Problem 75 - Prealgebra

Question

Divide the decimal by the given power of 10 . \(\frac{113.02}{1000}\)

Step-by-Step Solution

Answer

113.02 divided by 1000 is 0.11302.

Step 1: Understand Place Value Shift

Dividing by powers of 10 moves the decimal point to the left by the number of zeros in the power. For example, dividing by 1000 (which is 10 to the power of 3) will move the decimal three places to the left.

Step 2: Perform the Division

Start with the original number 113.02. Move the decimal point 3 places to the left to divide by 1000. The number becomes 0.11302.

Step 3: Confirm the Result

Ensure that after shifting the decimal point, the solution is in the correct form with the proper number of decimal places. Original: 113.02, New position of decimal after shifting: 0.11302.

Key Concepts

Place Value

Place value is crucial when working with decimal numbers because it tells us the value of each digit in a number based on its position. For example, in the decimal number 113.02, each digit has a specific value:

• The '1' stands for one hundred.
• The '1' next to it stands for ten.
• The '3' represents three units or ones.
• The digits after the decimal point, '0' and '2', represent tenths and hundredths, respectively.

Understanding place value helps us see the number as a sum of each digit multiplied by its respective positional value. This understanding plays an important role when performing operations like division by powers of ten, as it determines how far to move the decimal point and keeps the number's value consistent.

Powers of Ten

The concept of powers of ten is fundamental in mathematics, particularly when working with decimals. A power of ten means 10 raised to a particular exponent. For example:

• 10^1 = 10
• 10^2 = 100
• 10^3 = 1000, and so forth.

When you divide a number by a power of ten, such as 1000 in the example, it effectively makes the number smaller by shifting the decimal point to the left. Each power increases the number of positions the decimal shifts:

• Divide by 10 shifts one place left.
• Divide by 100 shifts two places left.
• Divide by 1000 shifts three places left.

Knowing the effect of these powers helps in efficiently calculating results without performing long division.

Decimal Point Movement

When dividing by powers of ten, moving the decimal point is a straightforward way to find the solution quickly. The number of places you move the decimal point equals the number of zeros in the divisor if it's a power of ten:

• For 10, move one place left.
• For 100, move two places left.
• For 1000, move three places left.

Let's take a real-world example: dividing 113.02 by 1000. To solve this, start by identifying how many zeros are in 1000, which tells us to move the decimal point three places to the left. Thus, 113.02 transforms into 0.11302.
This method of moving the decimal point is quick, efficient, and helps maintain accuracy in calculations without complicated steps. It's a handy technique whenever you deal with decimal divisions involving multiples of ten.