Problem 29 - 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. $$\frac{y}{12}=9$$

Step-by-Step Solution

Answer

The missing term is 108.

Step 1: Understand the Proportion

The given proportion is \( \frac{y}{12} = 9 \). This means that \( y \) divided by 12 equals 9.

Step 2: Rewrite the Proportion as an Equation

To find \( y \), we need to rewrite the proportion value into a simple equation by multiplying both sides by 12: \( y = 9 \times 12 \).

Step 3: Solve for y

Solve the equation \( y = 9 \times 12 \) to find the value of \( y \). By multiplying, we get \( y = 108 \).

Key Concepts

Finding Missing Terms in Proportions

Proportions are equations that express the equality between two ratios. If you are given a proportion and one term is missing, your goal is to determine the unknown term. Let's look at how to find the missing term in the proportion \( \frac{y}{12} = 9 \). In this equation, \( y \) is the missing term, and it is part of the ratio \( \frac{y}{12} \). This means \( y \) divided by 12 gives us 9. To find \( y \), we must manipulate the equation so that \( y \) stands alone on one side of the equation.Understanding where the missing term fits and how it relates to the other numbers in the proportion is key. Remember that a proportion states that two ratios are equivalent, and finding a missing term involves using this principle to create an equation that we can solve.

• Identify where the missing term fits in the proportion.
• Set up an equation to isolate the missing term.

Solving Proportions Step by Step

Once you’ve identified the missing term in a proportion, solving for it involves a few simple steps. Let's take a closer look using our example: \( \frac{y}{12} = 9 \).First, convert the statement of the proportion into an algebraic equation. You do this by performing the same operation on both sides of the equation to maintain equality. Here, you can clear the fraction by multiplying both sides of the equation by 12, the denominator of the fraction:\[ y = 9 \times 12 \]This transformation gives us a straightforward equation that directly relates \( y \) to the known numbers.Finally, solve the equation by performing the arithmetic operation. Multiply 9 by 12:\[ y = 108 \]By following each step methodically, you ensure that the relationship expressed in the proportion holds true in your final answer. This step-by-step approach simplifies complex problems into manageable parts, making it easier to find the correct solution.

• Transform the proportion into an equation.
• Multiply to solve for the unknown variable.

Equations from Proportions

In order to solve proportions, it is helpful to understand how to turn these into equations. A proportion is essentially a balance between two equal ratios, and equations are tools we use to find unknown values. Let's see how this works.With any given proportion, such as \( \frac{y}{12} = 9 \), you create an equation by eliminating the division. This equation can be solved using basic algebraic operations. Here, by eliminating the fraction through multiplication, you obtain an equation that is ready to be solved:\[ y = 9 \times 12 \]This process turns a potentially intimidating proportion problem into a simple multiplication problem. The conversion from proportions to equations applies to all sorts of proportional problems, not just involving unknown numerators or denominators, but also word problems involving direct or inverse relationships.

• Turn ratio expressions into algebraic equations.
• Solve using basic algebraic rules.

By practicing these conversions, you become adept at handling proportions confidently across different scenarios.