Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{16}{40}=\frac{22}{x}\)

Short answer

x = 55.

Step-by-step solution

Cross-Multiplying Proportion

In a proportion, the cross-products are equal. So, from \( \frac{16}{40} = \frac{22}{x} \), we can write the equation as: \(16 \cdot x = 40 \cdot 22\).

Perform the Multiplication

Calculate the product of the known values on the right side of the equation: \( 40 \times 22 = 880 \). So, the equation becomes \( 16x = 880 \).

Solve for x

To find \( x \), divide both sides of the equation by 16. It becomes \( x = \frac{880}{16} \).

Perform the Division

Carry out the division to calculate \( x = 55 \). Use the calculator to ensure the division is correct and note that if needed, we carry division two decimal places. However, the division resolves to a whole number: 55.

Key concepts

Cross-Multiplication Technique

Cross-multiplication is a common technique used in mathematics to solve proportions, where two ratios are equal. Let's start by understanding what cross-multiplication means. In a proportion like \(\frac{a}{b} = \frac{c}{d}\), cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other. This means:

• Multiply \(a\) by \(d\), which yields \(a \cdot d\).
• Multiply \(b\) by \(c\), which yields \(b \cdot c\).

The key principle here is that these multiplications are set equal to each other: \(a \cdot d = b \cdot c\). This allows us to form an equation that we can solve for the unknown variable. Cross-multiplication is particularly useful because it eliminates the fractions, making the problem simpler to solve. In the original exercise, applying cross-multiplication transformed the proportion \(\frac{16}{40} = \frac{22}{x}\) into the equation \(16 \cdot x = 40 \cdot 22\). The simplicity of multiplying whole numbers rather than dealing with ratios speeds up the process of finding the solution.

Solving Equations

Once we obtain a simple equation like \(16x = 880\) from the cross-multiplication step, the next step is to solve it for the unknown variable. Solving equations involves finding the value of the variable that makes the equation true. Here are the steps:

• Identify the variable, which is \(x\) in our case, and determine which operations are being performed with it.
• Use inverse operations to isolate the variable. Since \(x\) is multiplied by 16, we will use division as the inverse operation.

The process of solving this equation is straightforward:

• Divide both sides of the equation by 16 to isolate \(x\): \(x = \frac{880}{16}\).

When solving equations, always remember to perform the same operation on both sides to maintain equality. Solving it correctly, as demonstrated, will lead us directly to the value of \(x\) once all calculations are satisfactorily completed.

Division in Mathematics

Division is one of the four fundamental arithmetic operations and is used here to find the value of the variable \(x\). In our solved equation \(x = \frac{880}{16}\), performing the division will give us the final answer. Understanding division fully is crucial for a correct and accurate solution:

• The concept of division can be thought of as splitting a number into equal parts. Here, 880 needs to be divided into 16 equal parts.
• To calculate this division, 880 is divided by 16, which can be done using a standard calculator or long division method.

The result of \(x = \frac{880}{16}\) is a whole number 55. Sometimes in division, the result might not be a whole number, and you might be required to carry the division to two decimal places for precision. But in this scenario, \(x\) is exactly 55, showing that division can both simplify expressions and lead us directly to the precise solution.