1.04 Estimation and reasonable answers

Leading digit estimation

When we know how to round, we can use our rounded answers to perform leading digit estimation. The leading digit, is the first digit in our number. (Think of the word 'leading' as 'going first'). Once we know that, we can round our number, and work out an estimate to our problem.

Leading digit estimation basically means round to the nearest column of the leading digit.

Worked example

a) Round $5428$5428 and $3823$3823 to the nearest leading digit.

Think: Rounding to the nearest leading digit, means to round to the column of the first digit. So $5428$5428 has a first digit of $5$5 in the thousandths column, while $3823$3823 has a first digit of $3$3 in the thousandths column.

Do: Remember to round, we look to the digit to the right of the thousandths column. The number $5428$5428 has a $4$4 in the hundredths column, so it rounds down to $5000$5000. The number $3823$3823 has an $8$8 in the hundredths column, so it rounds up to $4000$4000.

b) Using a leading digit estimation what is $5428+3823$5428+3823?

Think: We can think of this question as $5000+4000$5000+4000, using the leading digit rounding we performed above.

Do: Using leading digit estimation, we would estimate $5428+3823$5428+3823 to be about $9000$9000.

 

Practice questions

Question 1

Question 2

Question 3

Checking reasonableness

Checking if an answer is reasonable means we check to see whether an answer is a good approximation to the question. This is really important because it allows us to check whether our calculations make sense.

For example, if a question was asked that said $\$10$$10 was shared between $2$2 people and they both got $\$50$$50, this would NOT be a reasonable calculation as my answer ($\$50$$50) is even more money than the total at the start ($\$10$$10)!

 

How do we check if an answer is reasonable?

We can check how reasonable an answer is by using leading digit estimation. Once you have rounded the numbers to easier values to work with, you can do a mental calculation. If the given answer is close to your estimate, then it is a reasonable answer.

You can also put the answer into context, and ask yourself does it make sense that the answer is this big, or this small?

Worked example

Is $6000$6000 a reasonable estimate for $57\times11$57×11?

Think: Let's round both these numbers to the nearest ten and see if it's close to $6000$6000.

Do: If we round $57$57 to the nearest ten, it's $60$60 and if we round $11$11 to the nearest ten, it's $10$10.

So, our estimate can be rewritten as $60\times10$60×10, which is equal to $600$600.

$600$600 is a lot less than $6000$6000 so we can say no, $6000$6000 is not a reasonable estimate!

 

Practice questions

Question 4

Question 5

Question 6

Question 7