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Chapter 6: Problem 7

In each problem, state what \(x\) represents, write an equation, and answer the question. In a certain fraction, the denominator is 6 more than the numerator. If 3 is added to both the numerator and the denominator, the resulting fraction is equivalent to \(\frac{5}{7} .\) What was the original fraction (not written in lowest terms)?

Short Answer

Expert verified
\( \frac{12}{18} \)

Step by step solution

01

- Define the variable

Let the numerator of the original fraction be represented by the variable \( x \).
02

- Express the denominator in terms of x

The denominator is 6 more than the numerator. Therefore, the denominator can be expressed as \( x + 6 \).
03

- Formulate the new fraction

If 3 is added to both the numerator and the denominator, the new numerator is \( x + 3 \) and the new denominator is \( x + 6 + 3 = x + 9 \). Thus, the new fraction is \( \frac{x+3}{x+9} \).
04

- Set up the equation

It is given that this new fraction is equivalent to \( \frac{5}{7} \). Therefore, we can write the equation: \[ \frac{x+3}{x+9} = \frac{5}{7} \]
05

- Solve the equation

Cross-multiply to solve for \( x \): \[ 7(x+3) = 5(x+9) \] Expand and simplify: \[ 7x + 21 = 5x + 45 \] Subtract \( 5x \) from both sides: \[ 2x + 21 = 45 \] Subtract 21 from both sides: \[ 2x = 24 \] Finally, divide both sides by 2: \[ x = 12 \]
06

- Find the denominator and original fraction

The original numerator is \( x = 12 \), and the denominator is \( 12 + 6 = 18 \). Therefore, the original fraction is \( \frac{12}{18} \), which is not in lowest terms.

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Key Concepts

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

fractions
In algebra, fractions represent a division of numbers or expressions. A fraction consists of a numerator and a denominator, separated by a horizontal line. The numerator is above the line, indicating how many parts we have. The denominator, below the line, shows the total number of equal parts. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator. This means we have 3 out of 4 equal parts.
cross-multiplication
Cross-multiplication is a method used to solve equations involving fractions. When we have an equation like \( \frac{a}{b} = \frac{c}{d} \), we can cross-multiply to eliminate the fractions. This involves multiplying the numerator of one fraction by the denominator of the other, and vice-versa. Thus, we write:
  • \( a \cdot d = b \cdot c \)
This creates a simpler equation without fractions, allowing us to solve for the variable more easily. For example, in our exercise, we have the equation \( \frac{x+3}{x+9} = \frac{5}{7} \). Cross-multiplying gives us:
  • \( 7(x + 3) = 5(x + 9) \)
We then solve this new equation.
variable representation
In algebra, variables represent unknown values that we aim to find. The variable is often denoted by letters like \( x \), \( y \), or \( z \). In our problem, we begin by letting \( x \) represent the numerator of the fraction. Once defined, we express other related quantities using this variable. For instance, the denominator is defined as '6 more than the numerator', which we write as \( x + 6 \). By setting up the problem with variables, we can form an equation based on the given conditions and solve accordingly.
simplifying equations
Simplifying equations involves performing operations to combine like terms and isolate the variable. In our example, once we have the cross-multiplied equation \( 7(x + 3) = 5(x + 9) \), we expand and simplify:
  • \( 7x + 21 = 5x + 45 \)
  • Subtract \( 5x \) from both sides: \( 2x + 21 = 45 \)
  • Subtract 21 from both sides: \( 2x = 24 \)
  • Divide both sides by 2: \( x = 12 \)
By following these steps, we simplify the equation and solve for the variable \( x = 12 \). Finally, substituting \( x \) back gives us the original numerator and denominator, allowing us to find the original fraction.

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Most popular questions from this chapter

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ 7-\frac{3}{5+\frac{2}{4-2}} $$

When solving an equation with variables in denominators, we must determine the values that cause these denominators to equal \(0,\) so that we can reject these values if they appear as proposed solutions. Find all values for which at least one denominator is equal to \(0 .\) Write answers using the symbol \(\neq\). Do not solve. $$ \frac{7}{x}+\frac{9}{x-4}=5 $$

Simplify each complex fraction. Use either method. $$ \frac{6+\frac{2}{r}}{\frac{3 r+1}{4}} $$

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These exercises involve factoring sums and differences of cubes. Write each rational expression in lowest terms. $$ \frac{x^{3}-27}{x-3} $$
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