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Chapter 1: Problem 72

Solve. $$ \frac{4}{5} t-\frac{3}{10}=\frac{2}{5} $$

Short Answer

Expert verified
\( t = \frac{7}{8} \)

Step by step solution

01

Isolate the variable term

Start by getting all variable terms on one side of the equation. Add \ \( \frac{3}{10} \) to both sides:\ \ \[ \frac{4}{5} t - \frac{3}{10} + \frac{3}{10} = \frac{2}{5} + \frac{3}{10} \] Simplifying this, we get: \ \[ \frac{4}{5} t = \frac{4}{10} + \frac{3}{10} \] Combine the fractions on the right-hand side: \ \[ \frac{4}{5} t = \frac{7}{10} \]
02

Solve for the variable

To solve for \( t \), multiply both sides of the equation by the reciprocal of \( \frac{4}{5} \). The reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \):\ \ \[ t = \frac{7}{10} \cdot \frac{5}{4} \] Simplify the right-hand side: \[ t = \frac{7 \cdot 5}{10 \cdot 4} \] \[ t = \frac{35}{40} \] Reduce the fraction: \[ t = \frac{7}{8} \]

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Key Concepts

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

isolating the variable
When solving linear equations, a key step is isolating the variable. This means we need to get the variable (in this case, \( t \)) on one side of the equation and all other terms on the opposite side.
Think of it as 'freeing' the variable to make it stand alone. This will allow us to directly find its value.
For example, in the problem \( \frac{4}{5} t - \frac{3}{10} = \frac{2}{5} \), we start by adding \( \frac{3}{10} \) to both sides to eliminate it from the left side:
  • Addition helps in neutralizing or canceling out subtracted fractions.
By combining like terms on the right side:
\( \frac{4}{5} t = \frac{2}{5} + \frac{3}{10} \).
Simplifying the right side further enables us to properly isolate the variable term.
fractions
In the provided equation, fractions are present on both sides, which can be tricky.
To handle them correctly, remember:
  • Always find a common denominator for addition or subtraction.

  • Simplify fractions whenever possible.
For instance, adding \( \frac{2}{5} \) and \( \frac{3}{10} \) requires us to find a common denominator (in this case, 10).
This converts \( \frac{2}{5} \) into \( \frac{4}{10} \):
  • \( \frac{4}{10} + \frac{3}{10} = \frac{7}{10} \)
This makes combining fractions and simplifying easier. Understanding this process ensures clarity in solving linear equations with fractions.
reciprocal
Once the variable is isolated, you often need to multiply by the reciprocal to solve for it. The reciprocal of a fraction is simply flipping its numerator and denominator.
Taking our example, the reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \):
  • To solve \( \frac{4}{5} t = \frac{7}{10} \), multiply both sides by \( \frac{5}{4} \).
  • This cancels the \( \frac{4}{5} \) on the left side, leaving \( t \).
Mathematically, it looks like:
  • \( t = \frac{7}{10} \times \frac{5}{4} \).
Simplify the multiplication:
\( t = \frac{7 \cdot 5}{10 \cdot 4} = \frac{35}{40} \).
Reducing the fraction gives us our solution: \( t = \frac{7}{8} \).

Understanding the concept of reciprocals is crucial for solving equations efficiently when dealing with fractions.

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Most popular questions from this chapter

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