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Chapter 4: Problem 71

Simplify the expressions. $$\frac{3}{4} a-\frac{1}{8} a$$

Short Answer

Expert verified
\( \frac{5}{8} a \)

Step by step solution

01

Identify Common Terms

Notice that both terms, \( \frac{3}{4} a \) and \( \frac{1}{8} a \), share the common variable \( a \).
02

Express Both Fractions with a Common Denominator

To combine the fractions, they must have a common denominator. The least common multiple of 4 and 8 is 8. Convert \( \frac{3}{4} \) to a fraction with denominator 8 by multiplying both the numerator and denominator by 2: \( \frac{3}{4} = \frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8} \). Hence, we have \( \frac{6}{8} a \).
03

Subtract the Fractions

Now, subtract \( \frac{1}{8} a \) from \( \frac{6}{8} a \). This results in \[ \frac{6}{8} a - \frac{1}{8} a \]. Since the denominators are the same, subtract the numerators: \[ \frac{6-1}{8} a = \frac{5}{8} a \].
04

Simplify

The expression is now simplified to \( \frac{5}{8} a \).

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Key Concepts

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

Common Denominators
When working with fractions, combining them often requires a common denominator. A common denominator is a shared multiple of the denominators of the fractions involved.
If you have fractions with denominators 4 and 8, the least common denominator is the smallest number both denominators can divide into without leaving a remainder.
  • The common denominator for 4 and 8 is 8 because it's the smallest number that both can evenly divide.
  • Transform each fraction to have the common denominator by multiplying the numerator and denominator of each fraction by the same number.
  • For example, to convert \(\frac{3}{4}\) to have a denominator of 8, multiply both the numerator and denominator by 2: \(\frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8}\).
This step makes it easier to perform operations like addition and subtraction on fractions.
Fraction Subtraction
Subtracting fractions becomes straightforward once they have a common denominator. Here's the process:
After ensuring all fractions have the same denominator, simply subtract the numerators while keeping the common denominator.
  • In our exercise, we have \(\frac{6}{8} a\) and \(\frac{1}{8} a\).
  • We subtract the numerators: 6 - 1 = 5, which gives us the new fraction \(\frac{5}{8} a\).
This is particularly important because the fractions must represent the same part-size of a whole for the subtraction to make sense.
So, remember always to find a common denominator first.
Algebraic Terms
Algebraic terms involve variables and coefficients. An algebraic term consists of:
  • A coefficient, which is a number that multiplies the variable. For example, in \(\frac{3}{4} a\), \(\frac{3}{4}\) is the coefficient.
  • A variable, which is a symbol that represents an unknown or a value that can change. Here, 'a' is the variable.
When simplifying expressions, you combine like terms, which are terms that have the same variables raised to the same power.
For our exercise, we combined \(\frac{3}{4} a\) and \(\frac{1}{8} a\) because they both have the variable 'a'. Once you have combined the fractions and simplified them, you are left with a simplified algebraic term, in this case, \(\frac{5}{8} a\).

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Most popular questions from this chapter

For a recent year, the population of the United States was reported to be \(296,000,000\). During the same year, the population of California was \(36,458,000\). a. Round the U.S. population to the nearest hundred million. b. Round the population of California to the nearest million. c. Using the results from parts (a) and (b), write a simplified fraction showing the portion of the U.S. population represented by California.

Simplify to lowest terms by first reducing the powers of 10. $$\frac{9800}{28,000}$$

Convert the improper fraction to a mixed number. $$-\frac{43}{18}$$

Simplify to lowest terms by first reducing the powers of 10. $$\frac{50,000}{65,000}$$

Plot the fraction on the number line. $$\frac{7}{6}$$
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