You know already that you may not find your mistakes, if you repeat a calculation in the same sequence.
To check your results, rearrange the steps in your calculation.
Addition
| start at the bottom | start at the top | |
| 1793.4 | +344.5 | |
| +667.4=+1448.9 | +667.4=+1011.9 | |
| +334.3=+781.5 | +334.3=+1346.2 | |
| +447.2 | +447.2=+1793.4 |
Subtraction
| subtract from the top | add to the result | |
| +997 | +997 | |
| -532 | ||
| +532 | ||
| +465 | +465 |
Multiplication
| interchange | factors | |
| 314·236 | 236·314 | |
| 628 | 708 | |
| ·942 | ·236 | |
| ··1884 | ··944 | |
| 74104 | 74104 |
Division
| 48 : 6 = 8 | multiplication test | 8 · 6 = 48 |
| 59 : 7 = 8 remainder 3 | 8 · 7 + 3 = 59 |
You check calculations by changes in the order of the steps.
Task: Check your evaluation of the expression 2a + 4b - a - 3b for a = 1/2, b =1/3!
| 2a + 4b - a - 3b, a = 1/2, b = 1/3 | 2a + 4b - a - 3b = 2·1/2 + 4·1/3 - 1/2 - 3/3 = 5/6 |
| Check: | 2a + 4b - a - 3b = a + b = 1/2 + 1/3 = 5/6 |
If the results of the first calculation and the test do not agree, you must recalculate.
Your friend says that 2.23·3.24 = 0.72252 and you say it is 72.252. You then compute 3·4 = 12, which is too large, and 1·2 = 2, which is too small. The result must lie in between. Hence both the results are wrong. The correct result lies between 2 and 12; it is 7.2252.
Such estimates let you to find errors.
Mechanization of method of computation
In the past, people used tables for products, percentages, taxes, etc. Now you have calculators.