In previous articles, I've given you techniques needed to figure out exact answers to math problems mentally. Sometimes, we don't need an exact answer and an estimation or "best guess" is satisfactory. Suppose you are getting quotes from different banks on a personal loan. All that is needed is a close approximation to the monthly payments. Another example where an estimation is satisfactory is settling a restaurant bill with some friends, where it's not important to calculate to the exact penny. This article will explain techniques to help you master mathematical estimation.
We'll first examine addition estimation. The trick is to round the
original numbers up or down. For example, 4,561 + 2,233 = 6,794. If we
round to the nearest hundred, rounding up at 50 or above and rounding
down below 50, we get 4,600 + 2,200 is approximately 6,800. If you
always round off to the nearest hundred, the estimate will always been
within 100 of the correct answer. This is within one percent of the
correct answer when the answer is 10,000 or more.
We can use this technique when shopping in a supermarket. Suppose you
want to approximate the total bill before the cashier rings up your
order. If you round all items to the nearest 50 cents, you'd be
surprised how accurate your estimation will be. Let's try it with these
prices: $1.69, $2.43, $0.79, $1.57, $0.40, $4.23, $1.75, $1.35, $2.65,
$0.89. The actual cost for these items is $17.75. When rounding off the
prices become $1.50, $2.50, $1.00, $1.50, $0.50, $4.00, $2.00, $1.50,
$2.50, $1.00. Adding these gives the estimation of $18.00, amazingly
close to the total cost.
Estimation involving subtraction is done the same way. The most
accurate estimation is when numbers are rounded to the nearest hundred.
For example, 9,251 - 3,771 = 5,480. The estimated answer is found by
taking 9,300 - 3,800 = 5,500.
For estimation involving multiplication problems, the process is
similar. We still want to round numbers. If we have a multiplication
problem with two, two-digit numbers, we can round each to the nearest
10. While that is quickest and easiest, it's not the most accurate. For
example, take 56 times 89. We round 56 to 60 and 89 to 90 to get 60
times 90 equals 5,400. The actual answer is 4,984.
A more accurate way to estimate the problem above is to round one
number up to the next 10 and the other down by the same amount the other
number was round up. For example, we have 56 times 89. If we round 89
up by one to 90, we round 56 down by one to 55. Now we have 90 times 55
equals 4,950. The estimation is only 34 off from the actual answer and
much easier to multiply 90 and 55 mentally than 89 and 56.
The same technique can be used when multiplying a three digit to a two
digit number and two three digit numbers. Note that while the
estimation technique will work, the multiplication is more difficult to
perform mentally. Master the mental multiplication techniques before
attempting problems above two digits.
There are other techniques for estimating answers to division problems
and square roots, which will be covered in a later article. I have
explained these techniques to my students over the past 14 years and
hope you find some use out of these as well.