Problem 20 - Algebra A Combined Function

Question

Multiply or divide as indicated. $$ \begin{array}{r} 0.079 \\ \times \quad 3.6 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The product is 0.2844.

Step 1: Eliminate Decimals

To simplify the multiplication, first eliminate the decimals. Convert 0.079 to a fraction by recognizing that it is equivalent to \( \frac{79}{1000} \). Similarly, convert 3.6 to \( \frac{36}{10} \).

Step 2: Multiply Fractions

Now multiply the two fractions. \( \frac{79}{1000} \times \frac{36}{10} = \frac{79 \times 36}{1000 \times 10} \). Calculate the numerator: \( 79 \times 36 = 2844 \) and the denominator: \( 1000 \times 10 = 10000 \).

Step 3: Convert Result to Decimal

Now convert the result back to decimal form. \( \frac{2844}{10000} \) is equivalent to moving the decimal four places from the right in the number 2844. This gives 0.2844.

Key Concepts

Fractions and Decimals

Fractions and decimals are two different ways to express numbers that are not whole. They are widely used to represent parts of a whole, or to show numbers more precisely than whole numbers can.
Understanding these two forms is essential for a smooth conversion between them, especially when performing operations such as multiplication and division.

• Fractions represent numbers by showing how many parts of a certain size make up the whole. For example, \( \frac{1}{2} \) means one part of something divided into two equal parts.
• Decimals, on the other hand, are based on powers of ten. They provide an alternative way to represent fractions, such as 0.5 representing \( \frac{1}{2} \).

To multiply decimals efficiently, we often convert them to fractions. This allows us to leverage our understanding of fraction multiplication to get the result.

Eliminating Decimals

When dealing with decimal multiplication, eliminating decimals helps simplify the operations by converting decimals into fractions. This method clarifies the calculation process and often makes it more manageable.

• To eliminate the decimal in 0.079, write it as a fraction: \( \frac{79}{1000} \). The digits right of the decimal point (079) over 1000 represents three decimal places.
• For 3.6, convert it by recognizing that it is the same as \( \frac{36}{10} \). This accounts for one decimal place, hence over 10.

Eliminating decimals early on ensures that we work with simpler numbers when multiplying fractions. It also provides a straightforward path back to decimals once multiplication is completed.

Converting Decimals to Fractions

Converting decimals back into fractions helps in simplifying calculations, especially during multiplication and division. Let’s look into how this conversion is executed and why it is helpful.

• To convert a decimal to a fraction, identify the place value of the farthest decimal digit. This determines the denominator. For instance, 0.079 becomes \( \frac{79}{1000} \) since the last digit is in the thousandth place.
• Another example is the decimal 3.6, which can be shifted into the fraction \( \frac{36}{10} \), acknowledging the single decimal place moves the number over ten.

By turning decimals into fractions, we can multiply them using simple fraction multiplication rules. Finally, once the result is obtained, we convert it back to its decimal form to make it more readable. This dual conversion ensures clarity during the mathematical operation and helps in verifying the final result.