Estimate √9.7+3.88×6.21

9514 1404 393

  a little more than 27

Step-by-step explanation:

Estimates are generally made by approximating the numbers using 1 or 2 significant digits. The idea is to do as much mental arithmetic as possible, not involving a calculator. Here, the estimate can be based on rounding the values like this:

  √9.7 +3.88 × 6.21 ≈ √9 +4 × 6 = 3 +24 = 27 . . . estimated expression value

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Another aspect of estimating is determining the approximate error involved.

We know that √9.7 will be more than √9 = 3, and less than √10 ≈ 3.16*. Its real value may be about 3.1.

3.88 is less than 4 by 0.12, or 3%. 6.21 is greater than 6 by .21, or 3.5%. So, the estimated product of 24 will be on the order of -3% +3.5% = 0.5% too low. Of course 1% of 24 is 0.24, so 1/2% will be about 0.12. Now, we are able to estimate that our estimate is about 0.2 units too low. (The actual value of the expression is an irrational number that rounds to about 27.2093.)

* The value of 3.16 for √10 is not so hard to remember, because it is just slightly more than π ≈ 3.14.

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Additional comment

Small percentages can usually be added and subtracted without introducing too much additional error. (The smaller the percentages, the less additional error is introduced.) Here, the error in our estimated product can be more accurately determined by computing ...

  (1 -3%)(1 +3.5%) = 1 +0.5% -0.105% = 1 +0.395% . . . . . . (not 1 +0.5%)

That is, our product's actual value is 24.0948, not 24.12.