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Chapter 6: Problem 7

Perform the following operations. \(\frac{5}{8}-0.513\)

Short Answer

Expert verified
The result of the operation is \(\frac{14}{125}\).

Step by step solution

01

Convert Decimal to Fraction

First, we need to convert the decimal number into a fraction for easy calculation. The decimal 0.513 can be written as \(\frac{513}{1000}\).
02

Find Common Denominator

Now, we need to find a common denominator for the fractions \(\frac{5}{8}\) and \(\frac{513}{1000}\). The least common multiple of 8 and 1000 is 1000. We will convert \(\frac{5}{8}\) to have the denominator of 1000 by calculating \(\frac{5 \times 125}{8 \times 125} = \frac{625}{1000}\).
03

Subtract the Fractions

Now that both fractions have a common denominator, we can subtract them: \(\frac{625}{1000} - \frac{513}{1000} = \frac{625 - 513}{1000} = \frac{112}{1000}\).
04

Simplify the Fraction

Reduce \(\frac{112}{1000}\) to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 8: \(\frac{112 \div 8}{1000 \div 8} = \frac{14}{125}\).

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Key Concepts

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

Decimal to Fraction Conversion
Changing a decimal to a fraction is relatively straightforward. All you need to do is recognize the place value of the decimal. For instance, in the decimal 0.513, "513" is in the thousandths place, so you can write it as \( \frac{513}{1000} \).
Understanding the position helps determine the denominator:
  • "0.5" is \( \frac{5}{10}\) or \( \frac{1}{2} \).
  • "0.05" is \( \frac{5}{100} \) or \( \frac{1}{20} \).
  • "0.005" is \( \frac{5}{1000} \).
This approach helps to turn any fraction into a form that鈥檚 easy to work with. Write the decimal over the base of ten, based on its place value, and simplify if necessary.
Common Denominator
When adding or subtracting fractions, having a common denominator is essential. This is the same denominator for both fractions so that you can easily do the arithmetic. To find a common denominator, you look for the least common multiple (LCM) of the denominators.
In the provided example, the denominators are 8 and 1000. The LCM of 8 and 1000 is 1000. This means both fractions should be expressed with 1000 as their denominator. For \( \frac{5}{8} \), you calculate \( \frac{5 \times 125}{8 \times 125} = \frac{625}{1000} \). Now, both fractions are ready for subtraction because they have the same denominator. This step is crucial to ensure the operations are carried out correctly.
Simplifying Fractions
Simplifying involves reducing the fraction to its smallest form, where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and use. To simplify, identify the greatest common divisor (GCD) for both terms and divide them by this number.
For example, with the fraction \( \frac{112}{1000} \), the GCD of 112 and 1000 is 8. Divide both the numerator and denominator by 8, leaving you with \( \frac{14}{125} \).
Keep in mind:
  • A fraction in its simplest form is often more practical for further calculation.
  • This step doesn't affect the value, just its representation.
Simplifying helps us understand the fraction without changing its value.

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Most popular questions from this chapter

Perform each operation and simplify. $$ 8.8 \div 19 \text { . Round to one decimal place. } $$

Convert each decimal to a proper fraction or a mixed number. \(1.09 \frac{1}{8}\)

Perform the following operations. Exercise 6.10 .26\((0.6)^{2} \cdot\left(\frac{1}{20}-\frac{1}{25}\right)\)

Write each decimal using digits. Fifty-six and thirty-five ten-thousandths.

Convert each decimal to a proper fraction or a mixed number. 0.05
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