Problem 17 - Prealgebra

Question

Multiply. $$248 \cdot \frac{1}{10} \cdot \frac{1}{10}$$

Step-by-Step Solution

Answer

The product is 2.48.

Step 1: Convert Fractions to Decimals

We begin the problem by expressing the fractions in decimal form. The fraction \( \frac{1}{10} \) is equivalent to 0.1. Therefore, \( 248 \cdot \frac{1}{10} \cdot \frac{1}{10} \) can be rewritten as \( 248 \cdot 0.1 \cdot 0.1 \).

Step 2: Multiply by 0.1

Now, multiply 248 by 0.1. To do this, shift the decimal place in 248 one place to the left, which gives us 24.8. So, \( 248 \cdot 0.1 = 24.8 \).

Step 3: Multiply the Result by 0.1 Again

Finally, we take our result from Step 2, which is 24.8, and multiply it by 0.1. Shift the decimal place in 24.8 one place to the left to get 2.48. Thus, \( 24.8 \cdot 0.1 = 2.48 \).

Key Concepts

Fractions to Decimals Conversion

When it comes to converting fractions to decimals, it's simpler than it seems. The fraction \( \frac{1}{10} \) is one of the easiest examples. To convert \( \frac{1}{10} \) to a decimal, think of what \( \frac{1}{10} \) represents—1 out of 10 parts, or 0.1. This converts straight to a decimal by simply moving the decimal point.

• Imagine 1 divided by 10, which equals 0.1.
• When fractions have a denominator of 10, 100, or 1000, conversion involves direct decimal placement.
• Understanding fractions as parts of a whole helps visualize decimal equivalents.

Whether you're dealing with a penny as part of a dollar (\( \frac{1}{100} = 0.01\)) or any similar fraction, the key is dividing the numerator by the denominator. In our example, \( \frac{1}{10} \) becomes 0.1, a straightforward decimal representation.

Multiplying Decimals

Multiplying decimals involves a simple twist on multiplying whole numbers. But, instead of focusing just on numbers, we also have to think about the decimal point's position.

• First, ignore the decimal points and multiply the numbers as if they were whole. For instance, multiplying 248 by 1 is 248.
• Next, count the total number of decimal places in the numbers you're multiplying. In this case, 0.1 has one decimal place.
• Adjust the final result by shifting the decimal point in the straightforward product by the total count of decimal places.

When you multiply \( 248 \times 0.1 \), you shift the decimal one place to the left. This changes 248 to 24.8. Repetition of this thought process ensures accuracy when multiplying any decimals.

Decimal Place Shifting

Decimal place shifting is crucial when working with decimals, particularly in multiplication and division.

• Shifting left moves the decimal over to the left, reducing the number and transforming it based on the place value.
• Every shift to the left signifies dividing the number by 10.
• For example, moving the decimal left once in 248 makes it 24.8 (dividing by 10), and doing it again changes 24.8 to 2.48.

Each multiplication by 0.1 factors into this shifting mechanism, translating 248 through a series of divisions by 10. Once to 24.8, and then to 2.48, clearly shows how place value plays a significant role in determining the final product. Understanding this will help you easily work through similar problems.