Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{1}{4}: \frac{1}{2}=1: x\)

Short answer

The value of \( x \) is 2.

Step-by-step solution

Set up the Proportion Equation

We start by setting up the proportion equation based on the given ratios: \( \frac{1}{4}: \frac{1}{2} = 1: x \). This can be rewritten as the equation \( \frac{1/4}{1/2} = \frac{1}{x} \).

Simplify the Left Side of the Equation

Next, simplify \( \frac{1/4}{1/2} \). Dividing fractions means multiplying by the reciprocal: \( \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} \). So, \( \frac{1/4}{1/2} = \frac{1}{2} \).

Convert the Equation into a Cross Multiplication Problem

Now we have \( \frac{1}{2} = \frac{1}{x} \). Cross-multiply to solve for \( x \): \( 1 \cdot x = 2 \cdot 1 \) resulting in \( x = 2 \).

Key concepts

Cross-multiplication

Cross-multiplication is a method used to solve equations that involve proportions, where two ratios are set equal to each other. This handy technique is often used because it transforms a proportion into a simple, solvable equation.
The idea behind cross-multiplication is to eliminate the fractions by multiplying the terms across the diagonal of the equal sign. For example, if you have a proportion like \(\frac{a}{b} = \frac{c}{d}\), the cross-multiplication principle states that \(a \times d = b \times c\).
In the given solution, the equation is simplified to \(\frac{1}{2} = \frac{1}{x}\). Applying cross-multiplication here, we set \(1 \times x = 2 \times 1\). The result, \(x = 2\), is easy to understand once you've mastered the method.

• This method requires care to ensure each side of the equation is multiplied correctly.
• Cross-multiplication is especially useful in word problems involving proportional relationships.
• Remember that this method only works if both sides of your equation are set as fractions or ratios. Ensure equation validation before cross-multiplying.

Fractions

Fractions are an important part of mathematics that allow us to express parts of a whole. In a fraction, there are two numbers: the numerator (top part) and the denominator (bottom part). When you're working with fractions, understanding how they interact is crucial.
In the exercise, the fractions \(\frac{1}{4}\) and \(\frac{1}{2}\) are part of a proportion. Simplifying these helps us solve the problem efficiently. Here, it began with the expression \(\frac{1/4}{1/2}\), which is a complex fraction. To simplify complex fractions, you multiply by the reciprocal of the denominator. So, \(\frac{1}{4} \times \frac{2}{1} = \frac{1}{2}\). This step simplifies the equation and prepares it for cross-multiplication.

• The numerators are the figures above the line in each fraction, representing how many parts we focus on.
• The denominators represent the total number of equal parts that form the whole.
• Fractions can be manipulated in many ways: adding, subtracting, multiplying, dividing, or even turning them into decimals.

Division

Division is a fundamental operation that is often paired with fractions and proportions. It allows us to split a number into equal parts. When working with fractions, division can be simplified by using the reciprocal.
In this exercise, we encountered division when simplifying a complex fraction. The expression \(\frac{1/4}{1/2}\) was approached by taking the reciprocal of \(\frac{1}{2}\) (which is \(\frac{2}{1}\)), and then multiplying it by \(\frac{1}{4}\) to obtain \(\frac{1}{2}\). This is effectively handling the concept of division of fractions by using multiplication.
Here are a few essentials to remember about division in dealing with fractions and proportions:

• Dividing by a number is the same as multiplying by its reciprocal.
• Always perform division after simplifying where possible, to avoid unnecessary complexity.
• With proportions, after simplifying, the use of division is crucial to find the unknown value once you cross-multiply.

Once mastered, division intertwines smoothly with fractions and proportions, serving as a pillar of understanding these mathematical ideas.