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Chapter 4: Problem 668

Translate and solve. The sum of two-thirds and \(n\) is \(-\frac{3}{5}\)

Short Answer

Expert verified
The value of \(n\) is \(-\frac{19}{15}\).

Step by step solution

01

Translate the sentence into an equation

Rewrite the given statement as an algebraic equation. The sentence 'The sum of two-thirds and n is -\frac{3}{5}' can be written as: \[ \frac{2}{3} + n = -\frac{3}{5} \]
02

Isolate the variable

Subtract \( \frac{2}{3} \) from both sides of the equation to isolate \( n \): \[ n = -\frac{3}{5} - \frac{2}{3} \]
03

Find the common denominator

Find a common denominator to combine the fractions. The least common denominator of 5 and 3 is 15:\[ n = -\frac{3}{5} - \frac{2}{3} = -\frac{9}{15} - \frac{10}{15} \]
04

Combine the fractions

Subtract the fractions now that they have a common denominator:\[ n = -\frac{9}{15} - \frac{10}{15} = -\frac{19}{15} \]
05

Simplify the result

The fraction \(-\frac{19}{15}\) is already in its simplest form.

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Key Concepts

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

Algebraic Equations
An algebraic equation is a mathematical statement that shows the equality of two expressions separated by an equal sign (=). In the given exercise, the statement 'The sum of two-thirds and \(n\) is \(-\frac{3}{5}\)' translates to the algebraic equation:

\[ \frac{2}{3} + n = -\frac{3}{5} \]

Understanding how to translate word problems into algebraic equations is crucial. Words like 'sum,' 'difference,' 'product,' and 'quotient' frequently appear in word problems and each has a corresponding arithmetic operation. In this exercise, 'sum' indicates that we are adding two quantities.
Fractions
Fractions represent parts of a whole and consist of a numerator (the top number) and a denominator (the bottom number). In our equation, we have fractions \( \frac{2}{3} \) and \( -\frac{3}{5} \).

When working with fractions in algebraic equations, it's essential to combine them correctly. You often need to find a common denominator before performing operations like addition or subtraction.

Let's not forget basic fraction rules:
  • To add or subtract fractions, they must have a common denominator.
  • The numerator dictates how many parts we have.
  • The denominator indicates how many parts make up a whole.
Understanding these rules will help you manipulate fractions in any algebra problem.
Least Common Denominator
The least common denominator (LCD) is the smallest number that both denominators of two fractions can divide into without leaving a remainder. Finding the LCD is essential when adding or subtracting fractions.

For our equation, we have fractions with denominators 3 and 5. The LCD for 3 and 5 is 15. To combine \( -\frac{3}{5} \) and \( \frac{2}{3} \), convert them to equivalent fractions with the denominator 15:

\[ -\frac{3}{5} = -\frac{9}{15} \]

\[ \frac{2}{3} = \frac{10}{15} \]

Now that the fractions have the same denominator, you can easily combine them.
Isolating Variables
Isolating the variable is a fundamental step in solving algebraic equations. Isolating \( n \) means getting \( n \) by itself on one side of the equation. Let's see this in our problem.

The equation is: \( \frac{2}{3} + n = -\frac{3}{5} \)

To isolate \( n \), subtract \( \frac{2}{3} \) from both sides of the equation:

\[ n = -\frac{3}{5} - \frac{2}{3} \]

Now simplify the right-hand side by finding a common denominator (which we did earlier):

\[ n = -\frac{19}{15} \]

Variable isolation is a key technique in solving any algebraic equation, helping to find the value of the unknown variable.

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Most popular questions from this chapter

In the following exercises, solve the equation. $$ y+\frac{3}{5}=\frac{7}{5} $$

In the following exercises, simplify. $$ \frac{18 r}{27 s} $$

In the following exercises, translate and solve. Three-eighths of \(y\) is 24

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x-\frac{2}{5}\) when (a) \(x=\frac{3}{5} \quad\) (b) \(x=-\frac{3}{5}\)

Convert the mixed number to an improper fraction. $$ 3 \frac{2}{7} $$
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