Here’s the whole series of QC tips:

Tip #1: Dealing with Variables

Tip #5: Estimation with a Twist

In my last post, we solved the following question:

A. The quantity in Column A is greater

B. The quantity in Column B is greater

C. The two quantities are equal

D. The relationship cannot be determined from the information given

Our approach was to first recognize that two of fractions are approximately 1/2 and two are approximately 1/3.

Now if we had used these approximations, we would have been left with:

This would have made us conclude (incorrectly) that the answer is C.

To solve this question using approximation, we applied a twist. We recognized that 213/428 is a little bit less than 1/2, which we denoted as 1/2^{–}. We also noticed that 3007/9101 is a little bit less than 1/3, which we denoted as 1/3^{–}.

And so on.

With these little twists, we were able to simplify the two columns as:

From here, it was clear that the correct answer is B

Okay, now let’s see if you can apply this approximation with a twist to solve the following question:

Aside: before you read my solutions, see if you can find additional ways to solve this question.

Okay, first we’ll solve the question using approximation with a twist, and then we’ll solve it using different approaches.

## Approximation with a twist:

First, let’s approximate as follows:

From here, we can drop 4 zeroes from each number to get:

At this point, I’ll apply a nice rule that says:

In other words, the product of two numbers is equal to the product of twice one number and half the other value.

So, in Column A, we’ll double 45^{+} and halve 64^{+} to get:

From here, when we compare the products in parts, we can see that Column A must be greater than Column B, so the answer is A.

## Alternative approach #1

As you might have guessed, the two original products are too large to work on a calculator. For example, (641,713)x(451,222)=289,555,023,286 and this number is too large for the GRE’s onscreen calculator. As such, the calculator would display an error message if you tried to perform this calculation.

There are, however, some ways to work around this constraint and still use the calculator.

For example, you could divide each number by 1000 to get:

** **

At this point, the products will still fit into the display of the onscreen calculator and you would clearly see that Column A is greater.

## Alternative approach #2

Another possible approach is to perform the same steps we performed earlier, but stop when we get to:

Originally, we applied that handy rule where we double one number and halve the other. However, we could also use the calculator at this point.

(64)x(45)=2880 and (90)x(32)=2880. This means that (64^{+})x(45^{+})=2880^{+} and (90^{–})x(32^{–})=2880^{–}.

So, we get:

Once again, we can see that the answer is A.

I just love this. Thank you so much

I am getting confused at the following point:

Column A:

641,713 x 451,222

Approximated to:

640,000 x 450,000

The approximations are slightly lesser than the original numbers.

Column B:

897,189 x 319,977

Approximated to:

900,000 x 320,000

The approximations are slightly greater than the original numbers.

Therefore, shouldn’t Column A have a negative sign applied and Column B a positive sign.

Please help me out here. What am I missing?

Hi Asmita!

In Column A, the estimates are a little lower than what the actual values are. Therefore, the

actual valuewill behigherthan the approximation. That’s why it gets a “+”. 🙂In Column B, the estimates are a little higher than what the actual values are. Therefore, the

actual valuewill belowerthan the approximation. That’s why this gets a “-“.Does that help clear things up? 🙂

Hi Brent

I correctly solved both problems before checking for the correct answers by rounding. Is there a weakness to rounding that is unforeseen (by me) that will cause me to get such QC answers incorrect? Thanks!

Hi Julia,

Rounding can be a powerful tool as long as you make sure the process of rounding doesn’t introduce error, like it would if small differences in rounding affect the final answer or if the answer choices are sufficiently close together that rounding could change your ultimate choice. Since you’re asking this question, though, I think you sound like you’ll be in good shape!

Why or how was a negative sign applied to column b?

This means that when this number was estimated, this column actually is a little bit ‘less than’ the written number (like, the saying ‘plus or minus’). That is why column A also has a +. That number is actually little more than the written number, which is an estimate. So, since the answers look to be the same (2880) when you estimate them, you can see that Column A is actually 2880 ‘plus’ and Column B is 2880 ‘minus’, so A is greater.

Brent!

Simple rules and your tricks are awesome! 🙂

That’s a terrific way of solving these type of problems. A * B = 2 A * 1/2B. Thanks very much

super cool!!

awesome! i love it when you teach us shortcut tricks..they are very helpful and fun too!

nice tip

Brent,

Fantastic system. I’m preparing myself for the GRE …….with you.

So you’re the one who speaks there , on the math part?

Love it

Tony

Thanks for the feedback, Tony.

Much appreciated.

Cheers,

Brent

Great idea – that also works!

Cheers,

Brent

Alternative approach:

Column (a) column (b)

641,713 x 451,222 897,189 x 319,977

div (a) with 897,189 and (b) with 641,713

=>

(1/2)+ (1/2)-

answer: A

Great idea – that also works!

Cheers,

Brent

Thank you for your invaluable help in math.

Thanks Erika.

Glad to help.

Cheers,

Brent

Hi sir, thank you for your tips

Thanks saranya!

Cheers,

Brent