Problem 10 - 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. $$\frac{3}{8}=\frac{9}{x}$$

Step-by-Step Solution

Answer

The missing term \( x \) is 24.

Step 1: Understand the Proportion

We are given the proportion \( \frac{3}{8} = \frac{9}{x} \). Our task is to find the value of \( x \) that makes this equation true. A proportion means that the two ratios are equivalent.

Step 2: Cross Multiply to Clear the Fraction

To remove the fractions, cross-multiply: multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction. This gives us:\[ 3 \cdot x = 8 \cdot 9 \]

Step 3: Simplify the Equation

Simplify the multiplication on both sides of the equation:\[ 3x = 72 \]

Step 4: Solve for x

Isolate \( x \) by dividing both sides of the equation by 3:\[ x = \frac{72}{3} \]

Step 5: Simplify the Solution

Divide 72 by 3 to find \( x \):\[ x = 24 \]. Thus, the solution in lowest terms is \( x = 24 \).

Key Concepts

Cross Multiplication

Cross multiplication is a powerful technique used to solve proportions, which are equations that show two ratios are equal. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross multiplication helps you eliminate the fractions.
In simpler terms, you multiply across the equation: the numerator of the first fraction (\(a\)) with the denominator of the second fraction (\(d\)), and the denominator of the first fraction (\(b\)) with the numerator of the second fraction (\(c\)). Put mathematically, you get:

• \( a \cdot d = b \cdot c \)

This simplification turns a fraction-based problem into a straightforward multiplication problem, making it much easier to solve for the unknown variable. Cross multiplication is especially useful when you need to solve real-world problems involving ratios and proportions.

Solving Equations

Once you have used cross multiplication to clear the fractions, the next step is to solve the equation. Imagine you've simplified the equation to something like \( 3x = 72 \).
To find the value of \(x\), you need to isolate it on one side of the equation. This usually involves dividing both sides by a number to solve for \(x\). In our example:

• Divide both sides by 3 to get \( x = \frac{72}{3} \).

The goal is to have \(x\) alone on one side of the equation so you know exactly what it equals. Solving the equation accurately gives you the answer to the problem at hand.
Remember, practice is key. Repeat these steps in multiple problems to strengthen your equation-solving skills.

Fractions in Lowest Terms

When solving proportions, often the final step is to simplify your answer, ensuring it's in its simplest form. This is known as reducing a fraction to its lowest terms.
For example, if your solution is \( \frac{72}{3} \), you simplify by performing the division to obtain \(x = 24\), which is already in its simplest form as there are no common divisors between 24 and 1.
Here are some tips to reduce a fraction to its lowest terms:

• Divide the numerator and the denominator by their greatest common divisor (GCD).
• If the result of the division is a whole number, then it is already in its simplest form.

Reducing fractions is important because it helps in comparing ratios, simplifying expressions, and ensuring answers are as straightforward as possible.