Problem 63 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{c}{10}=\frac{10}{c} $$

Step-by-Step Solution

Answer

The solutions are \( c = 10 \) and \( c = -10 \).

Step 1: Understand the Proportion

A proportion is when two ratios are set equal to each other. In this problem, \( \frac{c}{10} = \frac{10}{c} \) is a proportion, where you need to find the value of \( c \) that makes this equation true.

Step 2: Cross Multiply the Ratios

To solve \( \frac{c}{10} = \frac{10}{c} \), start by cross-multiplying the terms. This means multiplying the numerator of each ratio by the denominator of the other ratio, giving us: \( c \times c = 10 \times 10 \). This simplifies to \( c^2 = 100 \).

Step 3: Solve for c by Taking the Square Root

Now, solve \( c^2 = 100 \) by taking the square root of both sides of the equation. This gives \( c = \sqrt{100} \). Since \( \sqrt{100} = 10 \), we have \( c = 10 \). However, don't forget that taking the square root also gives a negative solution, so \( c \) could also be \( -10 \).

Step 4: Verify Solutions

Check both solutions in the original proportion to ensure they are correct. 1. For \( c = 10 \): \( \frac{10}{10} = \frac{10}{10} \), which is true. 2. For \( c = -10 \): \( \frac{-10}{10} = \frac{10}{-10} \), which simplifies to \( -1 = -1 \), which is also true. Both solutions satisfy the original equation.

Key Concepts

Cross Multiplication Explained

Cross multiplication is a technique used to solve equations involving two ratios set equal to each other, known as proportions. This method is particularly useful when you have an equation of the form \( \frac{a}{b} = \frac{c}{d} \). The goal is to find the unknown variable in the proportion by simplifying the expression into a basic algebraic equation.

To perform cross multiplication, follow these simple steps:

• Multiply the numerator of one ratio by the denominator of the other ratio. So, you'll have two products.
• Set the products equal to each other. This step transforms the proportion into a solvable equation.
• Solve the resulting equation for the unknown variable.

In the exercise \( \frac{c}{10} = \frac{10}{c} \), cross multiplication involves multiplying \( c \times c \) and setting it equal to \( 10 \times 10 \). This gives \( c^2 = 100 \), which is much easier to solve.

Remember, cross multiplication only works when two fractions are set equal to each other. It's a powerful tool for unraveling proportions and finding unknowns.

Understanding Square Roots

Square roots are fundamental in solving equations that result from methods like cross multiplication. A square root essentially undoes the action of squaring a number. If you have an equation such as \( x^2 = 100 \), taking the square root helps you find the original values of \( x \) that, when squared, will give you 100.

Key points to keep in mind when dealing with square roots:

• The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \).
• Square roots can yield both positive and negative solutions. For instance, \( \sqrt{100} \) results in both \( 10 \) and \( -10 \) because \( 10^2 = 100 \) and \((-10)^2 = 100\).
• Remembering this will help you consider all possible solutions to an equation.

In our exercise, \( c^2 = 100 \), taking the square root leads to two potential values for \( c \): \( 10 \) and \( -10 \). Noticing the dual nature of square roots is essential for finding all solutions in equations like these.

Verifying Solutions

Verification of solutions is a critical final step in solving algebraic problems. It ensures that the values you found are indeed correct and satisfy the original equation before proclaiming them as final solutions.

To verify, you substitute the found values back into the original equation and check if both sides are equal.

In our example, we found two values for \( c \): \( 10 \) and \( -10 \). For example:

• Substituting \( 10 \) in, the proportion becomes \( \frac{10}{10} = \frac{10}{10} \), which simplifies to \( 1 = 1 \).
• Substituting \( -10 \) in, the proportion turns into \( \frac{-10}{10} = \frac{10}{-10} \), simplifying further to \( -1 = -1 \).

Both checks confirm that the values of \( c \) play right into the original proportion, affirming that they are indeed correct solutions. Verifying not only boosts confidence in your solutions but also ensures mathematical accuracy.