Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{1}{2}: 2=\frac{1}{3}: x\)

Short answer

\( x = \frac{1}{3} \)

Step-by-step solution

Write the proportion as a fraction equation

The given proportion can be written as two equal ratios: \( \frac{1}{2} : 2 = \frac{1}{3} : x \). This can be rewritten as the fraction equation: \( \frac{1}{2} / 2 = \frac{1}{3} / x \).

Convert division to fraction multiplication

Convert the division in each ratio to multiplication by the reciprocal. For the equation \( \frac{1/2}{2} \), this becomes \( \frac{1}{2} \times \frac{1}{2} \), because dividing by 2 is the same as multiplying by \( \frac{1}{2} \):\[ \frac{1}{2} \times \frac{1}{2} = \frac{1}{3} \times \frac{1}{x} \]

Solve for x using cross-multiplication

Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other. So:\[ 1 \times 1 = 3 \times x \] This simplifies to:\[ x = \frac{1}{3} \]

Solution Verification

The solution is \( x = \frac{1}{3} \). Substitute back to the original proportion to verify:\( \frac{1}{2} : 2 = \frac{1}{3} : \frac{1}{3} \)Both fractions are equal, confirming the solution is correct.

Key concepts

Understanding Fraction Equations

Fraction equations are mathematical expressions where two fractions are set equal to each other. This type of equation often arises in problems involving proportions and ratios. When you have a proportion, like the one given in the exercise, it means that two fractions form an equivalence.To write a proportion as a fraction equation, you can rewrite it to display two equal fractions. For example, if you have the proportion \( \frac{1}{2} : 2 = \frac{1}{3} : x \), you can express this as: \( \frac{1}{2} / 2 = \frac{1}{3} / x \). This aligns both parts of the proportion into a clear mathematical equation, making it easier to solve.Working with fractions can seem tricky at first, but understanding how to manipulate them is a powerful skill. By reducing proportions to fraction equations, it becomes simpler to see the relationships between numbers.

Cross-Multiplication Technique

Cross-multiplication is a handy method used to solve fraction equations, especially when dealing with proportions. When two fractions are equal, their cross-products are also equal. This means that if you have fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), their equality implies that \( a \times d = b \times c \).In our exercise, you apply cross-multiplication to the equation: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{3} \times \frac{1}{x} \). Cross-multiplying gives us:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \( 1 \times x \)
• Then, multiply the numerator of the second fraction by the denominator of the first fraction: \( 1/3 \times 2 \)

This ultimately gives \( x = \frac{1}{3} \). Cross-multiplication is an effective strategy for solving equations quickly and accurately.

Reciprocal and Fraction Manipulation

The reciprocal of a number is simply 1 divided by that number. It's the number you get when you "flip" a fraction upside down. For instance, the reciprocal of \( 2 \) is \( \frac{1}{2} \), and the reciprocal of \( \frac{1}{3} \) is \( 3 \).When dealing with fractions in equations, we often use reciprocals to convert division into multiplication. In the given problem, you converted the division in the equation \( \frac{1}{2} / 2 = \frac{1}{3} / x \) by multiplying by the reciprocal of 2, which is \( \frac{1}{2} \), hence:\[ \frac{1}{2} \times \frac{1}{2} = \frac{1}{3} \times \frac{1}{x} \]Using reciprocals simplifies complex fraction problems and is an important technique in algebra that helps keep calculations manageable.

Step-by-Step Guide to Solve for x

To "solve for \( x \)" means finding the value of the variable that makes the equation true. When working with proportions, like in our provided exercise, this typically involves simplifying the equation using methods like cross-multiplication.Let's break down the steps:

• Express the original problem as a fraction equation: \( \frac{1}{2} / 2 = \frac{1}{3} / x \).


• Replace division with multiplication by using the reciprocal: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{3} \times \frac{1}{x} \).


• Cross-multiply to eliminate the fractions: \( 1 \times 1 = 3 \times x \).


• Simplify to find \( x \): \( x = \frac{1}{3} \).

By following these steps, you systematically tackle the equation, ensuring accuracy and making the task straightforward. Solving for \( x \) opens up the path to understanding relationships between numbers in various mathematical scenarios.