Decimal Worksheets - Free Printable

The image you've uploaded is a math worksheet titled "Understanding Multiplying Decimals" with 20 problems. Each problem follows a pattern: it gives a whole number multiplication fact (e.g., $9 \times 9 = 81$), and then asks you to compute a similar multiplication but with decimals (e.g., $0.09 \times 0.09$). The goal is to understand how multiplying decimals relates to multiplying whole numbers, especially in terms of decimal place value.

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🔍 How to Solve These Problems

Each problem uses the idea:

> If $a \times b = c$, then $(a \div 10^n) \times (b \div 10^m) = c \div 10^{n+m}$

Or more simply:
When you multiply two decimals, count the total number of decimal places in both factors, and place that many decimal places in the product.

Let’s go through a few examples to explain the logic.

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✔ Example 1:

> If $9 \times 9 = 81$, then $0.09 \times 0.09 = ?$

- $0.09 = 9 \div 100$, so two decimal places
- So $0.09 \times 0.09 = (9 \times 9) \div (100 \times 100) = 81 \div 10,000 = 0.0081$

But wait — the answer key says 0.081, which is incorrect.

Wait! Let's double-check.

Actually:
$0.09 \times 0.09 = ?$

- $0.09 = 9 \times 10^{-2}$
- So $0.09 \times 0.09 = 81 \times 10^{-4} = 0.0081$

But the answer key says 0.081, which is wrong.

Wait — look at the original:

> 1) If $9 \times 9 = 81$, then $0.09 \times 0.09 = 0.081$

This is incorrect. It should be:

$$
0.09 \times 0.09 = 0.0081
$$

But the answer key says 0.081 — that’s a mistake.

Wait — let's check the second problem:

> 2) If $2 \times 9 = 18$, then $0.2 \times 0.009 = ?$

- $0.2 = 2 \times 10^{-1}$
- $0.009 = 9 \times 10^{-3}$
- So product: $2 \times 9 = 18$, and decimal places: $1 + 3 = 4$
- So $18 \times 10^{-4} = 0.0018$

Answer key says 0.0018 → ✔ Correct

So why is #1 wrong?

Let’s recheck #1:

> $0.09 \times 0.09$

- $0.09$ has 2 decimal places → total of 4 decimal places
- $9 \times 9 = 81$
- Move decimal 4 places left: $0.0081$

But the answer key says 0.081 → That’s only 3 decimal places → ✘ Incorrect

So there’s an error in the answer key for problem #1.

Wait — maybe I misread the problem?

Look again:

> 1) If $9 \times 9 = 81$, then $0.09 \times 0.09 = ?$

Yes, $0.09 \times 0.09 = 0.0081$

But the answer key says 0.081

That’s off by a factor of 10.

Unless... was it supposed to be $0.09 \times 0.9$?

Then: $0.09 \times 0.9 = 0.081$ → yes!

So likely, the problem was meant to be:

> If $9 \times 9 = 81$, then $0.09 \times 0.9 = ?$

Because:
- $0.09 \times 0.9 = (9 \times 10^{-2}) \times (9 \times 10^{-1}) = 81 \times 10^{-3} = 0.081$

✔ That matches the answer key.

So either:
- The problem statement is wrong (it says $0.09 \times 0.09$), or
- The answer key is wrong (it says 0.081 instead of 0.0081)

But since the rest of the answers are correct, and this one stands out, we can assume a typo.

Let’s verify other problems.

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🔎 Check Problem #5:

> If $7 \times 8 = 56$, then $0.07 \times 0.8 = ?$

- $0.07 = 7 \times 10^{-2}$
- $0.8 = 8 \times 10^{-1}$
- Total decimal places: 3
- $7 \times 8 = 56$
- $56 \times 10^{-3} = 0.056$

Answer key says 0.056 → ✔ Correct

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🔎 Problem #11:

> If $6 \times 7 = 42$, then $0.006 \times 0.007 = ?$

- $0.006 = 6 \times 10^{-3}$
- $0.007 = 7 \times 10^{-3}$
- Product: $42 \times 10^{-6} = 0.000042$

Answer key says 0.000042 → ✔ Correct

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🔎 Problem #13:

> If $2 \times 8 = 16$, then $0.002 \times 0.08 = ?$

- $0.002 = 2 \times 10^{-3}$
- $0.08 = 8 \times 10^{-2}$
- Total: $16 \times 10^{-5} = 0.00016$

Answer key says 0.00016 → ✔ Correct

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🔎 Problem #14:

> If $3 \times 4 = 12$, then $0.3 \times 0.4 = ?$

- $0.3 \times 0.4 = 0.12$ → answer key says 0.12 → ✔ Correct

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🚩 Back to Problem #1:

> If $9 \times 9 = 81$, then $0.09 \times 0.09 = ?$

- $0.09 \times 0.09 = 0.0081$

But answer key says 0.081 → ✘ Incorrect

It seems like the answer key is wrong for problem #1.

Alternatively, perhaps the problem was meant to be:

> If $9 \times 9 = 81$, then $0.09 \times 0.9 = 0.081$

That would make sense.

So either:
- There’s a typo in the problem (should be $0.09 \times 0.9$), or
- There’s a typo in the answer key (should be 0.0081)

Given that all other answers are consistent, and the method is sound, we conclude:

> ✔ The correct answer to $0.09 \times 0.09$ is 0.0081, not 0.081.

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✔ General Rule Summary

To solve these problems:

1. Take the whole number multiplication result.
2. Count the total number of decimal places in the decimal factors.
3. Place the decimal point in the product so that it has that many decimal places.

For example:
- $0.006 \times 0.007$:
- $6 \times 7 = 42$
- 3 + 3 = 6 decimal places → $0.000042$

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✔ Final Answer: The worksheet is mostly correct, but Problem #1 has an error.

#### Corrected Answer Key:
| Problem | Correct Answer |
|--------|----------------|
| 1) | 0.0081 ← corrected (not 0.081) |
| 2) | 0.0018 ✔ |
| 3) | 0.008 ✔ |
| 4) | 0.004 ✔ |
| 5) | 0.056 ✔ |
| 6) | 0.0006 ✔ |
| 7) | 0.12 ✔ |
| 8) | 0.0002 ✔ |
| 9) | 0.0048 ✔ |
| 10) | 0.009 ✔ |
| 11) | 0.000042 ✔ |
| 12) | 0.000036 ✔ |
| 13) | 0.00016 ✔ |
| 14) | 0.12 ✔ |
| 15) | 0.0009 ✔ |
| 16) | 0.006 ✔ |
| 17) | 0.0072 ✔ |
| 18) | 0.056 ✔ |
| 19) | 0.007 ✔ |
| 20) | 0.00009 ✔ |

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📝 Conclusion

- The method used in the worksheet is correct: use known whole number products and adjust decimal places.
- Only problem #1 has an error: the answer should be 0.0081, not 0.081.
- Likely, the problem was intended to be $0.09 \times 0.9$, not $0.09 \times 0.09$.

If you're using this worksheet, correct problem #1 accordingly.

Let me know if you'd like a version with the corrected problem!