Approximating the Answer of an Operation

Approximating the answer to an operation is useful when solving a problem situation. Before performing a calculation, you can approximate it to get an idea of the magnitude of the answer.

So, depending on the context, you can quickly find out whether the answer you get makes sense.

For example, if you're looking for the cost of building a house, and most of the items to be considered in the price are in the thousands or tens of thousands, you'd expect to find a total cost in the tens of thousands or hundreds of thousands. If your approximation of the calculation you are about to make is much smaller or much larger, you've probably made a mistake.

Estimate the result of the operation |677 + 321.|

1. Round the addition terms.
   We can round these 2 terms to the nearest hundred.||\begin{align}677&\rightarrow700\\321&\rightarrow300\end{align}||

2. Perform the operation with the rounded terms.
   ||700+300=1000||

Answer: We estimate that the result of the addition |677+321| is about |1000.| The precise answer is actually |998.|

Estime le résultat de l’opération |789 - 657.|

1. Round the subtraction terms.
   We can round these 2 terms to the nearest hundred.||\begin{align}789&\rightarrow800\\657&\rightarrow700\end{align}||

2. Perform the operation with the rounded terms.
   ||800-700=100||

Answer: We estimate that the result of the subtraction |789 - 657| is about |100.| The precise answer is actually |132.|

In the previous example, we could have rounded the terms to the nearest ten rather than the nearest hundred, to get a more accurate approximation. In this case, the subtraction |790 - 660| would have given us |130,| which is much closer to the true value |(132).| However, this calculation is a little harder to do mentally.

It was also possible to change |657| to |650,| instead of rounding to the nearest hundred. This way, we are less far from the true value and have a number that's easier to calculate. |800 - 650| is |150.| This value is also more precise than |100.|

Estimate the result of the operation |48 \times 22.|

1. Round the multiplication terms.
   We can round these 2 factors to the nearest ten.||\begin{align}48&\rightarrow50\\22&\rightarrow20\end{align}||

2. Perform the operation with the rounded terms.
   ||50 \times 20=1000||

Answer: We estimate that the result of the multiplication |48 \times 22| is about |1000.| The precise answer is actually |1056.|

Estimate the result of the operation |511 \div 7.|

1. Round the division terms.
   We can round the dividend to the nearest hundred and modify the divisor.||\begin{align}511&\rightarrow500\\7&\rightarrow5\end{align}||

2. Perform the operation with the rounded terms.
   ||500 \div 5=100||

Answer: We estimate that the result of the division |511 \div 7| is about |100.| The precise answer is actually |73.|