Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(2: 26=4: x\)

Short answer

x = 52

Step-by-step solution

Set up the proportion equation

Given the proportion \( \frac{2}{26} = \frac{4}{x} \). We need to find the value of \( x \) that makes this proportion true.

Cross multiply

To solve the proportion \( \frac{2}{26} = \frac{4}{x} \), we cross multiply to eliminate the fractions: \( 2 \times x = 26 \times 4 \).

Simplify the equation

Calculate \( 26 \times 4 = 104 \), so the equation becomes \( 2x = 104 \).

Solve for x

Divide both sides of the equation by 2 to isolate \( x \): \( x = \frac{104}{2} \).

Calculate x

Perform the division: \( x = 52 \).

Key concepts

Cross Multiplication

Cross multiplication is a handy tool for solving proportion problems. In a proportion, you have two equal fractions like \( \frac{a}{b} = \frac{c}{d} \). Cross multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. In mathematical terms, you calculate \( a \times d = b \times c \). This step eliminates the fractions and provides a straightforward equation that you can solve.

• It's a great way to clear denominators and simplify equations.
• Ensures that you focus on one equation rather than two fractions.

In our example, with the proportion \( \frac{2}{26} = \frac{4}{x} \), cross multiplying gives us \( 2 \times x = 26 \times 4 \). This simplifies to \( 2x = 104 \), allowing us to proceed with solving the equation.

Solving Equations

Once we've applied cross multiplication, we're left with an equation like \( 2x = 104 \). Solving equations is about finding the value of the variable that makes the equation true. Here, \( x \) is our variable, and the solution involves isolating it.
**Steps to isolate x:**

• Identify the operation with \( x \). In this case, it's multiplication by 2.
• To undo this, divide both sides of the equation by 2.

This gives us \( x = \frac{104}{2} \). Calculating this, we find \( x = 52 \).
Solving for \( x \) becomes a straightforward process once the equation is simplified, as seen here.

Mathematical Reasoning

Understanding the logic behind each step is crucial in learning mathematics. Mathematical reasoning allows you not only to arrive at the answer but also to grasp why it is correct. In our problem, we employ mathematical reasoning from setting the proportion to solving the equation.
Consider:

• Reconnecting to the real-world situation: the proportional relationship between numbers.
• Ensuring each step logically follows from the previous one, removing any guesswork.
• Finding justifications for operations such as cross multiplication and division.

Through reasoning, we recognize that the operations performed aim to preserve equality and balance, leading us to the correct solution. By practicing reasoning, you build the foundation to tackle more complex mathematical problems.

Decimals in Division

When performing division, it's often necessary to carry the operation to decimal places for more precision. While our example results in a whole number, understanding decimals in division is essential for fractions or remainders.
**Considerations for decimal division:**

• Ensure the dividend or divisor is adjusted to reach a decimal solution.
• Carry division to two decimal places, rounding if necessary, to maintain accuracy.

For instance, say you perform \( \frac{104}{3} \). This division gives \( 34.666... \), rounded to two decimal places as \( 34.67 \).
This ability is vital in handling proportions involving non-integer results, expanding your skillset in solving various math problems.