By Matthew Canning, Become Better at Everything Founder
In just a few short weeks, my next book will be hitting shelves. Master the Language of the Universe takes you on a journey through practical math and shows you how to perform even complex calculations using only your head.
To celebrate the coming release, here’s another preview. This lesson is called, Roughing It (the Art of Quick, Imprecise Calculations).
“OMG HELP STUPID MATH QUESTION?…YOU WILL BE A HERO IF YOU ANSWER THIS WITH AND EXPLANATION SO PLEEEAAASE HELP ME”
—Title of a post on Yahoo! Answers (Canada)
Most of the time, you don’t need to know that 40 goes into 1,236 exactly 30.9 times; you just need to know that if you have 40 classrooms, 1,236 students, and a maximum capacity of 20 students per room, you’ve got a problem. This is when quick, imprecise calculations come in handy. Let’s learn some tricks for squeezing as much precision as you can out of loose calculations.
Ignore and Compensate
To use this trick, ignore the numbers’ smaller digits (usually the ones places or both the tens and ones places), and then make up for this by adjusting the answer based on the operation and direction you rounded.
Whenever you round down for addition or subtraction problems, you’ll know to round your final answer up. If you round up in order to solve the problem, you’ll know to round your final answer down. Take 4,809 + 664 for example. You can make two determinations with certainty:
The answer is “roughly” 5400: You know that 48 + 6 (from 4,809 + 664) is 54, so use this as the base of your rough answer.
The actual answer is going to be higher than the rough answer: Since you rounded 4,809 to 4,800 and 664 to 600 (both down), you must round final answer up, so you can say the final answer is “a bit more than 5,400.” You’d be right: the precise answer is 5,473.
Now that you’ve done an addition example, let’s look at a simple subtraction example. 554 − 441 equals 113. If you subtracted 44 from 55, you’d get 11 as the first two digits of your initial rough answer. Accounting for the fact that you rounded down, you should adjust the final answer up, to “a little more than 110,” which is accurate.
This method also applies to multiplication problems; here, too, the final answer is adjusted in the direction opposite the rounding. How would you handle 61 × 91? If the circumstances allow, you can rough it rather than perform the actual calculation; 6 × 9 = 54, so the answer could be expressed as “a bit more than 5,400.” It is important to note that when dealing with multiplication or division, the numbers lost in rounding can be significant. If the situation calls for a greater degree of precision, you must account for this.
Let’s look at 61 × 91 again, but this time more carefully. Given that you rounded down on both counts (to 60 × 90), you can assume that the actual answer is a fair amount higher than 5,400. In fact, using 91 instead of 90 would alone bring the answer up to almost 5,500 without even taking into account the 1 we ignored when rounding 61 down to 60. If you took the time to figure this out, you could instead say that the answer is “more than 5,500” (the real answer is 5,551). This illustrates an important point: The work you need to put in really depends on the precision required by the situation.
Division is more complicated, because minute changes in either number involved can cause major fluctuations in the answer. Let’s go back to the classroom/students problem presented in the introduction to this lesson: 1,236 ÷ 40.
Since 4 goes into 12 (from 40 into 1,236) exactly 3 times, the rough answer will begin with a 3. The smaller number (40) is two digits shorter than the larger number (1,236), so the answer will be either be two or three digits long. To figure out which, simply ask if the dividend (12) is more or less than ten times the divisor (4). Since it’s less, the answer will in most cases be two digits long. From this information, you can conclude that the answer is “roughly” 30. The actual answer is 30.9, so this seems like a reasonable response in many cases.
Another division example: how should you handle 850 ÷ 123? The actual answer is 6.91, but let’s look at the answers you get when addressing this problem with different precision tolerances:
- 8 ÷ 1 = 8
- 85 ÷ 12 = 7
With the actual answer being 6.91, would “about 8” have been acceptable? How about “around 7?” Again, it depends heavily upon context of the problem. This example illustrates another important point: When it comes to multiplication and division, this method is best used for smaller numbers.
Using the Ignore and Compensate method, do the following computations in your head. The correct answers are listed below. Don’t cheat yourself; figure out and write down the answers, then check them. Once finished, if there were any mistakes, try to figure out where you went wrong. Try to get as close to the correct answer as possible without spending too much time or brainpower.
- 65 × 44
- 82 × 15
- 190 × 16
- 440 ÷ 23
- 891 ÷ 66
- 333 + 741
- 186 + 230
- 8,392 − 901
- 764 − 88
One Up/One Down
Let’s explore a quick and easy method for addition. How would you quickly handle $6,431 + $1,777? You could use the method taught above, but since one number rests near the “bottom” of a hundreds-place (that is, 6,431 is closer to 6,000 than it is to 6,999) and the other number rests near the “top” (1,777 is closer to 1,999 than it is to 1,000), a rough estimate could be made by rounding one number down and the other up, beginning with the same digit or “place” in each (in this case, the hundreds place). Either direction works; you could round this to $6,400 + $1,800 or $6,500 + $1,700, both of which leave you with an answer of $8,200. An answer of “around $8,200” may be close enough to the actual answer, $8,208, depending on the context of the problem.
This method is obviously more precise if the two numbers are comparably distant from their rounded versions; for example, in the problem 331 + 269, both numbers are exactly 31 away from their nearest rounded counterparts. In some cases, however, one of the numbers may be close to the bottom of the hundreds place (e.g., 402) and the other closer to the middle (e.g., 458); no matter—this method is still great for extracting superficial precision from a quick calculation.
This won’t work if both numbers reside near the top (1,392 + 296) or bottom (1,329 + 226) of a number span (in this case, the hundreds place). For this, round both numbers either up or down, and then compensate as you learned before.
Using the One Up/One Down method whenever possible, do the following computations in your head. The correct answers are listed below. Don’t cheat yourself; figure out the answers, write them down, and then check them. Once finished, if your rough answer is unreasonably distant from the actual answer, try to figure out where you went wrong. Try to get as close to the correct answer as possible without spending too much time or brainpower.
- 23 + 438
- 75 + 213
- 754 + 7,892
- 8,243 + 907
- 288 (candidate for One Up/One Down)
Multiplication Between Two Targets
When a very rough answer will suffice, this multiplication method gets you an answer quickly.
Let’s learn by walking through an example: 65 × 271. You could round one of the numbers, so let’s say you choose to round 271. This number lies between 200 and 300, so you have two potential ways to round. Your answer will lie somewhere in between 65 × 200 and 65 × 300. Ignoring the zeroes for a moment, multiplying 65 × 2 gives you 130 and 65 × 3 gives you 195. Since 271 is closer to 300 than 200, place your rough guess somewhere slightly closer to 195 than 130. Let’s say 170. Then add the two zeros back to the end of your answer. This turns the answer into “around 17,000.”
How close did we get? The real answer is 17,615.
Let’s try another: 124 × 28. Let’s round 124 up to 125 to make it easier to work with. As 28 is closer to 30 than 20, multiply 125 × 2 (250) and 125 × 3 (375), and choose an answer that is much closer to the latter (say, 340). Adding the stripped 0 back in, you end up with “around 3,400.” The real answer is 3,472.
It’s important to use two targets because it gives you a sense of the scale of adjustment needed. Had you simply multiplied 125 × 30 and knew that you had to “drop down a bit,” you would have no sense of how much you needed to drop. But using two targets, you quickly realize that 28 is 2/10 of the way from 30 down to 20. With this knowledge, you can “come down roughly 2/10 of the way between 375 and 250,” which, though imprecise, is something you can grasp with practice.
We rounded 124 to 125 because it was easier to work with (due to currency, you should be used to dealing with numbers that end in 25). This illustrates an important point; once you’re comfortable with numbers, finding rough answers is more art than science. Get creative; find your way to a rough answer in any way that makes sense to you. You can simplify numbers as much as you’d like before performing calculations, as long as you adjust afterward.
If you were multiplying 81 × 5,300, you could choose to multiply 8 × 5 and 8 × 6 (answers: 40 and 48, respectively), and add the zeroes back in (400,000 and 480,000). Now you know the answer is “somewhere in the 400,000’s but less than 480,000.” Want more precision? Simply adjust for the fact that the 5,300 is a bit closer to 5,000 than it is 6,000, so the answer should be closer to 400,000 than 480,000. What do you think, maybe around 430,000? With the actual answer being 429,300, you’d be in good shape.
Using the Multiplication Between Two Targets method, do the following computations in your head. You’ll find the correct answers below. Don’t cheat yourself; figure out and write down the answers, then check them. Once finished, if your rough answer is unreasonably distant from the actual answer, try to figure out where you went wrong. Try to get as close to the correct answer as possible without spending too much time or brainpower.
- 32 × 12
- 83 × 28
- 28 × 29
- 17 × 55
- 65 × 891
When making rough, imprecise calculations, several strategies exist that can easily improve the precision of the answer, including the Ignore and Compensate, One Up/One Down, and Multiplication Between Two Targets methods.
If you enjoyed this, keep an eye out for the book.