So far we have been finding exact answers to arithmetical operations, and learning how to perform them quickly and accurately is an important skill. An equally important skill is being able to reasonably estimate answers - this is useful not only when speed is more important than accuracy, but also to check calculations to see if they are roughly on the right track.

Exploration

Suppose you and two friends had to split more than $\$10$$10 evenly between you. One of your friends counts it out, and by the end you have $\$1.50$$1.50. You would be able to rightly complain that something went wrong - if all three of you ended up with only $\$1.50$$1.50 then the total amount that was split would have been less than $\$5$$5 to begin with!

Importantly, you don't need to know exactly how much you started with to check the reasonableness of this calculation. Instead you can let your friends know that they need to try again - and you would watch a lot more closely this time.

One of the most important tools we use when checking reasonableness is rounding the numbers to make the calculation easier. This produces an estimate - the more you round, the more inaccurate your estimate becomes. If you round the numbers up, then the estimate will be larger than the true answer, and if you round down, the estimate will be smaller.

Careful!

The more you round, the less reasonable your calculation becomes - rounding each number in $6\times6$6×6 to the nearest $10$10 makes $10\times10$10×10, and $100$100 is not a good estimate for $6\times6=36$6×6=36!

Also, if you round some numbers up and some numbers down, you'll still produce an estimate, but you won't be able to easily tell if it's an overestimate or an underestimate. Rounding each number in $21\times19$21×19 to the nearest $10$10 makes $20\times20=400$20×20=400, which is a good overestimate. Rounding each number in $18\times23$18×23 to the nearest $10$10 also makes $20\times20=400$20×20=400, which is a good underestimate.

Worked example

Is $57\times9$57×9 close to $6000$6000?

Think: Let's round both numbers in the product up to the nearest ten to make an overestimate.

Do: $57$57 rounds up to $60$60, and $9$9 rounds up to $10$10. An overestimate for $57\times9$57×9 is $60\times10$60×10.

We now do the much easier multiplication $60\times10=600$60×10=600.

$600$600 is an overestimate for $57\times9$57×9, but $600$600 is far less than $6000$6000. We know that $6000$6000 is not a reasonable estimate for the original product.

Reflect: We never calculated the much more difficult product $57\times9$57×9, but we could still tell that $6000$6000 is far too big. Checking reasonableness is about speed, not accuracy.

Checking reasonableness

To find a reasonable estimate for a calculation, try rounding the numbers to the nearest $10$10, the nearest $100$100, or some other place value. This will make the calculation easier and give you a rough idea of what the answer should be.

If you round all the numbers up, your new calculation will be an overestimate (larger than the original).

If you round all the numbers down, your new calculation will be an underestimate (smaller than the original).

Practice questions

Question 1

Is this a reasonable statement?

"$67-29$67−29 is close to $40$40".

Yes
A
No
B

Question 2

Is this statement accurate?

"$4\times88$4×88 will be less than $320$320."

Yes
A
No
B

Question 3

Consider the product $37\times94$37×94.

Select all reasonable statements.
The product is less than $4000$4000.
A
The product is more than $2700$2700.
B
The product is more than $4000$4000.
C
The product is less than $2700$2700.
D

Outcomes

0580C1.9A

Make estimates of numbers, quantities and lengths and round off answers to reasonable accuracy in the context of a given problem.

0580C1.13

Use a calculator efficiently. Apply appropriate checks of accuracy.

0580E1.9A