Problem 29 - Prealgebra

Question

Solve each proportion. $$\frac{3}{14}=\frac{15}{m-3}$$

Step-by-Step Solution

Answer

The value of \( m \) is 73.

Step 1: Understand the Proportion

A proportion is an equation that shows two ratios are equal. In this case, the proportion is \( \frac{3}{14} = \frac{15}{m-3} \). We know that the cross products of a proportion are equal.

Step 2: Set up the Cross Products

The cross products of the proportion \( \frac{3}{14} = \frac{15}{m-3} \) are \( 3(m-3) = 15 \times 14 \).

Step 3: Calculate the Right-hand Side

Calculate \( 15 \times 14 \) to get \( 210 \). The equation becomes \( 3(m-3) = 210 \).

Step 4: Distribute on the Left-hand Side

Distribute the 3 through the parentheses: \( 3 \times m - 3 \times 3 = 210 \), which simplifies to \( 3m - 9 = 210 \).

Step 5: Solve for m

Add 9 to both sides to isolate the term with \( m \): \( 3m = 219 \). Then divide both sides by 3 to solve for \( m \): \( m = \frac{219}{3} = 73 \).

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve proportions. When you have a proportion, it means you are dealing with an equation that shows two ratios being equal. For example, in the problem \( \frac{3}{14} = \frac{15}{m-3} \), the two fractions are equal to each other.
To solve proportions using cross multiplication, you multiply the numerator of one fraction by the denominator of the other. This results in two cross products that you then set equal to each other. In this example:

• Multiply the numerator of the first fraction by the denominator of the second: \( 3(m-3) \)
• Multiply the numerator of the second fraction by the denominator of the first: \( 15 \times 14 \)

The newly formed equation is \( 3(m-3) = 15 \times 14 \), and these two products must be equal because that's what a proportion signifies. By solving this new equation, we can find the value of \( m \). This method is incredibly useful because it turns the problem into a simple equation with distributive steps you are already familiar with.

Equations

Equations are mathematical statements that assert the equality of two expressions. In the context of solving proportions, we create an equation through cross multiplication. A clear example is the equation we derive from the proportion \( \frac{3}{14} = \frac{15}{m-3} \) which becomes \( 3(m-3) = 210 \) after cross multiplying.

Understanding how to work with equations is fundamental because they are the language of algebra. You manipulate them by performing the same operations on both sides, maintaining balance. For this example, you first need to expand and simplify one side using distribution, and then isolate the variable "\( m \)" by performing mathematical operations such as addition or subtraction and division or multiplication.By practicing these steps, you become adept at solving for unknowns, which broadens your problem-solving skills across various math disciplines.

Ratios

Ratios are comparisons between two numbers showing the relative sizes of two quantities. They are often expressed as fractions, as seen in the exercise \( \frac{3}{14} \) and \( \frac{15}{m-3} \).

Understanding ratios is crucial for breaking down problems, such as proportions, into solvable components. When we state that two ratios are equal, we have a proportion, which implies a special relationship. In our example, the ratio of 3 candies to 14 kids is the same as 15 candies to \( m-3 \) kids. To solve it, understanding that ratios are essentially fractions, and thus can be treated with operations like cross multiplication, simplifies the task. Knowing how to work with ratios enables you to solve a wide array of problems from real-world scenarios, where direct comparisons are necessary.

Linear Equations

Linear equations are equations that make a straight line when graphed, typically represented in the form \( ax + b = c \). Solving proportions often leads to a linear equation, as seen when we equate the cross products of our proportion. For instance, the equation \( 3m - 9 = 210 \) is a linear equation.

To solve this linear equation, you need to isolate the variable. This process includes simplifying the equation on one side and performing basic operations to both sides to keep the equation balanced, such as:

• Adding or subtracting constants
• Multiplying or dividing by coefficients

In our example, after distributing, we add 9 to each side to get \( 3m = 219 \), which is a simpler linear equation. Finally, dividing each side by 3 results in \( m = 73 \). Mastery of solving linear equations not only aids in proportions but is a vital skill across all algebraic problem-solving.