Problem 13 - Prealgebra

Question

Solve each proportion. $$\frac{2}{15}=\frac{c}{72}$$

Step-by-Step Solution

Answer

The solution for \( c \) in the proportion is \( \frac{48}{5} \).

Step 1: Understand a Proportion

A proportion is an equation stating that two ratios are equivalent. In this case, we have \( \frac{2}{15} = \frac{c}{72} \), where \( c \) is the unknown we need to solve for.

Step 2: Cross Multiply

To solve the proportion, use cross multiplication. This means multiplying the numerator of the first ratio by the denominator of the second ratio, and multiplying the denominator of the first ratio by the numerator of the second ratio. This gives us: \( 2 \times 72 = 15 \times c \).

Step 3: Perform the Multiplications

Calculate both sides of the equation: \( 2 \times 72 = 144 \) and \( 15 \times c = 15c \). This simplifies the equation to: \( 144 = 15c \).

Step 4: Solve for c

To isolate \( c \), divide both sides of the equation by 15: \( c = \frac{144}{15} \).

Step 5: Simplify the Fraction

Simplify \( \frac{144}{15} \) by finding the greatest common divisor (GCD) of 144 and 15, which is 3. Divide both the numerator and the denominator by 3: \( \frac{144 \div 3}{15 \div 3} = \frac{48}{5} \).

Key Concepts

Cross Multiplication Magic

Cross multiplication is a powerful tool to tackle proportions with ease. When you're faced with an equation that compares two ratios, cross multiplication provides a straightforward way to find an unknown value.

Simply put:

• You multiply the "cross" terms across the equals sign.
• The numerator of the first fraction is multiplied by the denominator of the second.
• The denominator of the first fraction is multiplied by the numerator of the second.

In the proportion \( \frac{2}{15} = \frac{c}{72} \), applying cross multiplication yields \( 2 \times 72 = 15 \times c \). This operation helps us transform the proportion into an easier equation, \( 144 = 15c \), which can be solved to find the unknown \( c \).

This method is incredibly reliable and tends to be the go-to strategy for solving proportions.

Understanding Ratios and Proportions

Ratios and proportions are foundational concepts in math. They help us compare quantities and understand relationships in a meaningful way.
A **ratio** is simply a comparison of two numbers. For instance, the ratio \( \frac{2}{15} \) tells us that for every 2 of one quantity, there's 15 of the other.

When ratios are set to be equal, we call them **proportions**. So, when we say \( \frac{2}{15} = \frac{c}{72} \), we're stating that the two ratios represent the same relationship.

• If one part of a proportion changes, the other must adjust accordingly to maintain equality.
• Proportions are solved to find unknown values that maintain these relationships.

Recognizing proportions allows us to deal with real-world problems, such as scaling recipes or adjusting the terms in a contract.

Fraction Simplification Explained

Fraction simplification is all about making fractions cleaner and easier to work with. When we get to the stage where we have a fraction like \( \frac{144}{15} \), the goal is to reduce it.

The first step is finding the Greatest Common Divisor (GCD) of the numerator and the denominator. For \( 144 \) and \( 15 \), the GCD is 3. By dividing both the numerator and the denominator by this number, we simplify the fraction:

• Cuts down unnecessary complexity in ratios.
• Makes the numbers more manageable.

This gives us \( \frac{48}{5} \). Simplification helps in understanding and communicating numerical relationships more clearly.

What Are Equivalent Ratios?

Equivalent ratios are ratios that express the same relationship between numbers, even if the numbers themselves look different. Think of them as different-looking twins.
For example, let's use the ratios \( \frac{2}{15} \) and \( \frac{48}{360} \). Though not identical at first glance, they are equivalent because they simplify to the same ratio.

Finding equivalent ratios is about scaling up or down:

• If you multiply or divide both terms of a ratio by the same non-zero number, you get an equivalent ratio.
• Helpful in comparing different scenarios that express a similar ratio.

Knowing this helps us use ratios flexibly in various contexts, from deciding the size of a garden plot to scaling ingredients in cooking.