3.07 Rounding decimals

When we get a decimal quantity from a measurement or from a calculation, rounding allows us to express that quantity to a desired level of accuracy. It is most useful as a way to communicate only the amount of information that we think is necessary.

 

Exploration

If you ask someone their age, they will usually reply with a whole number of years: "I am $14$14." But unless it happens to be their birthday, we know that they are probably a little bit older than exactly $14$14 years, perhaps $14.38276$14.38276 years. By rounding their age to a whole number, what they mean is something like "I have lived for at least $14$14 years", or maybe "I have ridden the Earth for $14$14 whole loops around the Sun". The number of whole years is what is most important.

Now imagine you want to purchase some ham from the butcher using your bank card. Say the butcher is selling ham for $\$13.95$$13.95 per kg, and you want $0.25$0.25 kg. Then the cost would be given by $13.95\times0.25=\$3.4875$13.95×0.25=$3.4875. But we cannot transfer amounts of money less than $1$1 cent, so the cost will need to be rounded to two decimal places. This means you would ultimately pay $\$3.49$$3.49 for the $0.25$0.25 kg of ham.

What can we notice when comparing a rounded number with the original number? The examples above show that rounding a number makes it less precise. But the benefit of this precision loss is that usually we are not interested in the exact value of a quantity, so a rounded number is a simpler number.

 

Rounding to a place value

Let's look at what it means to round a decimal to the nearest whole number. If we choose any decimal number, it will either be exactly a whole number, or it will be somewhere in between two whole numbers. The rounded value is the whole number it is closest to. Use the applet below to see how this works.

Notice that the rounded whole number depends only on the value of the digit in the tenths place. If the digit in the tenths place is $0$0, $1$1, $2$2, $3$3, or $4$4 then the number rounds down, and if the digit in the tenths place is $5$5, $6$6, $7$7, $8$8, or $9$9 then the number rounds up. 

We use this same approach when rounding to any place value. For example, if we want to round a decimal to the nearest tenth, we would look at the digits in the tenths and hundredths places. Similarly, if we want to round a decimal to the nearest hundredth, then we look at the digits in the hundredths and thousandths places.

Worked examples

Example 1

Round $12.748$12.748 to the nearest tenth.

Think: This number is between $12.70$12.70 and $12.80$12.80, so we need to look at the digits in the tenths and hundredths places to round to the nearest tenth.

Do: Here is the number written in a place value table.

Looking at the tenths and hundredths places, we can see that we have $74$74 hundredths, which is closer to $70$70 hundredths than $80$80 hundredths, so we round down.

To simplify this process, we can see that the digit in the hundredths place is a $4$4, so $12.748$12.748 rounds down to $12.7$12.7.

Reflect: In this example we only needed to consider the values of the digits in the tenths and hundredths places. None of the other digits were changed by the rounding.

Example 2

Round $5.3175$5.3175 to the nearest hundredth.

Think: This number is between $5.31$5.31 and $5.32$5.32, so we need to look at the digits in the hundredths and thousandths places to round to the nearest hundredth.

Do: Here is the number written in a place value table.

Combining the hundredths and thousandths, we have $17$17 thousandths. This is closer to 20 thousandths than ten thousandths, so in this case we round up.

Reflect: Looking just at the digit in the thousandths column, we can see the digit is a $7$7, so $5.3175$5.3175 rounds up to $5.32$5.32.

Example 3

Round $2.95$2.95 to the nearest tenth.

Think: This number is between $2.9$2.9 and $3.0$3.0, so we need to look at the digit in the hundredths place to round to the nearest tenth.

Do: Here is the number written in a place value table.

The digit in the hundredths place is a $5$5, so $2.95$2.95 rounds up to $3.0$3.0.

Reflect: Unlike in previous examples, rounding $2.95$2.95 to the nearest tenth meant we also had to change the value of the digit in the ones place. This happens whenever we round a $9$9 up to a $10$10. We can think of this as rounding $29.5$29.5 tenths up to $30$30 tenths. 

 

Rounding to a number of decimal places

Instead of specifying that a number be rounded to the nearest whole number, or the nearest tenth or hundredth, we can simply state how many digits we want to keep in the decimal.

For example, rounding $5.3175$5.3175 to the nearest hundredth gave $5.32$5.32, and $5.32$5.32 has only two digits after the decimal point, so this rounding is to two decimal places. We can write this like so:

$5.3175=5.32$5.3175=5.32 (to 2 d.p.)

where "d.p." stands for "decimal place(s)". Let's look at some other examples.

Worked examples

Example 4

Round $6.287$6.287 to one decimal place.

Think: This number is between $6.2$6.2 and $6.3$6.3, so we will look at the digit in the second decimal place to round to one decimal place.

Do: The digit in the second decimal place is $8$8, so $6.287$6.287 rounds up to $6.3$6.3.

Example 5

Round $0.3901$0.3901 to three decimal places.

Think: This number is between $0.390$0.390 and $0.391$0.391, so we will look at the digit in the fourth decimal place to round to three decimal places.

Do: The digit in the fourth decimal place is $1$1, so $0.3901$0.3901 rounds down to $0.390$0.390.

Reflect: It is important to keep the final $0$0 in the rounded number, since this keeps track of the required level of precision. The number $0.39$0.39 could have been rounded from any number between $0.385$0.385 and $0.395$0.395, but the number $0.390$0.390 has been rounded from a much smaller range of between $0.3895$0.3895 and $0.3905$0.3905.

 

Practice questions

Question 1

Question 2

Question 3