Question

Solve for $x$ in the following proportions. Carry division two decimal places as necessary. $0.5:0.15=0.3:x$

Short answer

x = 0.09

Step-by-step solution

Set up the Proportion Equation

Given the proportion $0.5:0.15=0.3:x$, we can express it as an equation with fractions: $\frac{0.5}{0.15}=\frac{0.3}{x}$. This allows us to find the value of $x$.

Cross-Multiply to Eliminate Fractions

To solve for $x$, cross-multiply the fractions. This gives: $0.5\cdot x=0.15\cdot 0.3$.

Perform the Multiplications

Calculate $0.15\times 0.3$: $0.045$. Substitute this back into the equation: $0.5x=0.045$.

Solve for x

Divide both sides of the equation by $0.5$ to isolate $x$: $x=\frac{0.045}{0.5}$.

Calculate the Division

Perform the division: $0.045÷0.5=0.09$. Thus, $x=0.09$.

Key concepts

Cross-Multiplication

Cross-multiplication is a method used in algebra to solve equations involving fractions. It's a handy technique especially when dealing with proportions. A proportion is an equation that states two fractions are equal. When you have a proportion like $\frac{a}{b}=\frac{c}{d}$, you cross-multiply to eliminate the fractions. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. You end up with an equation $a\cdot d=b\cdot c$. With the fractions gone, solving for the variable becomes much easier. Cross-multiplication essentially simplifies complex-looking problems into straightforward multiplication and division tasks.
This method is especially useful when the denominator of one fraction includes the variable you need to solve for, as it allows you to clear it from the fraction entirely.

Solving Equations

Solving equations is all about finding what value of the variable makes the equation true. After cross-multiplying, you'll often end up with a simple equation to solve. Let's say you have an equation like $0.5x=0.045$. To solve it, you need to isolate $x$. This means getting $x$ alone on one side of the equation. Usually, you will do this by performing the same operation on both sides of the equation. In our example, since $x$ is multiplied by $0.5$, you divide both sides by $0.5$ to find $x$.
Remember that whatever operation you do to one side, you must do to the other to keep the equation balanced.

Fractions

Fractions represent parts of a whole and are a crucial part of understanding proportions. A fraction $\frac{a}{b}$ is made up of a numerator $a$ and a denominator $b$. Proportions often compare two fractions to find out if they are equivalent.
In many math problems, like our original exercise, you start with a proportion expressed by fractions. Understanding how to manipulate fractions through methods like cross-multiplication is essential. Always simplify fractions when you can, to make calculations easier and the numbers more manageable.

Division

Division is one of the key arithmetic operations used to solve equations, especially when you've isolated the variable. In our step-by-step solution for $x$, after performing multiplication, you should divide to complete the solution.
In the equation $0.5x=0.045$, you divide both sides by $0.5$. This means splitting $0.045$ into portions that fit into $0.5$. The result of this division tells us what the value of $x$ must be. Division requires careful attention to detail, particularly when you're dealing with decimal points.
Always double-check your decimal placement and consider rounding if necessary, as specified in the problem instructions.