Rounding
Rounding is a way to simplify our numbers, to help with estimating. With whole numbers, we might round to the nearest ten, or hundred. With decimals, we might round to the nearest unit, tenth or hundredth.
Rounding $13$13 to the nearest ten is the same as asking if $13$13 is closer to $10$10, or closer to $20$20. As we know that $15$15 is halfway between $10$10 and $20$20, and $13$13 is less than halfway, $13$13 must be closer to $10$10.
Rounding $175$175 to the nearest hundred is the same as asking if $175$175 is closer to $100$100 or $200$200. As we know that $150$150 is halfway between $100$100 and $200$200, and $175$175 is greater than $150$150, then $175$175 is closer to $200$200 than $100$100.
It's this notion of "halfway-ness" that leads us to the following rule for rounding.
When rounding, look at the digit in the column to the right of the column you are rounding to. If that digit is $5$5 or more (i.e. halfway or more than halfway to the next $10$10), then round up. If it is less than $5$5, (i.e. less than halfway to the previous $10$10), then round down.
Practice questions
Question 1
Round $54$54 to the nearest ten.
Question 2
Round $370$370 to the nearest hundred.
Question 3
Round $96501$96501 to the nearest thousand.
Rounding decimals
Exploration
When we round a number to the nearest thousandth, we first want to know about its decimal values. A number like $0.0537$0.0537 can be written in a place value table as shown below.
| Units | . | Tenths | Hundredths | Thousandths | Ten-thousandths |
|---|---|---|---|---|---|
| $0$0 | $.$. | $0$0 | $5$5 | $3$3 | $7$7 |
We can see that the thousandths is the third decimal place.
To round this number, we look at the ten thousandths, or the fourth decimal place. If the place value is less than $5$5 we round down, and if it's greater or equal to $5$5 we round up.
The number $0.0537$0.0537 lies between $0.053$0.053 and $0.054$0.054, and because the place value in the fourth decimal place is $7$7, we round up to $0.054$0.054.
We can round to the nearest ten-thousandth, or to four decimal places, using the same procedure.
Worked examples
Example 1
Round $45.43021$45.43021 to four decimal places.
Think: We want to look at the place value in the fifth decimal place.
Do: The place value in the fifth decimal place is $1$1. So we round the number $45.43021$45.43021 down to $45.4302$45.4302.
example 2
Round $45.09997$45.09997 to four decimal places.
Think: We want to look at the place value in the fifth decimal place.
Do: The place value in the fifth decimal place is $7$7. So we round the number $45.09997$45.09997 up.
Ordinarily, this means we increase the place value by $1$1, but whenever there is a $9$9, we change it to zero $0$0 and add the $1$1 to the place value to the left.
So the number $45.09997$45.09997 rounds up to $45.1000$45.1000.
Reflect: Just as we have done before, we can compare $45.09997$45.09997 with the numbers that have four decimal places closest to $45.0997$45.0997.
The decimal $45.09997$45.09997 lies between $45.1000$45.1000 and $45.0999$45.0999, and so when we want to round to four decimal places, we round up to $45.1000$45.1000. This is why we can't ordinarily increase the place value in the fourth decimal place by $1$1.
So just like the rounding with whole numbers, we can round decimals in the same way.
To round a decimal value to a certain number of decimal places, look at the next decimal place value to the right.
If it's less than $5$5 then round down and if it's greater than or equal to $5$5 then round up.
Practice questions
question 4
Round $95.19$95.19 to the nearest tenth.
question 5
Round $85.1446$85.1446 to the nearest hundredth.
question 6
Round $430.4875$430.4875 to the nearest thousandth.
question 7
Round $99.097906$99.097906 to four decimal places.
question 8
Round $22.678$22.678 up to the nearest whole number.
Rounding cash to the nearest 5c
Consider the problem of buying goods at the supermarket that total $\$56.79$$56.79 . We can either pay using some form of bank card, or we can pay with cash. If we pay with card we will be charged exactly $\$56.79$$56.79, but if we pay using cash it's not possible to pay exactly $79$79 cents as we no longer have $1$1 and $2$2 cent coins. Therefore the shop assistant will round the to the nearest $5$5 cents and charge us $\$56.80$$56.80.
If the cent value is $1$1 or $2$2 then you round down to the nearest ten cents. So $\$85.72$$85.72 rounds to $\$85.70$$85.70
If the cent value is $3$3 or $4$4 then you round up to the nearest five cents. So $\$12.34$$12.34 rounds to $\$12.35$$12.35
If the cent value is $6$6 or $7$7 then you round down to the nearest five cents. So $\$6.76$$6.76 rounds to $\$6.75$$6.75
If the cent value is $8$8 or $9$9 then you round up to the nearest ten cents. So $\$123.39$$123.39 rounds to $\$123.40$$123.40
Practice questions
Question 9
Luigi is paying with cash at the supermarket, and his shopping bill totals $\$337.57$$337.57.
How much does he pay?
Question 10
Irene is paying cash at the petrol station and the petrol metre reads $\$75.79$$75.79.
How much does she pay?
Question 11
Round $\$199.98$$199.98 to the nearest five cents.