a proportion is not a proportion. Sometimes, cross multiplying is not the tool you want to be pulling out of your GMAT tool belt.

The problem, however, is that a lot of students don’t have another tool to call on.

Fear not! I’m about to give you one. Let’s take a closer look at the difference between direct and inverse variation problems on the GMAT, and which mathematical tool to use for solving each.

Direct Variation

GMAT algebra direct variation problemsDirect variation questions are GMAT ratio problems where the two quantities move in the same direction. Think about a chocolate chip cookie recipe. Two of the ingredients in the recipe, flour and sugar, are directly proportional to each other and move in the same direction. If you double the recipe, you must double both the amount of flour and sugar. If you cut the recipe in half, the amount of flour and sugar you need is also cut in half. The quantities of the two ingredients vary proportionally.

Key to solving direct variation problems: Set up a proportion and cross multiply to solve for the missing variable. Here’s the generic formula (where Q and R are the two quantities) to use:

Q₁  =  Q₂
R₁       R₂

This is the tool that most GMAT students already have in their proverbial tool belt (but incorrectly try to use for all ratio problems, as we’ll see in a moment). But be careful. Remember that when setting up a proportion to solve these types of problems, make sure you’re comparing the same units, in the same order, on both sides of the equal sign.

Here’s an example:

If a furnace uses 40 gallons of oil in a week, how many gallons, to the nearest gallon, does it use in 10 days?

This is a direct variation problem, right? The amount of oil that the furnace requires over a certain amount of time is fixed. For the furnace to run for more time, it requires more oil; less time requires less oil.

To solve this problem, then, you set up a simple proportion and solve for the missing number of gallons. (Note: Be sure to convert weeks to days first so that you’re comparing the same units when solving the final equation).

40 gallons  =  40 gallons  =  X gallons
  1 week             7 days              10 days

Cross multiply and you get that 40 gallons * 10 days = X gallons * 7 days —> 400 gallon days = 7x days —> X = 400/7 57 gallons.

Shortcut: Sometimes, rather than actually cross multiplying and taking the time to do the math to solve for x, you can just see what the multiplier is to scale the ratio up or down. If you can “see” what you have to multiply the numerator (or denominator) on the left side of the equation by to get the corresponding numerator (or denominator) on the right side of the equation, that becomes your multiplier and you can use it to quickly solve for the missing variable.

For example:

If a car can drive 25 miles on two gallons of gasoline, how many gallons will be needed for a trip of 150 miles?

Because this is a direct variation problem, you can set up the following proportion:

25 miles  =  150 miles
2 gallons      X gallons

But what do you notice? What would you need to multiply the numerator on the left by to get the numerator on the right? Six, because 25 * 6 = 150. So the ratio on the left essentially needs to be scaled up by a factor of 6 (it’d be like if you had a cookie recipe and needed to multiply all of the ingredients by 6 to make enough for a large party).

Very quickly, you can just multiply the denominator by six (6) to solve for X and get the answer. Two gallons * 6 = 12 gallons, which is your answer.

Indirect Variation

Indirect variation questions are GMAT ratio problems where the two quantities change in opposite directions. For example, think about the relationship between the number of workers and the number of days required to complete a job. As one quantity goes up, the other goes down, right? I mean, if you have more workers, then you’ll need fewer days to complete the job. Conversely, if you have fewer workers, then you’ll need more time (days) to complete the job. The two quantities are inversely related to each other.

Yet, a lot of students see questions like this and instinctively set up a normal proportion and cross multiply to solve them. That is a mistake. You need another tool for your tool belt.

Key to solving indirect variation questions: Multiply the first quantity by the second and set the products equal to each other. In other words, the total relational quantities must be maintained. Here’s the generic formula you need to solve indirect variation questions (where Q and R are the two quantities):

Q₁ * R₁ = Q₂ * R₂

Consider this example:

If a case of cat food can feed 5 cats for 4 days, how long would the case of cat food feed 8 cats?

Using the above formula, you start by multiplying the first and second quantities. In other words, if the case feeds 5 cats for 4 days, then there’s enough food for 5 * 4 = 20 cat days. That’s the fixed total that must be maintained.

So if the case has enough food for “20 cat days,” then use the rest of the formula to find out how many days this amount will feed 8 cats:

20 cat days = 8 cats * X days

Divide by 8 to solve for X: 20 / 8 = 2.5 days. So, that same case that would feed 5 cats for 4 days would feed 8 cats for 2.5 days.

Does that make sense?

Be sure not to confuse direct variation problems (where you set up a traditional proportion to solve) with these indirect variation questions (where you need to use the Q1*R1 = Q2*R2 equation) on test day. Now that you have this additional tool in your GMAT tool belt, you’re better equipped to dominate the GMAT!

To prove it, try your hand at this final sample question and write your answer (or questions) in the “Comments” area below.

Application Example: A school has enough bread to feed 30 children for 4 days. If 10 more children are suddenly enrolled in the school, how many days will the bread last?