Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{2}{x}=\frac{13}{52}\)

Short answer

The solution is \( x = 8 \).

Step-by-step solution

Understand the Proportion

We are given the equation \( \frac{2}{x} = \frac{13}{52} \). This means that the ratio of 2 to \( x \) is equal to the ratio of 13 to 52. To find \( x \), we will use the property of cross-multiplication for proportions.

Cross-Multiply

Using the property of cross-multiplication, we multiply the numerator of the first fraction with the denominator of the second and vice versa. This gives us:\[ 2 \times 52 = 13 \times x \] which simplifies to give the equation:\[ 104 = 13x \]

Solve for \( x \)

To isolate \( x \), divide both sides of the equation by 13:\[ x = \frac{104}{13} \] By performing the division:\[ x = 8 \]

Verify the Solution

Substitute \( x = 8 \) back into the original proportion \( \frac{2}{8} = \frac{13}{52} \). Simplify both fractions:\[ \frac{2}{8} = \frac{1}{4} \] and \[ \frac{13}{52} = \frac{1}{4} \]These are equal, verifying that our solution \( x = 8 \) is correct.

Key concepts

Understanding Cross-Multiplication

When dealing with proportions, cross-multiplication is a handy tool. A proportion states that two ratios are equal. For example, in the equation \( \frac{2}{x} = \frac{13}{52} \), each fraction represents a part-to-whole relationship.
To find the unknown value, like \( x \), you can cross-multiply. This means you multiply the numerator of one fraction by the denominator of the other and vice versa.

• Multiply 2 (the numerator of the first fraction) by 52 (the denominator of the second): \( 2 \times 52 \)
• Multiply \( x \) (the denominator of the first fraction) by 13 (the numerator of the second): \( 13 \times x \)

Once you perform these multiplications, equate the products to each other. This step gives you a simple equation. It's straightforward and directs you towards isolating the variable.

Exploring Fractions

Fractions represent part of a whole. In any fraction, the number above the line is called the numerator, and the number below is the denominator.
Each fraction in a proportion tells us how one part relates to another. In our example \( \frac{2}{x} = \frac{13}{52} \), we have two fractions equating to each other.
Fractions can be simplified by dividing both numerator and denominator by their greatest common divisor.
This simplification helps in understanding and confirming our solutions.

• The fraction \( \frac{13}{52} \) simplifies to \( \frac{1}{4} \), as both 13 and 52 are divisible by 13.
• Checking \( \frac{2}{8} \), we simplify it down to \( \frac{1}{4} \).

This simplification technique aids in verifying solutions. Simplifying fractions is essential for clear comparisons or to confirm equality in proportions.

Solving Equations

Once you have cross-multiplied in a proportion problem, you typically find yourself with an equation to solve. Our cross-multiplication gave us \( 104 = 13x \).
To solve for \( x \), we need to isolate \( x \) on one side of the equation. This usually involves dividing both sides of the equation by the coefficient accompanying \( x \).

• In our solution, we divided 104 by 13, giving \( x = 8 \).
• This operation effectively isolates \( x \) by eliminating the coefficient, offering a clear value of \( x \).

Solving equations in this linear form is straightforward. It requires simple arithmetic and ensures that \( x \) is left by itself, making it easier to comprehend and verify the answer.

Mathematical Verification

Verification is the process of confirming the correctness of your solution. Once you've found \( x \), you should substitute it back into the original problem to check your work.
In our case, we found \( x = 8 \) and needed to verify by plugging it back in: \( \frac{2}{8} = \frac{13}{52} \).
Simplifying both sides gave us \( \frac{1}{4} \) and \( \frac{1}{4} \), confirming the equality.

• This clear comparison ensures both sides of our original equation balance.
• Verification acts as a safety net, catching errors that may have occurred during calculations.

Always verify your solutions in mathematical problems, especially with proportions. It reassures accuracy and builds confidence in solving future equations correctly.