Year 7AC9M7N03

Powers of 10 and Estimation

Represent natural numbers in expanded notation using place value and powers of 10. Rounding and estimation are used to check the reasonableness of solutions (AC9M7N05).

Expanded notation shows what each digit is worth. Estimation uses the same idea to check whether an answer is sensible before you trust it.

Method

Each place is a power of 10

10^0 = 1 \quad 10^1 = 10 \quad 10^2 = 100

Moving left multiplies by 10 each time.

Expanded notation splits the number up

4275 = 4\times10^3 + 2\times10^2 + 7\times10 + 5

Every digit written as its true value.

Estimate by rounding to one significant figure

Then check your exact answer sits near the estimate.

Worked example

e.g.

\text{Write } 4275 \text{ in expanded notation, then estimate } 4275 \times 19.

Write down what each digit is worth. $4000 + 200 + 70 + 5$
Rewrite each part using a power of 10. $4\times10^3 + 2\times10^2 + 7\times10^1 + 5$
To estimate, round each number to one significant figure. $4275 \approx 4000 \quad 19 \approx 20$
Multiply the rounded numbers. $4000 \times 20 = 80\,000$
Compare with the exact value. It should be close. $4275 \times 19 = 81\,225 \quad\checkmark$

Practice

Expand, or estimate then check.

\text{Expand } 3061

Answer

3\times10^3 + 6\times10 + 1

\text{Estimate } 612 \times 48

Answer

600 \times 50 = 30\,000

\text{Estimate } 3981 \div 21

Answer

4000 \div 20 = 200

\text{A calculator gives } 61 \times 39 = 2379. \text{ Is it sensible?}

Answer

60\times40 = 2400 \quad\checkmark

\text{A calculator gives } 4{,}2 \times 8 = 3{,}36. \text{ Is it sensible?}

Answer

\text{No — expect about } 32. \text{ The point is misplaced.}