/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 Solve each equation. $$\frac{7... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 60

Solve each equation. $$\frac{7}{n}=0.25$$

Short Answer

Expert verified
The solution is \( n = 28 \).

Step by step solution

01

Understand the Equation

We start with the equation \( \frac{7}{n} = 0.25 \). This means that 7 divided by some number \( n \) equals 0.25. Our goal is to find the value of \( n \).
02

Eliminate the Fraction

To isolate \( n \), we can multiply both sides of the equation by \( n \) to remove the fraction. This gives us: \[ 7 = 0.25n \]
03

Solve for n

Now, we need to solve for \( n \) by isolating it on one side. Divide both sides by 0.25:\[ n = \frac{7}{0.25} \]
04

Calculate the Value

Finally, calculate the division \( \frac{7}{0.25} \). Dividing by a fraction is the same as multiplying by its reciprocal, so this is equivalent to:\[ n = 7 \times 4 = 28 \]

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Key Concepts

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

Fraction to Decimal Conversion
When you come across a fraction in an equation, it's often helpful to understand its decimal form. A decimal provides a clear, easily readable form of the fraction, especially useful when solving equations. For instance, in our problem, we are given the fraction \( \frac{7}{n} = 0.25 \). Here 0.25 is already in decimal form, representing one quarter.
  • Fractions consist of a numerator (top number) and a denominator (bottom number). To convert a fraction to a decimal, divide the numerator by the denominator.
  • Examples include: \( \frac{1}{4} = 0.25 \), \( \frac{1}{2} = 0.5 \), and \( \frac{3}{4} = 0.75 \).
Converting fractions to decimals can simplify arithmetic operations and make it easier to solve equations.
Equation Isolation
Isolating the variable in an equation is a crucial step in solving for that variable. It's like uncovering the mystery number hidden by layers of operations!

In the equation \( \frac{7}{n} = 0.25 \), we aim to find out what \( n \) is. We start by removing any fractions or variables that are not part of our desired solution:
  • Initially, multiply both sides of the equation by \( n \), eliminating the fraction. This gives us the equation \( 7 = 0.25n \).
  • Then, divide both sides by 0.25 to isolate \( n \) on one side of the equation, resulting in \( n = \frac{7}{0.25} \).
This step ensures that \( n \) is all by itself, ready to be processed further to find its value.
Reciprocal Multiplication
Once you've isolated the variable, you need to solve it, often requiring the multiplication by a reciprocal. This is a key aspect when dealing with division in equations.
  • The reciprocal of a number is essentially one divided by that number, flipping numbers in a fraction. For example, the reciprocal of \( 0.25 \) is \( \frac{1}{0.25} = 4 \).
  • Applying this to our equation \( n = \frac{7}{0.25} \) results in \( n = 7 \times 4 \).
  • This multiplication step simplifies the fraction problem, avoiding division altogether, and gives the clear result \( n = 28 \).
Understanding reciprocal multiplication helps simplify many mathematical processes, offering an intuitive way to handle division by fractions.

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