How to Determine the Missing Number when Multiplying Decimals by Powers of 10
TL;DR: Multiplying a decimal by a power of 10 has the most satisfying shortcut in arithmetic: the decimal point just slides to the right. So 3.45 times 10 becomes 34.5, times 100 becomes 345, times 1000 becomes 3450. When a problem asks you to find the missing factor, count exactly how many places the decimal point hopped between the original and the final number. That count tells you which power of 10 did the job. Easy detective work, every time.
Key takeaways:
- Multiplying by 10 moves the decimal point ONE place right.
- Multiplying by 100 moves it TWO places right; by 1000, THREE places.
- The number of zeros in the power of 10 tells you the number of places to shift.
- To find a missing factor or product, count the decimal-point shift.
- Example: if \(2.4\times \square=240\), the shift is 2 places, so \(\square=100\).
Multiplying decimals by powers of 10 is a fundamental concept in mathematics. The beauty of powers of 10 is their ability to shift the decimal point, making multiplication straightforward. Let’s explore how to determine the missing number when given a multiplication sentence involving decimals and powers of 10.
Key Insight:
Type the problems from this lesson and check each step.
When multiplying by \(10\), the decimal point moves one place to the right. For \(10^2\) (or 100), it moves two places to the right, and so on.
Multiplying by a Power of 10 with Decimals: Determine the Missing Number
Example 1:
Complete the multiplication sentence: \(0.25 \times 10 = \_\).
Solution Process:
Move the decimal point in \(0.25\) one place to the right because we are multiplying by \(10\).
Answer:
\(0.25 \times 10 = 2.5\).
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Example 2:
Complete the multiplication sentence: \(3.45 \times 100 = \_\).
Solution Process:
Move the decimal point in \(3.45\) two places to the right because we are multiplying by \(10^2\) (or 100).
Answer:
\(3.45 \times 100 = 345\).
Multiplying decimals by powers of 10 is a powerful tool that simplifies complex calculations. By understanding the relationship between decimals and powers of 10, you can quickly determine missing numbers in multiplication sentences. This skill deepens your mathematical prowess and boosts your confidence in handling decimals. So, the next time you encounter a decimal multiplication problem involving powers of 10, remember the key insight and tackle it with ease!
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Practice Questions:
1. Determine the missing number: \(0.56 \times 10 = \_\).
2. Complete the multiplication sentence: \(4.78 \times 1000 = \_\).
3. Determine the missing number: \(0.009 \times 100 = \_\).
4. Complete the multiplication sentence: \(5.67 \times 10^3 = \_\).
5. Determine the missing number: \(0.123 \times 10^2 = \_\).
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Answers:
1. \(5.6\)
2. \(4780\)
3. \(0.9\)
4. \(5670\)
5. \(12.3\)
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For a workbook that drills decimal multiplication and the place-value rules behind it, the Grade 5 Math for Beginners covers decimals and powers of 10 with worked examples. For broader pre-algebra coverage, the Pre-Algebra for Beginners connects decimal place value to scientific notation and unit conversions.
Frequently Asked Questions
How do you multiply decimals by powers of 10?
Move the decimal point to the right one place for each zero in the power of 10. So 0.68 × 10 = 6.8, 0.68 × 100 = 68 and 0.68 × 1000 = 680. Add zeros as placeholders if you run out of digits. To find a missing factor, count how far the point moved: in 1.25 × ? = 125, it moved two places, so the missing number is 100.
What is powers of 10 in math?
A power of 10 is 10 multiplied by itself a certain number of times, like 10, 100, 1000 and 10,000. Each one is ten times the one before it. Because our number system is based on ten, multiplying by a power of 10 simply shifts every digit to a higher place value, which is why the decimal point appears to slide right.
What is powers of 10 notation?
Powers of 10 notation writes these numbers with an exponent: 10¹ = 10, 10² = 100, 10³ = 1000, and so on. The exponent tells you how many zeros follow the 1, and also how many places the decimal point moves when you multiply. So 4.7 × 10³ means move the point three places right, giving 4700. This is the basis of scientific notation.
How do you multiply and divide powers of 10?
When you multiply by a power of 10, the decimal point moves right; when you divide, it moves left by the same number of places. For example, 5.2 × 100 = 520, while 5.2 ÷ 100 = 0.052. A quick check: multiplying by 10, 100 or 1000 makes a number bigger, and dividing makes it smaller, so the answer should change in that direction.
How many powers of 10 is 1000?
1000 is 10 to the third power, written 10³, because 10 × 10 × 10 = 1000. That means it has three zeros, and multiplying a decimal by 1000 moves the decimal point three places to the right. For example, 0.042 × 1000 = 42. In a missing-number problem, a three-place shift always points to 1000.
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