Problem 27 - Prealgebra

Question

Multiply. $$0.3(360)$$

Step-by-Step Solution

Answer

The product of 0.3 and 360 is 108.

Step 1: Understand the Expression

We need to multiply the decimal number 0.3 by the whole number 360. We'll perform this multiplication by treating 0.3 as a fraction and then multiplying by 360.

Step 2: Convert the Decimal to a Fraction

The number 0.3 can be converted to the fraction \( \frac{3}{10} \). This conversion will facilitate the multiplication process by allowing us to work with fractions instead.

Step 3: Set Up the Fraction Multiplication

Now we multiply the fraction \( \frac{3}{10} \) by 360. This is expressed as: \( \frac{3}{10} \times 360 \). We know that multiplying by a fraction is the same as dividing by the denominator and then multiplying by the numerator.

Step 4: Divide First

Divide 360 by the denominator of the fraction, which is 10. This calculation is: \( 360 \div 10 = 36 \).

Step 5: Multiply the Result by the Numerator

Now, multiply the result from the previous step by the numerator of the fraction: \( 36 \times 3 \). Calculate this multiplication: \( 36 \times 3 = 108 \).

Step 6: Present the Final Result

The product of \( 0.3 \times 360 \) is 108. We determined this by multiplying \( \frac{3}{10} \times 360 \) following basic arithmetic operations.

Key Concepts

Fraction Conversion

Decimal numbers and fractions are two different ways to express parts of a whole. Converting a decimal into a fraction often simplifies computations, such as multiplication. For example, the decimal 0.3 can be represented as the fraction \( \frac{3}{10} \). To do this conversion:

• Identify the place value of the decimal. Here, 0.3 means 3 tenths because the number 3 is in the tenths place.
• Write the decimal as a fraction with the numerator as the digit (3) and the denominator as the place value (10).
• Hence, 0.3 becomes \( \frac{3}{10} \).

This makes it easier to multiply with other numbers by converting the process into simple fraction multiplication.

Arithmetic Operations

Arithmetic operations like addition, subtraction, multiplication, and division form the basis of mathematics. In this exercise, we focus on multiplication, which combines quantities to find their total size.

• When multiplying fractions, you multiply the numerators together and the denominators together. However, when multiplying a fraction by a whole number, it simplifies the operation.
• First, divide the whole number by the denominator of the fraction. This gives you an equivalent value scaled down by the fractional part.
• Next, multiply the result by the numerator to find the final result.

By breaking down the multiplication into smaller steps involving division and multiplication, we make complex calculations more manageable.

Multiplication Process

The multiplication process used in this exercise demonstrates how to handle operations with decimals using a fraction approach. Let's break down the steps:First, convert the decimal 0.3 into the fraction \( \frac{3}{10} \). This helps simplify the process because fractions can be easier to manage.

• Start by dividing the whole number 360 by the denominator, which is 10. This division simplifies the number down to 36 as it scales by one-tenth.
• Then, multiply the result, 36, by the numerator, which is 3. This scales the value back up, resulting in 108.

This pattern of scaling a number down via division and then back up via multiplication exemplifies the neat symmetry of arithmetic operations, ensuring an accurate result.