Estimation of answers

No one ever disputes that estimating when dealing with potentially complex calculations is a good approach, but an unspoken assumption is that while you may save a bit of time by estimating, it isn’t absolutely crucial to do so. After all, how long does it take to do a little arithmetic? The problem is that, under pressure, hard arithmetic can cause us to freeze. To illustrate this  “quick, what’s 1.3 divided by 3.2?” This is usually greeted by blank stares or nervous laughter. But when  asked “okay, what’s 1 divided by 3?” they see the point: trying to solve 1.3/3.2 won’t just be time-consuming, but can easily lead to a careless mistake prompted by arithmetical paralysis.

Here is the full question:

The age of the Earth is approximately 1.3 * 10^17 seconds, and one year is approximately 3.2 * 10^7 seconds. Which of the following is closest to the age of the Earth in years?

  1. 5 * 10^9
  2. 1 * 10^9
  3. 9 * 10^10
  4. 5 * 10^11
  5. 1 * 10^11

Most test-takers quickly see that in order to convert from seconds to years, we have to perform the following calculation: 1.3 * 10^17 seconds * 1 year/3.2 * 10^ 7 seconds or (1.3 * 10^17)/(3.2 * 10^ 7.)

It’s here when many test-takers freeze. So let’s estimate. We’ll round 1.3 down to 1, and we’ll round 3.2 down to 3. Now we’re calculating or (1* 10^17)/(3 * 10^ 7.) We can rewrite this expression as (1/3) * (10^17)/(10^7.) This becomes .333  * 10^10. If we borrow a 10 from 10^10, we’ll get 3.33 * 10^9. We know that this number is a little smaller than the correct answer, because we rounded the numerator down from 1.3 to 1, and this was a larger change than the adjustment we made to the denominator. If 3.33 * 10^9 is a little smaller than the correct answer, the answer must be B.  (Similarly, if we were to estimate 13/3, we’d see that the number is a little bigger than 4.)

This strategy will work just as well on tough Data Sufficiency questions:

If it took Carlos ½ hour to cycle from his house to the library yesterday, was the distance that he cycled greater than 6 miles? (1 mile = 5280 feet.)

  1. The average speed at which Carlos cycled from his house to the library yesterday was greater than 16 feet per second.
  2. The average speed at which Carlos cycled from his house to the library yesterday was less than 18 feet per second.

The fact that we’re given the conversion from miles to feet is a dead-giveaway that we’ll need to do some unit conversions to solve this question. So we know that the time is ½ hour, or 30 minutes. We want to know if the distance is greater than 6 miles. We’ll call the rate ‘r.’ If we put this question into the form of Rate * Time  = Distance, we can rephrase the question as:

Is r * 30 minutes > 6 miles?

We can simplify further to get: Is r > 6 miles/30 minutes or Is r > 1 mile/5 minutes?

A quick glance at the statements reveals that, ultimately, I want to convert into feet per second. I know that 1 mile is 5280 feet and that 5 minutes is 5 *60, or 300 seconds.

Now Is r > 1 mile/5 minutes? becomes Is r > 5280 feet/ 300 seconds. Divide both by 10 to get Is r > 528 feet/30 seconds. Now, let’s estimate. 528 is pretty close to 510. I know that 510/30 is the same as 51/3, or 17. Of course, I rounded down by 18 from 528 to 510, and 18/30 is about .5, so I’ll call the original question:

Is r > 17.5 feet/second?

If we get to this rephrase, the statements become a lot easier to test. Statement 1 tells me that Carlos cycled at a speed greater than 16 feet/second. Well, that could mean he went 16.1 feet/second, which would give me a NO to the original question, or he could have gone 30 feet/second, so I can get a YES to the original question. Not Sufficient.

Statement 2 tells that his average speed was less than 18 feet/second. That could mean he went 17.9 feet/second, which would give me a YES. Or he could have gone 2 feet/second, which would give me a NO.

Together, we know he went faster than 16 feet/second and slower than 18 feet/second. So he could have gone 16.1 feet/second, which would give a NO, and he could have gone 17.9, which would give a YES, so even together, the statements are not sufficient, and the answer is E.

The takeaway: estimation isn’t simply a luxury on the CAT; on certain questions, it’s a necessity. If you find yourself grinding through a host of ungainly arithmetical calculations, stop, and remind yourself that there has to be a better, more time-efficient approach.