Problem 61 - Algebra 2

Question

PREREQUISITE SKILL Use cross products to solve each proportion. $$ \frac{4}{20}=\frac{1}{m} $$

Step-by-Step Solution

Answer

The value of \( m \) is 5.

Step 1: Understanding the Proportion

The given proportion is \( \frac{4}{20} = \frac{1}{m} \). A proportion is an equation stating that two ratios are equal.

Step 2: Cross Multiply to Eliminate Fractions

To solve the proportion \( \frac{4}{20} = \frac{1}{m} \) using cross multiplication, multiply the numerator of the first fraction by the denominator of the second and vice versa: \( 4 \times m = 20 \times 1 \). This results in the equation \( 4m = 20 \).

Step 3: Solve for the Variable

To find the value of \( m \), divide both sides of the equation \( 4m = 20 \) by 4: \( m = \frac{20}{4} \).

Step 4: Calculate the Result

Divide 20 by 4 to solve for \( m \): \( m = 5 \).

Key Concepts

Proportions

Proportions are mathematical expressions that show two ratios are equal. They form the basis of many real-world applications, such as scaling recipes or converting units. A proportion looks like two equal fractions. In simpler terms, if you have two different ratios that express the same relationship, then those ratios form a proportion. For example, in our exercise, we had \( \frac{4}{20} = \frac{1}{m} \), indicating that the two fractions are equivalent.

To understand proportions better, let's look at some everyday instances:

• When you double a recipe, the proportion of ingredients remains the same.
• Height and shadow length under the same light source are proportional.

Recognizing proportions can help in solving many practical problems with ease.

Ratios

Ratios help us compare two or more quantities. They can be written in three common ways: using the word "to," a colon, or as a fraction. For example, the ratio of 4 to 20 can be expressed as 4:20 or \( \frac{4}{20} \). Ratios can show relationships like part-to-part or part-to-whole and are fundamental in understanding proportions.

In our problem, the given ratio \( \frac{4}{20} \) is compared with another ratio \( \frac{1}{m} \). These two ratios are deemed equal, forming the basis for a proportion.

Some key points about ratios:

• Ratios are dimensionless, focusing solely on the comparison aspect.
• You can simplify ratios just like fractions. For example, simplifying 4:20 gives us 1:5, meaning one part to every five parts.

Understanding ratios aids not only in solving mathematical problems but also in interpreting data in real life.

Solving Equations

Solving equations means finding the value of the unknown that makes the equation true. In the context of our proportion problem, solving the equation involved using cross-multiplication to get rid of the fraction setup. We transformed \( \frac{4}{20} = \frac{1}{m} \) into the equation \( 4m = 20 \).

Here are some basic steps for solving linear equations like these:

• Use cross-multiplication to simplify the equation, which means multiplying the numerator of one fraction by the denominator of the other.
• Once you have a simple equation like \( 4m = 20 \), isolate the variable \( m \) by performing arithmetic operations.
• Divide both sides by the coefficient of \( m \) (here, it's 4) to solve for \( m \).

Through these steps, we found \( m = 5 \). Mastering basic equation-solving techniques is essential for tackling more complex math problems.