Problem 28 - Prealgebra

Question

Solve each proportion. $$\frac{n}{20}=\frac{15}{50}$$

Step-by-Step Solution

Answer

The value of \( n \) is 6.

Step 1: Understand the Proportion

A proportion states that two ratios are equal. In this problem, we have \( \frac{n}{20} = \frac{15}{50} \). Our goal is to find the value of \( n \) that makes this proportion true.

Step 2: Cross-Multiply

To solve the proportion, use cross-multiplication. Multiply the numerator of the first ratio by the denominator of the second ratio and vice versa. This gives us the equation: \( n \times 50 = 20 \times 15 \).

Step 3: Simplify the Multiplications

Simplify the multiplication on both sides of the equation from Step 2. Calculate \( 20 \times 15 \) to get 300, so the equation is \( 50n = 300 \).

Step 4: Solve for n

To solve for \( n \), divide both sides of the equation \( 50n = 300 \) by 50. This results in \( n = \frac{300}{50} \).

Step 5: Simplify the Result

Calculate \( \frac{300}{50} \), which simplifies to \( 6 \). Therefore, \( n = 6 \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful technique used to solve problems involving proportions or equal ratios. It is a method where you multiply the numerator of one ratio by the denominator of the other ratio, setting the two products equal to each other. This technique helps to eliminate fractions, converting the problem into a simpler equation. For example, in the proportion \( \frac{n}{20} = \frac{15}{50} \), you perform cross-multiplication as follows:

• Multiply the numerator 'n' from the first ratio by the denominator '50' of the second ratio, giving us \( n \times 50 \).
• Multiply the numerator '15' of the second ratio by the denominator '20' of the first ratio, resulting in \( 20 \times 15 \).

This results in the equation \( 50n = 300 \), making it easier to solve for \( n \). Cross-multiplication is especially useful in ensuring the two ratios or fractions represent the same value, helping you to find unknown variables effectively.

Ratios

Ratios compare two quantities, providing a way to show how much one value relates to another. A ratio can be written in different forms such as \( a:b \), \( \frac{a}{b} \), or using the word 'to', as in 'a to b'. Ratios are foundational to understanding proportions, as they allow us to set up equations demonstrating how different quantities scale in relation to each other. In our exercise, the ratio \( \frac{n}{20} \) was compared to \( \frac{15}{50} \). This indicates the relationship between the two quantities and sets the stage for using cross-multiplication to solve for \( n \). Knowing how to interpret and manipulate ratios is integral to solving mathematical problems, especially those involving direct comparisons between different quantities.

Solving Equations

Solving equations is about finding the value of an unknown variable that makes the equation true. In the context of proportions, once cross-multiplication has converted the proportion into an equation, the next step is to solve for the unknown variable. Let's look at our example:

• After performing cross-multiplication, we get \( 50n = 300 \).
• The goal is to isolate the variable \( n \). This can be done by performing operations that simplify the equation, such as division or subtraction.
• Dividing both sides of the equation by \( 50 \) gives us \( n = \frac{300}{50} \).

Finally, simplify the resulting fraction to find that \( n = 6 \). Solving equations requires understanding the operations needed to keep the equation balanced, ensuring that both sides of the equation yield an equivalent expression. Mastering this skill allows you to solve various mathematical problems with confidence.