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Chapter 11: Problem 32

Solve using the addition principle. $$ t+\frac{3}{8}=\frac{5}{8} $$

Short Answer

Expert verified
t = \( \frac{1}{4} \)

Step by step solution

01

- Understand the Equation

The given equation is \[ t + \frac{3}{8} = \frac{5}{8} \]. The goal is to solve for the variable \( t \).
02

- Apply the Addition Principle

To isolate \( t \), subtract \( \frac{3}{8} \) from both sides of the equation. This gives: \[ t = \frac{5}{8} - \frac{3}{8} \].
03

- Simplify

Subtract the fractions on the right side: \[ t = \frac{5 - 3}{8} \]. This simplifies to: \[ t = \frac{2}{8} \].
04

- Reduce the Fraction

Reduce the fraction \( \frac{2}{8} \) by dividing the numerator and the denominator by 2: \[ t = \frac{2 \div 2}{8 \div 2} = \frac{1}{4} \].

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Key Concepts

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

Solve Equations
Solving equations is all about finding the value of the variable that makes the equation true. To do this, apply various algebraic principles, such as the addition principle, which can help isolate the variable. The main goal is to perform operations that simplify the equation, ultimately determining the variable's value.
In our example, we started with the equation \( t + \frac{3}{8} = \frac{5}{8} \). By isolating \( t \), we reveal its value through a series of algebraic steps.
Fractions
A fraction represents a part of a whole and consists of two numbers: the numerator and the denominator. Understanding fractions is important when solving equations that contain them. In our example, we dealt with fractions \( \frac{3}{8} \) and \( \frac{5}{8} \), which share the same denominator.Being comfortable with fractions prepares you for operations like addition, subtraction, multiplication, and division involving fractions.
Simplifying Fractions
Simplifying fractions means expressing them in their simplest form. This often makes calculations easier and the results clearer. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
In our example, simplifying \( t = \frac{5-3}{8} \) leads to \( t = \frac{2}{8} \). Focusing on maintaining simplicity gives a much clearer understanding of the numbers involved.
Reducing Fractions
Reducing fractions involves bringing a fraction to its lowest terms. This is done by finding the greatest common divisor (GCD) of both the numerator and the denominator. After identifying the GCD, divide both the numerator and denominator by it.
For instance, in our example, we reduced \( \frac{2}{8} \ to \frac{1}{4} \). The GCD of 2 and 8 is 2, so dividing both by 2 yields the simplest form, \frac{1}{4} \. Reducing fractions not only makes them simpler but also easier to understand and work with in further calculations.

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