ACARA v9 CONTENT DESCRIPTION “multiply and divide decimals by multiples of powers of 10 without a calculator, applying knowledge of place value and proficiency with multiplication facts; using estimation and rounding to check the reasonableness of answers”
Builds on: Adding and Subtracting Decimals (AC9M6N04). Place-value thinking with decimals extends here to multiplying and dividing by 10, 100 and 1000 — where the digits shift instead of the point.
Multiplying by 10
Multiplying a decimal by 10 makes every digit ten times bigger, which moves each digit one place to the left in the place-value columns. The simplest way to see this is that the decimal point appears to move one place to the right: 3.5 times 10 becomes 35, and 0.42 times 10 becomes 4.2. No long multiplication is needed, because place value does the work. Understanding that multiplying by 10 shifts digits, rather than adding a zero, is the foundation for multiplying and dividing by any power of 10.
Multiplying by 10
Multiplying a decimal by 10 moves the decimal point one place to the right.
Press to multiply by 10 and watch the digits shift.
Powers of 10
Ten, one hundred and one thousand are powers of 10, and each extra zero means one more place to move. Multiplying by 100 moves the point two places right, and by 1000 moves it three, because 100 has two zeros and 1000 has three. So 2.5 times 100 is 250, and 0.6 times 1000 is 600. The rule is wonderfully simple: count the zeros in the power of 10, and move the decimal point that many places. This pattern works for any power of 10, however large.
Powers of 10
The number of zeros in 10, 100 or 1000 is how many places the decimal point moves.
2.5 × 100 = 250 — the number of zeros in the power of 10 tells you how many places to move the point.
Dividing by powers of 10
Dividing by a power of 10 works the same way, but in the opposite direction: the digits become smaller, so the decimal point moves left instead of right. Dividing by 10 moves it one place left, by 100 two places, by 1000 three. So 45 divided by 10 is 4.5, and 320 divided by 100 is 3.2. When there are not enough digits, zeros fill the gap: 7 divided by 10 is 0.7. Multiplying moves the point right and dividing moves it left, by the number of zeros each time.
Dividing by powers of 10
Dividing by 10, 100 or 1000 moves the decimal point left, by the number of zeros.
Press to divide and watch the point move the other way.
Multiplying in practice
Putting the rule to work, multiplying a decimal by a power of 10 is just a matter of counting zeros and shifting the point right. For 3.5 times 10, one zero means one place, giving 35; for 0.4 times 100, two zeros give 40; for 1.2 times 1000, three zeros give 1200. The key is to track the point carefully and to write in extra zeros where places run out. Because no calculator is needed, this is a fast, reliable method that rests entirely on understanding place value.
Multiplying by powers of 10
Move the point right by the number of zeros in the power of 10.
What is 3.5 × 10? Pick A, B or C.
Dividing in practice
Dividing a decimal or whole number by a power of 10 follows the mirror-image rule: count the zeros and shift the point that many places left. So 45 divided by 10 is 4.5; 600 divided by 100 is 6; and 8 divided by 10 is 0.8, with a zero written in front to hold the ones place. Watching where the point lands, and adding leading zeros when needed, keeps the value correct. Dividing by powers of 10 is exactly as quick as multiplying, once the direction of the shift is clear.
Dividing by powers of 10
Move the point left by the number of zeros in the power of 10.
What is 45 ÷ 10? Pick A, B or C.
Estimating to check
Because it is easy to move the point the wrong number of places, estimating is a vital check. A quick estimate, using rounding and place value, tells you the size the answer should be: 4.9 times 10 should be about 50, so an answer of 49 is reasonable while 490 would be far too big. A common error is treating multiplying by 100 as simply adding two zeros, which fails for decimals; place-value reasoning shows the point must move two places. Checking each answer against an estimate catches these slips before they cause trouble.
Estimating to check
A rough estimate and place-value reasoning tell you whether an answer is sensible.
Use estimation and place value to judge the answer. Pick A, B or C.
Where this leads
Multiplying and dividing by powers of 10 is the engine behind the metric system, scientific notation, and everyday work with money and measurement, where converting between units means shifting the decimal point. The place-value thinking built here, that each power of 10 moves the point a fixed number of places, carries directly into multiplying and dividing decimals by any number, and into the powers and indices of later mathematics. It is a small rule with very wide reach, used constantly in science, finance and daily life.
Quick self-check
1. What is 3.5 × 10?
2. When you multiply by 100, the decimal point moves...
3. What is 45 ÷ 10?
4. What is 0.6 × 1000?
5. A student says 3.2 × 100 = 32. Using estimation, this answer is...