Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{1}{4}: 1.6=\frac{1}{8}: x\)

Short answer

x = 0.8

Step-by-step solution

Understand the Proportion

The problem is asking us to solve for \(x\) in the given proportion: \(\frac{1}{4} : 1.6 = \frac{1}{8} : x\). This can also be written in fractional form as \(\frac{1/4}{1.6} = \frac{1/8}{x}\).

Set Up the Equation

Translate the proportion directly into an equation: \(\frac{1/4}{1.6} = \frac{1/8}{x}\). This equation shows that the two ratios are equal.

Cross-Multiply

To solve this equation, cross-multiply to eliminate the fractions: \((1/4) \times x = (1/8) \times 1.6\).

Calculate the Products

Calculate each side of the equation. On the right, \((1/8) \times 1.6 = 0.2\). This gives us the equation: \((1/4) \times x = 0.2\).

Solve for x

Divide both sides of the equation by \(1/4\) to solve for \(x\): \(x = \frac{0.2}{1/4}\).

Perform the Division

Perform the division to solve for \(x\): \(x = 0.2 \times 4 = 0.8\).

Key concepts

Cross-Multiplication

When solving proportion problems, cross-multiplication is a fundamental technique. It involves multiplying the numerator of one fraction with the denominator of the other and vice versa. This allows us to eliminate fractions from the equation and turn it into a simpler form, making it easier to solve.
For instance, if you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication gives you the equation \( a \times d = b \times c \). By using this method, we can easily compare or solve proportions because we transform them into a basic multiplication equation.
In the original exercise, the problem \( \frac{1/4}{1.6} = \frac{1/8}{x} \) becomes \((1/4) \times x = (1/8) \times 1.6\) after cross-multiplying. This step is crucial as it takes us a big step closer to finding the value of \(x\).

Solving Equations

Once you've set up your proportion as an equation through cross-multiplication, solving it becomes the next task. The objective here is to isolate the variable you're solving for.
In our exercise, after cross-multiplying, we obtained the equation \((1/4) \times x = 0.2\). The goal now is to solve for \(x\). This might involve dividing or multiplying both sides of the equation by the same number to get \(x\) alone on one side of the equation.
In this specific case, you simply divide both sides by \(1/4\) to isolate \(x\): \(x = \frac{0.2}{1/4}\). This gives us \(x = 0.8\) when the division is performed. By systematically performing these operations, you can solve the equation and find the unknown term in a proportion problem.

Fractions

Fractions are a key element when working with proportions, particularly in the cross-multiplication process. It's vital to understand how to manage fractions to navigate through proportion problems.
In the initial step of the exercise, the given proportion \( \frac{1}{4} : 1.6 = \frac{1}{8} : x \) translates into a fractional equation \( \frac{1/4}{1.6} = \frac{1/8}{x} \). Fractions represent parts of a whole and can be challenging if you're not familiar with multiplying or dividing them.
Here are a few tips to handle fractions effectively:

• Convert mixed numbers or improper fractions if needed for easier computation.
• Find a common denominator when adding or subtracting fractions.
• Multiply straight across the numerators and denominators when multiplying fractions.
• Remember dividing by a fraction is the same as multiplying by its reciprocal.

When dividing fractions, like in this exercise, keep these essential rules in mind to accurately compute values and solve the problem.