Multiplying by 10, 100 and 1000
Multiplying by these numbers is very useful in mathematics because you do not need a calculator or complicated calculations. It is based on shifting place values.
When we multiply a whole number (a number without a decimal point), we simply add as many zeros to the end as there are zeros in the multiplier:
| Multiply by | Add | Example |
| $10$ | one zero | $5 × 10 = 50$ |
| $100$ | two zeros | $5 × 100 = 500$ |
| $1000$ | three zeros | $5 × 1000 = 5000$ |
This also works for larger numbers – we simply add the zeros after the last digit:
| $23 × 10 = 230$ | $23 × 100 = 2300$ | $23 × 1000 = 23000$ |
| $147 × 10 = 1470$ | $147 × 100 = 14700$ | $147 × 1000 = 147000$ |
Multiples of tens, hundreds and thousands
The same principle works when we do not multiply directly by 10, 100 or 1000, but by a multiple of them, such as 50 or 700. We can split such numbers into factors to make the multiplication easier to see.
For example, $50 = 5 × 10$ and $700 = 7 × 100$. Therefore:
A quick method is therefore to multiply the numbers without the ending zeros ($5 × 7 = 35$) and then add all the zeros from both factors (one from 50 and two from 700, so three altogether): $35000$.
Hidden zeros
Be careful about one common mistake. Sometimes a zero already appears when we multiply the numbers without their ending zeros – and it is easy to overlook it. Consider this example:
Step 1. Multiply the numbers without the ending zeros: $5 × 6 = 30$. The result already ends in a zero – this is the hidden zero. It does not come from the zeros in the original factors; it appears as a result of the multiplication.
Step 2. Add the zeros from the original factors: 50 has one zero and 60 has one zero, so there are two zeros altogether.
Step 3. Add these two zeros after $30$: $3000$.
If we overlook the hidden zero, we might write $300$ – but that is ten times too small. We can check our answer: $50 × 60$ must be greater than, for example, $50 × 50 = 2500$, so $300$ cannot be correct.
Hidden zeros appear whenever the product of the numbers without their ending zeros ends in zero, for example $2 × 5 = 10$, $4 × 5 = 20$, $5 × 8 = 40$ or $25 × 4 = 100$:
| Example | Without ending zeros | Result |
| $50 × 60$ | $5 × 6 = 30$ | $3000$ |
| $20 × 500$ | $2 × 5 = 10$ | $10000$ |
| $40 × 50$ | $4 × 5 = 20$ | $2000$ |
| $250 × 40$ | $25 × 4 = 100$ | $10000$ |
The rule is simple: we always add the zeros from the original factors after the complete product of the numbers without their ending zeros – including any zeros that are already part of that product. We never replace or remove zeros that already appear in the product.
