Problem 28 - Prealgebra

Question

Multiply. $$0.4(360)$$

Step-by-Step Solution

Answer

The product is 144.

Step 1: Understand the Operation

The task is to multiply a decimal number, 0.4, by an integer, 360. This means finding a number that is 0.4 times the size of 360.

Step 2: Convert Decimal to Fraction

Convert 0.4 to a fraction for easier multiplication. The decimal 0.4 can be written as \( \frac{4}{10} \) which simplifies to \( \frac{2}{5} \).

Step 3: Setup the Multiplication

Now that we have 0.4 as \( \frac{2}{5} \), rewrite the expression as \( \frac{2}{5} \times 360 \).

Step 4: Multiply the Fractions

Multiply the fraction by the number: \( \frac{2}{5} \times 360 = \frac{2 \times 360}{5} = \frac{720}{5} \).

Step 5: Simplify the Result

Divide the numerator by the denominator: \( \frac{720}{5} = 144 \).

Key Concepts

Conversion of Decimals to Fractions

When dealing with decimal numbers, converting them to fractions can simplify arithmetic operations like multiplication. To convert a decimal to a fraction, consider the place value of the decimal.
For instance, the decimal 0.4 is equivalent to 4 tenths. This means it can be converted into the fraction \(\frac{4}{10}\).
Once converted, fractions can often be further simplified.

• Look for the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by the GCD to simplify the fraction.

In the case of \(\frac{4}{10}\), both numbers can be divided by 2, the GCD, resulting in \(\frac{2}{5}\).
This gives us the simplest form of the decimal 0.4.

Simplifying Fractions

Simplifying fractions is a crucial step in fraction arithmetic as it makes the numbers easier to work with and understand.
Simplification involves reducing a fraction to its lowest terms.
Here's how:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

Let's break it down using our example of \(\frac{4}{10}\):- The GCD of 4 and 10 is 2.- Divide both 4 and 10 by 2 to get \(\frac{2}{5}\).
Now, the fraction is in its simplest form, making subsequent mathematical operations straightforward.

Multiplication of Fractions

Multiplying fractions is straightforward once you understand the process. Unlike adding or subtracting fractions that require a common denominator, multiplying fractions doesn't need one.
To multiply fractions, follow these steps:

• Multiply the numerators together.
• Multiply the denominators together.

In our case with \(\frac{2}{5} \times 360\), treat 360 as \(\frac{360}{1}\): - Multiply the numerators: \(2 \times 360 = 720\).- Multiply the denominators: \(5 \times 1 = 5\).
This results in the fraction \(\frac{720}{5}\).
This fraction will need simplifying, but that becomes easier after the multiplication.

Basic Arithmetic Operations

Understanding basic arithmetic operations is fundamental when working with any kind of numbers. These operations include addition, subtraction, multiplication, and division.
In multiplication, the focus is on scaling one number by another.
Here's a simplified breakdown:

• Identify the numbers you need to multiply.
• Apply any necessary conversions (like decimals to fractions).
• Perform the multiplication operation.
• Simplify the result if needed.

For example, in multiplying decimals and whole numbers like 0.4 and 360, convert the decimal to a fraction (\(\frac{2}{5}\)) for clarity.
Then multiply it with 360, as fractions, to get a straightforward product, simplifying at the end.