Sunday, April 13, 2014

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.

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