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GRE Arithmetic: Decimals

GRE Arithmetic DecimalsWe broadly talk about “arithmetic” as being one of the major mathematical concepts tested on the GRE, and within the world of arithmetic, decimals are a major player. Decimals can drive students crazy, but the reality is that decimals are just a way of quantifying and expressing numbers, often non-integers. Thought of another way, they’re the other side of the coin from fractions and percents.

Take seventy-five percent, for example. As a percent, 75% represents how much out of 100 something is (in this case, 75 out of 100). Relationally, we can express the same 75% as a fraction, relating the part to the whole — either as 75/100 or reduced to 3/4. And finally, we can express that same relationship as a decimal. When you actually divide 3 by 4, you get 0.75. Thus the decimal equivalent of 75% is 0.75.

Converting numbers to decimal form is useful on the GRE, especially when it comes to comparing the relative size of numbers on Quantitative Comparison questions, as we’ll see in a moment. But another way the GRE likes to test your understanding of decimals is through the terminology of the different place holders in a decimal. Let’s take a closer look.

Decimal Places Terminology

It may seem pretty basic, but it’s worthwhile to go back to the fundamentals of math and remind ourselves of the terms used to describe each of the place holders in a decimal. Believe it or not, you may encounter this on the GRE. Imagine that a question pops up on your screen on test day and it asks, “What is the units digit of…?” You may know how to do the math itself perfectly well, but if you don’t know what the units (or tens, or hundredths, etc.) digit is, you’ll have a tough time getting the question right.

So let’s review.

Consider the following number: 72.641

Very simply, the units digit = 2

The tens digit = 7

And the tenths digit = 6

Those three terms are the most commonly-tested decimal place holders on the GRE. One decimal place to the left of the decimal point is the ones (or units) place. Two decimal places to the left of the decimal point is the tens place. And one decimal place to the right of the decimal place is the tenths place.

In the interest of thoroughness, here’s a table showing additional decimal terminology. Keep your eye on the 4 to see where the decimal places fall.
 

Descriptor Example
thousands 4,000.0
hundreds 400.0
tens 40.0
ones / units 4.0
tenths 0.4
hundredths 0.04
thousandths 0.004

 
So what else do we need to know about decimals and how does this terminology help us on the GRE? Let’s take a look at an example.

Application Example

Consider the following sample GRE quantitative comparison question:
 

Quantity A Quantity B
The tenths digit in the
decimal representation of
a number less than 1/3
The tenths digit in the
decimal representation of
a number greater than 1/4

 

So what do you think? I’ll keep you in suspense for a moment longer. Let’s talk through this together.

A good place to start is to imagine that quantity A instead asks for the tenths digit of a decimal that equals 1/3. Remember that our goal when answering QC’s is to make an initial determination about the quantities so that we can begin eliminating answer choices, and initially thinking about Quantity A as 0.3333333 (repeating indefinitely) helps us do that. Likewise with quantity B, assume that it asks for the tenths digit of the decimal representation of 1/4, which is 0.25 in decimal form.

So if you were actually just comparing the tenths digits of 1/3 and 1/4, this question would be simple, right? Or would it? Well if you didn’t know what the tenths digit is, you’d be in trouble! But thanks to this article, now you do. So it is in fact simple to compare 3 (the tenths digit of the decimal representation of 1/3) to 2 (the tenths digit of the decimal representation of 1/4) and see that in our hypothetical revision of the question, quantity A is greater since 3 > 2.

(Note: You may be saying to yourself, “But wait, Brett, quantity A says “less than” 1/3, not “equal to”. And quantity B says “greater than” 1/4. So why are you assuming they’re equal?!” Okay, okay, call off the dogs. I hear you. If you’re uncomfortable with assuming they’re equal as a starting place for this question, then write out a decimal that’s VERY close to equal but that satisfies the “less than” and “equal to” conditions. For a number less than 1/3, for example, you might go with 0.333333333332. And for a number greater than 1/4, you might go with 0.250000000001. The point is that the tenths digits are still 3 and 2, respectively, so it still gives us that initial determination about the relationship that’s so important when solving QC questions on the GRE).

So according to the ideal methodology for solving GRE quantitative comparison questions, you can now eliminate answer choices B and C straight away. But the question now becomes, will quantity A be greater all the time, no matter what?

The answer is no. Quantity A says that the decimal in question simply has to be less than 1/3, so it’s entirely possible, for example, that it could be something like 0.125. In that case, the tenths digit (1) would be less than that of quantity B (assuming we’re still going with 0.25000000001), giving us a conflicting outcome that now enables us to eliminate answer choice A as well. The correct answer is D.

So there you have it — decimal terminology as it’s tested on the GRE. Certainly there’s a lot more to learn about decimals, fractions, and percents on the GRE so check out our a-la-carte GRE Algebra course to truly dominate these topics. But in the meantime, please post your comments or questions below and I’ll look forward to continuing to help you master this important topic!