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Chapter 4: Problem 544

Mixed Practice, In the following exercises, solve. $$\frac{7}{8} m=\frac{1}{10}$$

Short Answer

Expert verified
m = \(\frac{4}{35}\).

Step by step solution

01

Isolate the Variable

To solve for the variable, isolate it on one side of the equation. In this case, the equation is \ \frac{7}{8} m = \frac{1}{10}. To isolate \(m\), multiply both sides of the equation by the reciprocal of \(\frac{7}{8}\), which is \(\frac{8}{7}\).
02

Multiply Both Sides

Multiply both sides of the equation by \(\frac{8}{7}\). This gives: \ \left(\frac{8}{7}\right) \cdot \frac{7}{8} m = \left(\frac{8}{7}\right) \cdot \frac{1}{10} \ Since \(\frac{8}{7} \cdot \frac{7}{8} = 1\), the left side simplifies to \(m\): \ m = \frac{8}{70}.
03

Simplify the Fraction

Simplify \(\frac{8}{70}\) by finding the greatest common divisor (GCD) of 8 and 70, which is 2. Divide the numerator and the denominator by 2: \ m = \frac{8 \div 2}{70 \div 2} = \frac{4}{35}.

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Key Concepts

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

Isolating the Variable
The first thing we need to understand when solving an equation is how to isolate the variable. Isolating the variable means getting it alone on one side of the equation. This helps us see what value the variable must hold to make the equation true. For the equation \(\frac{7}{8} m = \frac{1}{10}\), we need to isolate \(m\). To do this, we can multiply both sides of the equation by the reciprocal of \(\frac{7}{8}\), which is \(\frac{8}{7}\). The reciprocal is simply flipping a fraction upside down. So, \(\frac{7}{8}\) becomes \(\frac{8}{7}\). When we multiply both sides by \(\frac{8}{7}\), we essentially cancel out the fraction on the left, leaving \(m\) isolated.
Multiplying Reciprocals
Now, let's talk more about multiplying reciprocals. A reciprocal of a fraction is what you get when you flip the fraction's numerator and denominator. For instance, the reciprocal of \( \frac{7}{8} \) is \( \frac{8}{7} \). To solve our equation by isolating the variable, we need to multiply both sides by this reciprocal: \(\frac{8}{7} \times \frac{7}{8} m = \frac{8}{7} \times \frac{1}{10} \). When you multiply a number by its reciprocal, the result is always 1. Therefore, on the left side 鈥� \( \frac{8}{7} \times \frac{7}{8} = 1 \), and we are left with \( m \). The equation now simplifies to: \( m = \frac{8}{70} \).
Simplifying Fractions
The last step involves simplifying fractions. It's important to simplify fractions to their lowest terms for a cleaner and more understandable solution. Our fraction here is \( \frac{8}{70} \). To simplify it, we need to find the greatest common divisor (GCD) of the numerator (8) and the denominator (70). The GCD of 8 and 70 is 2. By dividing both the numerator and the denominator by their GCD, we get the simplest form of the fraction: \( \frac{8 \div 2}{70 \div 2} = \frac{4}{35} \). Now, the value of \( m \) is clearly \( \frac{4}{35} \). Simplifying fractions makes them easier to work with and better understood.

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