Problem 26 - 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. $$\frac{8}{4}=\frac{4}{n}$$

Step-by-Step Solution

Answer

The missing term \( n \) is 2.

Step 1: Understanding the Proportion

We are given the proportion \( \frac{8}{4} = \frac{4}{n} \). This means the ratios on both sides of the equation should be equivalent.

Step 2: Cross-Multiply to Eliminate Fractions

To solve for \( n \), we use cross-multiplication. Multiply the numerator of the first ratio by the denominator of the second ratio, and vice versa. This gives us:\[ 8 \times n = 4 \times 4 \]

Step 3: Solve the Equation

Calculate the product on the right side:\[ 8n = 16 \]Next, solve for \( n \) by dividing both sides of the equation by 8:

Step 4: Simplify the Result

Divide both sides by 8 to isolate \( n \):\[ n = \frac{16}{8} \]Simplify the fraction:\[ n = 2 \]

Key Concepts

Cross-Multiplication

Cross-multiplication is an effective method used to solve equations that contain proportions. It works by eliminating the fractions, making it easier to find the unknown value. In the given example, \( \frac{8}{4} = \frac{4}{n} \), cross-multiplication involves multiplying the numerator of each fraction with the denominator of the other fraction.

• This results in the equation \( 8 \times n = 4 \times 4 \).
• These products are equivalent, enabling us to isolate and solve for \( n \).

Cross-multiplication helps to quickly move from a fractional setting to a simpler algebraic equation, making calculations straightforward.
In general, always ensure both sides of your proportion are cross-multiplied correctly to maintain balance in your equation.

Simplifying Fractions

Simplifying fractions is improving the clarity of your answer by reducing the fraction to its simplest form. When we get a fraction like \( \frac{16}{8} \), it can be simplified because both the numerator and denominator share a greatest common divisor (GCD).

• The key step is to divide both the numerator and denominator by their GCD. In this case, the GCD is 8.
• This reduces \( \frac{16}{8} \) to \( n = 2 \).

Fully simplified fractions are often preferable in mathematics as they provide the answer in its clearest and most concise form.
Also, simplified fractions aid in avoiding calculation errors in further steps, should additional calculations be required.

Ratios in Mathematics

Ratios in mathematics are a way of comparing two quantities. They are expressed as either \( a:b \) or \( \frac{a}{b} \). In proportions, ratios indicate that two fractions are equivalent.

• For example, the problem \( \frac{8}{4} = \frac{4}{n} \) describes two equivalent ratios, implying there is a direct relationship we need to resolve.
• Understanding ratios is critical because it helps in many contexts like scaling recipes in cooking, converting currencies, or determining speed.

Recognizing proportions and solving them enhances problem-solving skills in a wide range of mathematical applications by highlighting the interconnectedness of different values and quantities.