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Chapter 5: Problem 564

In the following exercises, multiply. $$ (0.3)(0.7) $$

Short Answer

Expert verified
0.3 \times 0.7 = 0.21

Step by step solution

01

Understanding the Problem

The task is to multiply two decimal numbers: 0.3 and 0.7.
02

Convert Decimals to Fractions

Convert 0.3 to a fraction: \(0.3 = \frac{3}{10}\). Convert 0.7 to a fraction: \(0.7 = \frac{7}{10}\).
03

Multiply the Fractions

Multiply the numerators: \(3 \times 7 = 21\). Multiply the denominators: \(10 \times 10 = 100\). So, \(\frac{3}{10} \times \frac{7}{10} = \frac{21}{100}\).
04

Convert the Fraction Back to a Decimal

Divide the numerator by the denominator: \( \frac{21}{100} = 0.21 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convert Decimals to Fractions
To multiply decimal numbers more easily, one handy technique is to convert them into fractions. Decimals and fractions are closely related; a decimal is just another way to represent a fraction.

For instance, to convert the decimal 0.3 to a fraction, think about the number's place value. The 3 is in the tenths place, which means 0.3 is equivalent to \(\frac{3}{10}\).

Likewise, 0.7 has the 7 in the tenths place. Therefore, 0.7 can be written as \(\frac{7}{10}\). Converting both these decimals to fractions allows you to work more easily with multiplication.
Multiplying Fractions
Once the decimals are converted to fractions, the process of multiplication becomes straightforward. You need to follow these steps:

\bullet Multiply the numerators (the top numbers).
\bullet Multiply the denominators (the bottom numbers).

In our example, after converting 0.3 and 0.7 to fractions, we get \(\frac{3}{10}\) and \(\frac{7}{10}\). To multiply these fractions:

\bullet Multiply the numerators: \(\frac{3}{10} \times \frac{7}{10}\). \(\frac{3 \times 7}{10 \times 10} = \frac{21}{100}\).

Voila! We've multiplied the fractions successfully. Next, we need to convert the result back to a decimal for easier reading.
Converting Fractions Back to Decimals
After obtaining the fraction from the multiplication process, the final step is to convert it back to a decimal. Doing this is simple and straightforward as well.

The fraction we have from our multiplication is \(\frac{21}{100}\). To convert a fraction into a decimal, divide the numerator by the denominator. Here, we divide 21 by 100: \[ \frac{21}{100} = 0.21 \]

So, \(\frac{21}{100}\) is equivalent to 0.21 as a decimal. This means that multiplying 0.3 and 0.7 results in 0.21.

Following these steps ensures that you can multiply decimals with ease and accuracy by converting them to fractions, multiplying the fractions, and then converting the result back to a decimal.

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