Estimating a product well means choosing numbers that are easy to multiply and knowing which direction you have pushed the answer. An estimate you cannot place above or below the true value is much less useful than one you can.
Three ways to estimate
| Method | Calculation | Estimate | Direction |
|---|---|---|---|
| Round both up | 50 x 70 | 3,500 | too high |
| Round both down | 40 x 60 | 2,400 | too low |
| Round one each way | 50 x 68 is awkward, use 50 x 70 | 3,500 | usually high |
| Compatible numbers | 48 x 70 is close and easy to halve | 3,360 | close, slightly high |
Knowing the direction
The rule is simple and worth saying out loud. Round both factors up and the estimate is above the true product. Round both down and it is below. Round one up and one down and the direction depends on which change was larger, so the estimate is no longer a guaranteed bound.
Bounding a product without computing it
29 x 41 low bound: 20 x 40 = 800 high bound: 30 x 50 = 1,500 so the answer is between 800 and 1,500 the exact product is 1,189
When an estimate is the answer
Plenty of real questions never need the exact product. Whether 24 boxes of 18 tiles will cover a floor needing 400 tiles is settled by 24 x 18 being a little over 400, and the exact figure of 432 adds nothing to the decision.
Common mistakes
Rounding both factors the same way and calling it close
50 x 70 for 47 x 68 is nearly ten percent high, because both errors push the same direction and then multiply together.
Using the estimate to correct the exact answer
If they disagree, redo the exact calculation. Never nudge an exact answer toward an estimate.
Check yourself
Estimate 62 x 39 and say whether your estimate is high or low.
Rounding both up gives 70 x 40 = 2,800, which is high. Rounding both down gives 60 x 30 = 1,800, which is low. The exact product is 2,418.
Give a low bound and a high bound for 88 x 52.
Low 80 x 50 = 4,000, high 90 x 60 = 5,400. The exact product, 4,576, sits between them.
Which is the better estimate for 47 x 68, 3,500 or 3,360?
3,360, from the compatible numbers 48 and 70. It is 164 away from the exact 3,196, while 3,500 is 304 away.
Why can rounding one factor up and one down give an unpredictable direction?
Because the two changes push opposite ways and rarely cancel exactly. Whichever rounding moved further has the larger effect, and which one that is depends on both numbers.
See also
Sources
- Estimation strategies for multiplication as taught in United States upper elementary mathematics. Examples on this page are our own.