Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(0.12: 0.8=0.6: x\)

Short answer

x = 4.

Step-by-step solution

Understand the Proportion

A proportion states that two ratios are equal. Given the problem, the proportion is written as \(\frac{0.12}{0.8} = \frac{0.6}{x}\). This means the two ratios, \(\frac{0.12}{0.8}\) and \(\frac{0.6}{x}\), are equivalent.

Cross-Multiply

To solve for \(x\), we cross-multiply the proportion: \(0.12 \times x = 0.8 \times 0.6\). This simplifies to an equation: \(0.12x = 0.48\).

Solve for x

To isolate \(x\), divide both sides of the equation by \(0.12\): \(x = \frac{0.48}{0.12}\). This gives \(x = 4\).

Key concepts

Cross Multiplication

Cross multiplication is a technique used to simplify solving proportions. A proportion compares two ratios and states that they are equivalent, like the example given: \( \frac{0.12}{0.8} = \frac{0.6}{x} \). Cross multiplication involves multiplying the numerator of one ratio by the denominator of the other ratio. This results in two products that are equal. For instance, in the provided problem, you multiply \( 0.12 \) by \( x \) and \( 0.8 \) by \( 0.6 \). Therefore, you get:

• \( 0.12 \times x = 0.8 \times 0.6 \)

Cross multiplication helps us quickly establish an equation that we can solve to find the unknown variable. It transforms the proportion into an equation without fractions, making the problem easier to handle.
This step essentially allows us to "cross" out the denominators, focusing on solving a straightforward equation. Remember, the order of multiplication doesn't affect the outcome.

Solving Equations

Once you use cross multiplication to establish an equation, the next step is to solve it. In this context, solving an equation means isolating the unknown variable, which is \( x \) in our example. After cross-multiplying, we have the equation \( 0.12x = 0.48 \).To isolate \( x \), perform the inverse operation of multiplication, which in this case is division. Divide both sides of the equation by the coefficient of \( x \), which is \( 0.12 \).

• \( x = \frac{0.48}{0.12} \)

Perform the division to simplify, resulting in \( x = 4 \). It's essential to keep the equation balanced by performing all operations on both sides. Always check your solution by substituting it back into the original proportion to confirm both sides are equal.

Equivalence of Ratios

The concept of equivalence of ratios is foundational in understanding proportions. Two ratios are considered equivalent if they express the same relationship between numbers, just like fractions. For example, with the given proportion \( \frac{0.12}{0.8} = \frac{0.6}{x} \), the ratios on both sides are equal.Determining equivalence is crucial as it enables us to set up equations that reflect these equal relationships. Ratios can vary in appearance based on the numbers but still represent the same proportion. For instance:

• \( \frac{0.12}{0.8} \) can be simplified further, but its equivalence to \( \frac{0.6}{4} \) (after solving) underscores that they describe the same relationship.

Understanding this concept thoroughly helps in identifying and working with proportional relationships. It also underlines the significance of ratios, which are often used to compare different quantities in everyday mathematical problems. Recognizing equivalent ratios allows us to solve more complex problems by breaking them down into manageable proportions.