How to round numbers in order to estimate the result of a math problem? Are there exact rules?
I have gotten several questions from my customers over the years about estimating and how to round numbers in order to get (supposedly) the "correct" estimate.
One person wrote:
My daughter is working through the Light Blue Series book 5A Ch 1. ... It seems a little arbitrary as to how she should be estimating. ... I realize that different people will estimate differently, but when looking through the answers, it seems inconsistent as to how those estimations are made. Do you have a system you recommend? Or is there a reasoning that we should be aware of (estimate to the one thousands if it's in the thousands, estimate to the tens if it's less than 100, etc???)?
Another person wrote:
In 6A, Rounding and Estimating, the dollar amounts in problem 5 and 6 led me to a question. In question 5 the dollar amount is rounded to the nearest TENTH, but in question 6 it is rounded to the nearest ONE for the apples and TENTH for the cheese. This is our first book we have used and I was wondering if there was a rule we should be following for this procedure? Thank you so much! These books are awesome.
As a rule of thumb, your estimate is more accurate, the less you have to round your numbers (this is just common sense). So, you cannot look at rounding rules in a rigid manner, when you're estimating the result of a calculation. In other words, you cannot say that you should always round to the nearest ten, or to the largest place value, or to use front-end estimation, or use any other rule.
Instead,
- Consider your mental math skills. Round as little as possible, based on what YOU can calculate in your head.
- And, consider also how the rounding up or down will affect the final result. Sometimes it is better to round one number up and the other down, even though the technical rounding rules would dictate otherwise.
Let's look at some examples.
Elisa bought 7 lb of apples for $1.19 per pound and three blocks of cheese for $11.45 a piece. Estimate the total cost.
Whether you round to the nearest dollar or nearest ten cents in this problem is up to you. Check what kind of numbers you get — are they easy to calculate with mentally?
Personally, I would probably round $1.19 to the nearest dollar, to $1, and $11.45 to $11.50, or perhaps to $12. If using $11.50, the estimate would be 7 × $1 + 3 × $11.50 = $7 + $34.50 = $41.50. Or, if using $12, the estimate is $7 + $36 = $43.
When it has to do with shopping and estimating the total cost, you probably don't want to underestimate, but rather make sure you don't round the majority of the prices down.
The amount $1.19 could have also been rounded to $1.20, if you can do 7 × $1.20 in your head.
(For reference, the exact amount is $42.68. We can see that the estimate that used $12 was the closest — because rounding $1.19 down to $1 creates some error, and rounding the other price up all the way to $12 compensated for that nicely.)
Estimate: 4 × 22,399
Here, you could round 22,399 to 20,000, to 22,000, or to 22,400. Which you use will depend on your mental multiplication skills and how accurate an estimate you wish to get.
Most people would probably go with 22,000 and get an estimate of 88,000.The rent is $128.95 per month. Estimate the total rent for 6 months.
Again, you could round $128.95 to $100, to $130, or to $129. I personally would use $130 to get 6 × 130 = $600 + $180 = $780.
This creates a very small error of estimation, since $128.95 is very close to $130 (the error or the difference is only 6 × $1.05 = $6.30).
Estimate the answer to 146 cm × 2 ½.
One useful tip is, if you're multiplying, and you round one factor DOWN, round the other factor UP (if feasible).
That is because if you round both factors UP, you will overestimate, and if you round both down, you will underestimate — possibly by quite a bit, since this is multiplication and not addition.
So when estimating the answer to 146 cm × 2 ½, if you round 2 ½ to 3 (UP), then it makes sense to round 146 cm DOWN to 140 cm. We get 140 cm × 3 = 420 cm.
Alternatively, you could estimate this as 150 cm × 2 = 300 cm.
Neither is very good, considering that the exact answer is 365 cm. In this case, a very good estimate would be provided by only rounding 146 cm to 150 cm, and not rounding 2 ½ at all. Then we get 150 cm × 2 ½ = 300 cm + 75 cm = 375 cm.
Another way to estimate this would be to take the AVERAGE of 2 ½ × 100 cm and 2 ½ × 200 cm. That would give us the average of 250 cm and 500 cm, which also is 375 cm.
How many 0.024 kg papers can you mail with a 0.400 kg weight limit?
Converting the amounts to grams, we get 24 g and 400 g. The idea is to find out how many times 24 goes into 400, which we CAN find out by estimating.
But which gives you less of an error, rounding 24 to 25, or rounding 24 to 20?
Rounding 24 to 25 is only a difference of one... whereas rounding 24 all the way to 20 is a difference of 4. Also, using 25 still allows us to use mental math as it is easy to figure out how many 25's there are in 400.
Since there are four 25s in 100, there are sixteen 25s in 400, so our estimate says you can mail 16 such papers. The exact calculation also gives an answer of 16 (since 400 ÷ 24 = 16.666...).
See also
Estimation tips and tricks from MathsIsFun.com
By Maria Miller
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