Problem 11 - Prealgebra

Question

Find the missing term in each of the following proportions. Set up each problem like the examples in this section. Write your answers as fractions in lowest terms. $$\frac{1}{y}=\frac{5}{12}$$

Step-by-Step Solution

Answer

The missing term is \( y = \frac{12}{5} \).

Step 1: Cross-multiply the terms

To find the missing term in the proportion \( \frac{1}{y} = \frac{5}{12} \), we'll start by cross-multiplying the numerators and denominators. This gives us the equation: \( 1 \times 12 = 5 \times y \).

Step 2: Simplify the equation

Simplify the equation from Step 1: \( 12 = 5y \). This step involves checking our multiplication and writing out the equation clearly.

Step 3: Solve for \( y \)

To find \( y \), divide both sides of the equation by 5: \( y = \frac{12}{5} \).

Step 4: Check if the fraction is in lowest terms

The fraction \( \frac{12}{5} \) is already in its lowest terms because there are no common factors between 12 and 5 other than 1.

Key Concepts

Cross-Multiplication

Understanding cross-multiplication is a pivotal part of solving proportions. Cross-multiplication is a method used to solve equations that have two fractions set equal to each other, such as \( \frac{a}{b} = \frac{c}{d} \). Here’s how it works:

• Take the numerator of the first fraction \(a\) and multiply it by the denominator of the second fraction \(d\), giving you \( a \times d \).
• Then, take the numerator of the second fraction \(c\) and multiply it by the denominator of the first fraction \(b\), giving you \( b \times c \).

This approach offers a straightforward path to finding unknown values within proportions by transforming a two-fraction equation into a simple linear equation. For the equation \( \frac{1}{y} = \frac{5}{12} \), the cross-multiplication is calculated as: \[1 \times 12 = 5 \times y\]This method essentially eliminates the fractions, making it easier to solve for any variable in the equation.

Solving for a Variable

Solving for a variable means isolating it on one side of the equation to find its value. Once you've performed cross-multiplication, you're left with a linear equation like \( 12 = 5y \). The goal is to get "\(y\)" alone on one side of the equation. Here's how:

• First, identify the operation being applied to the variable. In this case, \(y\) is being multiplied by 5.
• To isolate \(y\), you'll need to perform the inverse operation. Since \(y\) is multiplied by 5, you divide both sides of the equation by 5: \[\frac{12}{5} = y\]

Now, \(y\) is isolated, and you have successfully solved the equation. This step illustrates a crucial algebraic technique that you will use frequently in various math problems. Practice and familiarity with this process will help you gain confidence and accuracy in solving equations involving variables.

Simplifying Fractions

After solving an equation for a variable, you often encounter fractions that need to be simplified. To simplify a fraction, you find the greatest common divisor (GCD) of the numerator and the denominator. Then, divide both by their GCD to reduce the fraction to its simplest form.
For instance, \( \frac{12}{5} \) needs to be checked if it can be simplified. Here's how to do it:

• List the factors of the numerator (12): 1, 2, 3, 4, 6, 12.
• List the factors of the denominator (5): 1, 5.
• Identify any common factors. The only common factor between 12 and 5 is 1.

Since the only common factor is 1, \( \frac{12}{5} \) is already in its simplest form. Simplifying fractions is a key skill to ensure that answers are neat and easy to understand, making your solutions clearer and more professional.