1.11 Rounding

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.

 

Practice questions

Question 1

Question 2

Question 3

Rounding decimals

 

Worked examples

Example 1

Round  $0.0537$0.0537 to the nearest thousandth.

Think: $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. The question could have also been written as "round to three decimal places". Our answer will have exactly three numbers after the decimal point. The answer will either be $0.053$0.053 or $0.054$0.054. To decide if we round up to $0.054$0.054 we must consider the value in the ten-thousandths or next 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.

Do: The value in the fourth decimal place is $7$7, therefore 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.

Example 2

Round $45.43021$45.43021 to four decimal places.

Think:  Our answer will have exactly four numbers after the decimal place. It will either be $45.4302$45.4302 or $45.4303$45.4303. To decide if we round up or down we must 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 3

Round $45.09997$45.09997 to two decimal places (nearest hundredth).

Think:  Our answer will have exactly two numbers after the decimal point. It will either be $45.09$45.09 or round up to the next number with two decimal places which is $45.10$45.10. To decide if we round up or down we must look at the place value in the third decimal place.

The place value in the third decimal place is $9$9. So we round 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 (think of rounding the $9$9 up to $10$10)

Do: So the number $45.09997$45.09997 rounded to two decimal places is  $45.10$45.10.

Reflect: The zero is important! $45.10$45.10 is not the same as  $45.1$45.1. If the question says two decimal places then there must be exactly two numbers after the decimal point. 

 

 

Practice questions

question 4

question 5

question 6

question 7

question 8

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. 

 

 

Practice questions

Question 9

Question 10

Question 11