Approximation and Rounding a Number

Approximation means finding a value that is as close as possible to a given number.

Estimating consists of determining an approximate value for an unknown quantity.

If we count the number of visible olives in this jar, we count around |35.| We can then estimate that there are twice as many in the whole jar, so around |70| olives.

Rounding means giving a value that is close to a certain number.

A rounded number is less precise than the actual number, but easier to reuse later. It is rounded to a certain place value (units of thousands, tens, hundredths, etc.). To do this, first locate the digit at the place value being rounded. Then, analyze the digit directly to its right.

• If it's a 0, 1, 2, 3 or 4, the number at the place value to be rounded does not change.

• If it's a 5, 6, 7, 8 or 9, add 1 to the number in the position being rounded.

All digits to the right of the selected place value are replaced by |0.|

Round |27\ 841| to the nearest hundred.

Locate the number in the hundreds position.||27\ \boldsymbol{\color{#3b87cd}8}41||

The digit directly to its right is a |4.| Since this digit is less than |5,| we don't add a hundred. We now change all the digits to the right of the hundreds to |\boldsymbol{\color{#3a9a38}{0}}.|||\begin{align}27\ &\boldsymbol{\color{#3b87cd}{8}}\underline{4}1\\ &\:\!\Big\downarrow \\27\ &\boldsymbol{\color{#3b87cd}{8}} \boldsymbol{\color{#3a9a38}{00}}\end{align}||

Answer: The number |27\ 841| rounded to the nearest hundred is |27\ 800.|

Round |299| to the nearest ten.

Locate the digit in the tens position.||2\boldsymbol{\color{#3b87cd}9}9||

The digit directly to its right is a |9.| Since this digit is greater than or equal to |5,| we add a ten. Since we end up with |10| tens, we carry over to the hundreds position. Then we change the digit to the right of the tens with a |\boldsymbol{\color{#3a9a38}{0}}.|||\begin{align}\overset{\boldsymbol{\color{#fa7921}{\large1}}}{2}&\boldsymbol{\color{#3b87cd}{9}}\underline{\boldsymbol9}\\&\Big\downarrow\small{+1}\\3&\boldsymbol{\color{#3b87cd}{0}}\boldsymbol{\color{#3a9a38}{0}}\end{align}||

Answer: The number |299| rounded to the nearest ten is |300.|

Note: The number |299| rounded to the nearest hundred is also |300.|

When we round a number, we need to determine if it is closer to a lower limit or an upper limit.

• We find the lower limit by truncating the number at the desired place value.

• We then get the upper limit by adding |1| to the desired place value.

We then decide whether to round up or down, depending on the value of the number.

• If it is closer to the lower limit, we round down.

• If it is closer to the upper limit, we round up.

Round |17\ 683| to the nearest thousand.

Locate the number in the thousand units position.||1\boldsymbol{\color{#3b87cd}7}\ 683||

Next, we determine the 2 possible limits for rounding. For the 1st, we truncate |17\ 683| to the thousand unit. For the 2nd, we add one thousand to the lower limit.||1\boldsymbol{\color{#3b87cd}{7}}\ 683\begin{cases}\begin{gather}1\boldsymbol{\color{#3b87cd}{8}}\ 000\\\text{or}\\1\boldsymbol{\color{#3b87cd}{7}}\ 000\end{gather}\end{cases}||

The number directly to the right of the thousand unit is a |6.| Since this number is greater than or equal to |5,| we round up.||1\boldsymbol{\color{#3b87cd}{7}}\ \underline{\boldsymbol6}83\begin{cases}\begin{gather}1\boldsymbol{\color{#3b87cd}{8}}\ 000 \\[3pt]\color{#ec0000}{\cancel{\color{black}{17\ 000}}}\end{gather}\end{cases}||

Answer: The number |17\ 683| rounded to the nearest thousand is |18\ 000.|

Round |34.876| to the nearest tenth.

Locate the number at the tenths position.||34.\boldsymbol{\color{#3b87cd}8}76||

The digit directly to its right is a |7.| Since this digit is greater than or equal to |5,| we add one tenth. Because the digits to the right of the tenths are after the decimal point, we can simply remove them instead of replacing them with |0.|||\begin{align}34.&\boldsymbol{\color{#3b87cd}{8}}\underline{7}6\\ &\:\!\Big\downarrow \small{+1}\\34.&\boldsymbol{\color{#3b87cd}{9}} \end{align}||

Answer: The number |34.876| rounded to the nearest tenth is |34.9.|

Round |-457.9| to the nearest hundred.

Locate the number in the hundreds position.||-\boldsymbol{\color{#3b87cd}4}57.9||

The number directly to its right is a |5.| Since this number is greater than or equal to |5,| we add one hundred. Be careful! The "−" is ignored when adding the hundred. There are |4| hundreds, not |-4| hundreds. So, when we add one, there are |5.| We then replace the digit in the tens position and the digit in the units position with |0|s and remove the digit after the decimal point.||\begin{align}-&\boldsymbol{\color{#3b87cd}{4}}\underline{5}7.9\\ &\:\!\Big\downarrow\small{+1} \\-&\boldsymbol{\color{#3b87cd}{5}}\boldsymbol{\color{#3a9a38}{\,0\,0}} \end{align}||

Answer: The number |-457.9| rounded to the nearest hundred is |-500.|

As in earlier examples, sometimes the rounding unit is clearly indicated. However, this is not always the case. Sometimes you have to rely on the context of the problem.

• Some contexts almost always call for the same rounding unit. Therefore, it is not specified.
  For example, money is often rounded to the nearest cent (¢), or hundredth of a dollar |($).| Therefore, |\$58.988| would be rounded to |\$58.99.|
  
  Also, a number of objects, animals or people must be rounded off to a whole number, since we can't have, for example, |26.8| chairs in a room. Instead, there would be |26| or |27| chairs.

• Some contexts always require rounding up.
  For example, if we calculate that, for an event, we need |52.4| volunteers, we'd have to find |53| volunteers even if |52.4| is closer to |52| because otherwise, the work won't be completed on time.

• Some contexts always require rounding down.
  For example, this is what we do when we use the concept of the greatest integer.

• Depending on the context, the rounding expected may change.
  For example, to find the population of Canada, we round up to the nearest million, or about |40| million. However, the population of a small municipality like Senneterre, with around |2200| inhabitants, is more appropriately rounded to the nearest hundred.

Truncating (cutting) means giving a value close to a number by simply eliminating the digits after a given position, without changing the digit at the chosen position.

If we truncate the number |3.456723134| to |3| decimal places, we get the number |3.456.|
If we had chosen to round this same number to the nearest thousandth, we would instead get |3.457.|

If we truncate the number |765| at the hundreds position, we get |700.|
If we had instead decided to round it to the nearest hundred, we would get |800.|

If we truncate the number |183.1| at the ones position, we get |183.|
If we had instead decided to round it to the nearest unit, we would also get |183.|