1. Number

Middle Years

Lesson

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.

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.

Round $54$54 to the nearest ten.

Round $370$370 to the nearest hundred.

Round $96501$96501 to the nearest thousand.

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.

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.

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.

Remember!

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.

Round $53.36$53.36 to the nearest tenth (one decimal place).

Round $22.2539$22.2539 to the nearest hundredth (two decimal places).

Round $558.4415$558.4415 to the nearest thousandth (three decimal places).

Round $92.018649$92.018649 to four decimal places (nearest ten-thousandth).

Round $22.678$22.678 up to the nearest whole number.

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.

Rounding to the nearest 5 cents

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

Luigi is paying with cash at the supermarket, and his shopping bill totals $\$337.57$$337.57.

How much does he pay?

Irene is paying cash at the petrol station and the petrol metre reads $\$75.79$$75.79.

How much does she pay?

Round $\$199.98$$199.98 to the nearest five cents.