Learning how to estimate accurately serves a number of practical purposes, from making better financial decisions to predicting the answers to the problems of day-to-day life; the former is certainly the most appealing, but the latter, once you’ve honed your abilities, is just plain fun.
Following is a guide on how to estimate in a variety of situations involving addition, multiplication and division, the most useful functions in daily life. Our two points of interest are saving time while maintaining accuracy. Estimation even has tons of professional applications. Serious approximators can learn estimating and cost control from this five-star course on building and construction.
When To Estimate
As I mentioned above, estimation has professional applications, but I hardly need to point out that it isn’t always appropriate; if you need an exact answer, estimation isn’t going to cut it. We want to use estimation when an exact answer is unnecessary and when we want to save time by quickly generating a figure that will serve our purposes.
Estimating With Addition
Whether you’re adding items at the grocery store double-checking a bill, addition can save you time and cash. Pick up even more practical tips with this blog post on 6 mental math tricks for all intents and purposes.
This first addition problem is fairly complex, but it does have “trickle down” properties; that is, once you know how to add a string of relatively large numbers, you will more easily be able to add smaller numbers, even those with decimals. But I will show you how to do all of this.
First . . .
Let’s generate a list of numbers: 333 + 284 + 95 + 767 + 425 = ???
What you want to do is round each number to the nearest hundreds digit. This might seem like it leaves a lot to chance, but let’s just see what we get: 300 + 300 + 100 + 800 + 400 = 1900
You don’t even have to write all the numbers out before you add them; with numbers this easy to add together, you can work your way along the original equation and add as you go.
Out actual answer, in this case, is 1904. That is exceptionally close and, believe it or not, I wrote down these numbers as they popped into my head. This example is probably too ideal; the vast majority of the time you will not get within 5 on such a large addition problem. Still, it shows you how close you can get with a very simple trick.
Studying for the SAT? Invest in this awesome five-star Beat-The-SAT math course for topic by topic questions and explanations guaranteed to raise your score.
Flaws In The Plan
1. Cozying Up To 50
The obvious flaw in this form of estimation is one you probably already noticed: many of the numbers I generated were far from middle ground; in other words, none of the tens digits were close to 50. If you had a list of numbers such as 345 + 540 + 649 + 239, you would risk a very inaccurate estimate. In this case, our estimate would be 1,600 while our actual answer is 1,773.
That’s not nearly as close as our first example. The reason is that all of these numbers round down by nearly as much as they possibly can; once you go over fifty, you start rounding up. Look what happens when we marginally change the two numbers that are closest to 50: 345 and 649. If we make these 355 and 651, then our estimate changes to 1,800 and our actual answer is 1,785. Now that’s more like it.
Once you have a little practice under your belt, you can easily avoid these mistakes. For example, if you notice that a series of large numbers is just under or just over 50, then you might want to round to the nearest 50 instead; this would make our first example 350 + 550 + 650 + 250, which is easy enough to add together to get 1,800, which is a darn good estimate.
2. Smaller Numbers
But what if the numbers aren’t so large? What if we’re dealing with numbers like 57 + 98 + 13 + 34? Or even $3.55 + $14.61 + $9.23 + $3. 01?
In the first case, we would find it most accurate to round to the nearest ones digit: 60 + 100 + 10 + 30. So our estimate is 200 while the real answer is 202. That’s more than enough to get us by when estimating a bill. Then again, if you’re practicing for the ACT, you might need something that can get you the exact answer; refer to this top-rated ACT Math course for a step-by-step guide to boosting your score.
In the second case – you might call this our grocery store example – you can actually treat the numbers like our initial examples that dealt with numbers in the hundreds. They have three decimal places, so we can be fairly confident that things won’t get too out of hand. But if we wanted to play it safe, we know there’s nothing wrong with rounding to the nearest ones digit, even if it does require a little more brain power.
Estimating With Multiplication And Division
We use similar techniques for both multiplication and division. Rounding is the best way to quickly generate an estimate. But as with addition, there are flaws in the plans that need to be addressed.
Let’s look at a relatively simple example: 3.4 * 8.7 * 2.1. At this point we should already know what to do: we round the decimal to the nearest number: 3 * 9 * 2 = 54. Our actual answer is 62.12.
If you don’t know your times tables, check out this tutorial from Math Is Fun. There’s only so much you can do without a strong grasp of your times tables. However, if you’re proficient with multiplication, computing the above equation would only take about five seconds.
If you really want to blow someone’s mind with math tricks, check out this slick post on clever number tricks and how to produce the illusion of mind reading.
Refining The Technique
Our margin of error was just a little too large for comfort. So how can we make our estimate more exact? While there is no easy way to generate a better number than 54, we can know whether or not our number is above or below the actual answer. When multiplying an odd set of numbers (in this case, three), if a majority of the numbers are rounded up, then the estimate will very likely be higher than the actual answer; if a majority is rounded down, as in our case, then the answer is likely to be lower.
Now we should be able to breeze through the division work. Let’s get right into an example:
33.9 / 4.1
The first thing we should notice (if we know our multiplication tables) is that these numbers are close to 4 and 32, of which 8 is a common factor. So right off the bat we know that our estimate should be close to 8; in fact, you could even use 8 as your estimate if you’re truly in a hurry. But if we round the numbers, like we did for addition and multiplication, we should get 34 / 4.
Now, we know that the answer is going to be slightly higher than 8 since 34 is greater than 32. If this were a math test and you had an extra ten seconds to try to get the closest possible answer, then this is what you would do:
We know 32 / 4 = 8. We need to find the remainder: 34 – 32 = 2. Since we know 4 goes into 34 at least 8 times, we then divide the remainder by 8: 2/8. This is equal to 0.25. Now all we have to do is add 8 and 0.25 to get an extremely accurate estimate: 8.25. The actual answer? 8.268.
As you can see, a little extra time and effort can make your estimate all but perfect. Take your estimating and quantitative problem solving skills to the next level with this GRE and GMATH math course: so easy a child could do it.