Question

Multiply the following numbers by moving the decimal. \(0.849 \times 1,000=\) ______

Short answer

849

Step-by-step solution

Understand Decimal Multiplication

When multiplying a decimal number by 1,000, we shift the decimal point to the right by the number of zeros in 1,000. Since 1,000 has three zeros, we'll move the decimal three places to the right.

Identify Current Decimal Position

The number given is 0.849, which has a decimal point after the first zero. Currently, the decimal is in the thousandths place (three places from the right of the last digit).

Move the Decimal Point

Shift the decimal point three places to the right. Start with 0.849, and move the decimal after the digit '9'. This gives us 849.

Confirm the New Number

The new number, after moving the decimal point, is 849, since moving the decimal three times fully shifts it past the numerals without any remainder.

Key concepts

Decimal Point Movement

When you multiply decimals by powers of ten, understanding decimal point movement is key. Each zero in the power of ten indicates a shift in the decimal position. For instance, multiplying by 10 (one zero) shifts the decimal one place to the right. Similarly, multiplying by 1,000 (three zeros) moves the decimal three places. Moving the decimal involves identifying its initial position and determining the new location after the shift. This method ensures the number is correctly transformed without altering the significant digits. So if you multiply 0.849 by 1,000, moving the decimal point three places to the right gives you 849.

Place Value in Decimals

Place value is crucial in understanding decimals. Each digit in a decimal number has a specific value based on its position relative to the decimal point.
To break this down:

• The first number right after the decimal is in the tenths place.
• The second is in the hundredths place.
• The third is in the thousandths place, and so on.

Knowing these positions helps you properly assign value when multiplying or moving decimal points. When a decimal point moves, these place values shift like seats in a row, ensuring each digit remains in the correct order. Therefore, when moving the decimal for 0.849 by three places to 849, you're effectively transitioning from a thousandths place to units and hundreds, increasing the overall value.

Multiplying by Powers of Ten

Multiplying by powers of ten is a technique used to simplify calculations. This method significantly influences the number's magnitude by altering the decimal point location.
Here’s how it works:

• Identify the power of ten you're multiplying by, like 10, 100, or 1,000.
• The exponent or number of zeros in the multiplier tells you how many places to move the decimal point.
• After the decimal has shifted, the number's scale has increased, corresponding to the scale of multiplication.

For example, multiplying the number 0.849 by 1,000 converts it into 849. The decimal movement process effectively multiplies the initial value by 1,000 by repositioning the decimal instead of digit aggregation, providing clarity and efficiency in solving problems.