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Chapter 5: Problem 79

Find each of the following sums and differences. (Add or subtract.) $$8 \frac{5}{10}-2 \frac{4}{100}$$

Short Answer

Expert verified
The result is \(\frac{323}{50}\).

Step by step solution

01

Convert mixed numbers to improper fractions

Start with the mixed numbers \(8 \frac{5}{10}\) and \(2 \frac{4}{100}\). Convert them into improper fractions. For \(8 \frac{5}{10}\), multiply 8 (whole number) by 10 (denominator) and add 5 (numerator). This gives \(\frac{85}{10}\). For \(2 \frac{4}{100}\), multiply 2 (whole number) by 100 (denominator) and add 4 (numerator), resulting in \(\frac{204}{100}\). Thus, the problem becomes \(\frac{85}{10} - \frac{204}{100}\).
02

Find a common denominator

To subtract these fractions, find a common denominator. The denominators are 10 and 100. The least common denominator is 100. Convert \(\frac{85}{10}\) to \(\frac{850}{100}\) by multiplying both the numerator and denominator by 10.
03

Subtract the fractions

Now that both fractions have a common denominator, subtract them: \(\frac{850}{100} - \frac{204}{100} = \frac{646}{100}\).
04

Simplify the fraction

Simplify the resulting fraction \(\frac{646}{100}\). Both the numerator and the denominator are divisible by 2. Dividing, we get \(\frac{323}{50}\). Thus, \(8 \frac{5}{10} - 2 \frac{4}{100} = \frac{323}{50}\).

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Key Concepts

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

Understanding Improper Fractions
Improper fractions are a way to express numbers where the numerator is greater than or equal to the denominator. They are commonly used when dealing with mixed numbers. A mixed number consists of a whole part and a fractional part, like in the example problem \(8 \frac{5}{10}\) and \(2 \frac{4}{100}\). To convert mixed numbers into improper fractions, follow these steps:
  • Multiply the whole number by the denominator of the fraction part.
  • Add the result to the numerator.
  • Write the sum as the new numerator over the original denominator.
By doing this, \(8 \frac{5}{10}\) converts to \(\frac{85}{10}\), and \(2 \frac{4}{100}\) turns into \(\frac{204}{100}\). Improper fractions help in the arithmetic operation of fractions, simplifying processes like addition and subtraction.
Finding a Common Denominator
To perform addition or subtraction with fractions, they must have the same denominator. A common denominator is essential since it allows you to compare or combine fractions properly. In this exercise, the denominators 10 and 100 needed to be unified to perform the subtraction.
  • Determine the least common denominator (LCD) by finding the smallest number that's a multiple of both denominators.
  • For denominators 10 and 100, the LCD is 100.
Next, scale the fractions accordingly so that both share this common denominator. Convert \(\frac{85}{10}\) into \(\frac{850}{100}\) by multiplying both the top and bottom by 10. A common denominator makes it easier to combine the fractions effectively and allows for easy subtraction.
Subtracting Fractions
With a common denominator in place, you can subtract fractions by simply subtracting their numerators and keeping the same denominator. In this case, we transformed the problem to \(\frac{850}{100} - \frac{204}{100}\). Here's how you can solve it:
  • Subtract the numerator of the second fraction from the numerator of the first fraction: \(850 - 204\).
  • Keep the same common denominator: 100.
  • The result was \(\frac{646}{100}\).
Remember, the subtraction of fractions is only manageable when the denominators are identical. After performing the subtraction, always check if the resulting fraction can be simplified further to make it easy to interpret. In this solution, dividing both the numerator and denominator by 2 gives the simplified fraction \(\frac{323}{50}\), which is the final answer.

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Most popular questions from this chapter

Write each fraction as an equivalent fraction with denominator \(100 .\) $$\frac{17}{20}$$

Carry out cach of the following divisions only so far as needed to round the results to the nearest hundredth. $$0 . 0 5 9 \longdiv { 0 . 6 9 }$$

Problems Work each of the following problems on your calculator. If rounding is necessary, round to the nearest hundred thousandth. $$0.0503 \div 0.0709$$

Perform each of the following divisions. $$0 . 0 5 2 \longdiv { 0 . 2 3 7 1 2 }$$

How many hours does a person making \(\$ 6.78\) per hour have to work in order to earn \(\$ 257.64 ?\)
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