Problem 81 - Prealgebra

Question

Solve each proportion. $$\frac{m}{10}=\frac{9.4}{100}$$

Step-by-Step Solution

Answer

The value of \( m \) is \( \frac{47}{50} \).

Step 1: Understand the Proportion

A proportion means that two ratios or fractions are equal to each other. Here, you have \( \frac{m}{10} = \frac{9.4}{100} \). Our goal is to find the value of \( m \) that makes this equation true.

Step 2: Cross Multiply

To solve the equation \( \frac{m}{10} = \frac{9.4}{100} \), we can use cross multiplication. This means we multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the numerator of the second fraction multiplied by the denominator of the first fraction. So, we get:\[ m \times 100 = 9.4 \times 10 \]

Step 3: Simplify the Equation

Now perform the multiplications on both sides of the equation. Multiply \( m \) by 100 to get \( 100m \), and multiply 9.4 by 10 to get 94. So, the equation becomes:\[ 100m = 94 \]

Step 4: Solve for m

Divide both sides of the equation by 100 to solve for \( m \). This gives us:\[ m = \frac{94}{100} \]

Step 5: Simplify the Fraction

Finally, simplify the fraction \( \frac{94}{100} \) by finding the greatest common factor of 94 and 100, which is 2. Divide both the numerator and the denominator by 2 to get \( \frac{47}{50} \). So, \( m = \frac{47}{50} \).

Key Concepts

Cross Multiplication

Proportions are fascinating because they show that two fractions are equal to each other. When confronted with a proportion like \( \frac{m}{10} = \frac{9.4}{100} \), cross multiplication becomes a very handy tool to find unknown variables, such as \( m \). This method is often used because it eliminates the fractions, making the equation easier to solve.

Cross multiplication works by multiplying across the equal sign, diagonally. Here's the step-by-step approach:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.
• Set these two products equal to each other to form a new equation without fractions.

In this exercise, the multiplication results in the equation \( m \times 100 = 9.4 \times 10 \), simplifying the process and paving the way to solve for \( m \). It's like magically transforming a fraction equation into a linear one, which can be simpler to handle!

Ratios

Ratios are the backbone of proportions and everyday life comparisons. They tell us how much of one thing there is compared to another. Consider the ratio \( \frac{m}{10} \), it's saying "for every 10 parts, there are \( m \) parts," and similarly, \( \frac{9.4}{100} \) states "for every 100 parts, there are 9.4 parts."

Understanding ratios can clarify many real-world situations:

• Ratios are useful in cooking, where you may need to double or halve a recipe.
• In business, ratios help analyze financial health, such as comparing assets to liabilities.
• In classrooms, ratios describe the student-to-teacher ratio.

In this problem, recognizing \( \frac{m}{10} = \frac{9.4}{100} \) as two connected ratios helps us utilize techniques like cross multiplication to find the missing value \( m \), thereby balancing the equation.

Solving Equations

Solving equations is all about finding unknown values that make a mathematical statement true. Once you have used cross multiplication to remove the fractions in the equation \( 100m = 94 \), the task now shifts to isolating the unknown variable \( m \).

Here's how we continue solving the equation:

• Perform arithmetic operations to isolate \( m \). In this case, divide both sides by 100.
• This results in \( m = \frac{94}{100} \).
• Finally, to simplify the fraction, find the greatest common factor of the numerator and the denominator, which is 2 in this case.
• Divide both by 2 to simplify the fraction to \( \frac{47}{50} \).

This simplification process is critical as it reduces the fraction to its simplest form, making the solution clear and concise. Mastery in solving equations, especially involving fractions, provides strength in mathematical reasoning and problem-solving.