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Chapter 5: Problem 691

In the following exercises, simplify each expression. $$ (0.64)(0.3) $$

Short Answer

Expert verified
0.64 脳 0.3 = 0.192

Step by step solution

01

- Understand the values

Identify the numbers that need to be multiplied. Here they are 0.64 and 0.3.
02

- Convert to fractions

Convert the decimal values to fractions: 0.64 = \(\frac{64}{100}\) and 0.3 = \(\frac{3}{10}\).
03

- Multiply the fractions

Multiply the numerators and the denominators: \(\frac{64}{100} \times \frac{3}{10} = \frac{64 \times 3}{100 \times 10} = \frac{192}{1000}\).
04

- Simplify the fraction

Simplify the fraction \(\frac{192}{1000}\) by finding the greatest common divisor (GCD). The GCD of 192 and 1000 is 8. \(\frac{192 \rightarrow 24}{1000 \rightarrow 125} = \frac{24}{125}\).
05

- Convert back to decimal

Convert \(\frac{24}{125}\) back to a decimal:\(\frac{24}{125} = 0.192\).

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Key Concepts

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

Converting Decimals to Fractions
To make math easier, we often convert decimals to fractions. For instance, let's take the decimal 0.64.

First, recognize that it has two decimal places. Therefore, we can write it as 64 over 100, because 0.64 is the same as 64 hundredths. In fraction form, it becomes \(\frac{64}{100}\).

The same goes for 0.3, which has one decimal place. It's 3 over 10, resulting in \(\frac{3}{10}\).

This conversion helps us perform operations like multiplication more easily.
Fraction Simplification
Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).

For example, consider the fraction \(\frac{192}{1000}\). To simplify it, find the GCD of 192 and 1000. Here, it's 8. Divide both the numerator and the denominator by 8:
\(\frac{192 \div 8}{1000 \div 8} = \frac{24}{125}\).

Now, \(\frac{24}{125}\) is in its simplest form.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is the largest number that can exactly divide two numbers. Finding the GCD helps in simplifying fractions.

To find the GCD of 192 and 1000, list the factors:
  • Factors of 192: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192
  • Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000

Both numbers share these common factors: 1, 2, 4, and 8.

The largest of these common factors is 8, making it the GCD. Using the GCD simplifies the fraction effectively.

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