Solve the following proportion for x:
$\frac{2}{3}=\frac{3 x+4}{15}$
The objective is to solve the linear equation given by the proportion \(\frac{2}{3} = \frac{3x + 4}{15}\). To find the value of \(x\), we need to isolate the variable on one side of the equation. A proportion \(\frac{a}{b} = \frac{c}{d}\) can be solved using the cross-multiplication property, which states that \(a \cdot d = b \cdot c\). Applying this property to our equation allows us to remove the denominators and simplify the expression into a standard linear form.
We start with the equation:
\[\frac{2}{3} = \frac{3x + 4}{15}\]
Cross-multiplying yields:
\[2 \cdot 15 = 3 \cdot (3x + 4)\]
This simplifies to:
\[30 = 9x + 12\]
Now, we subtract 12 from both sides of the equation to isolate the term containing \(x\):
\[30 - 12 = 9x\]
\[18 = 9x\]
Finally, divide both sides by 9 to solve for \(x\):
\[x = \frac{18}{9}\]
\[x = 2\]
Proportions are widely used in everyday life, for example, in cooking or mixing chemicals. If a recipe for a cake uses 2 cups of sugar for every 3 cups of flour, and you decide to use 15 cups of flour, you can use the same proportional reasoning to determine how much sugar is needed. By setting up the ratio \(\frac{2}{3} = \frac{x}{15}\), you can easily calculate that you need 10 cups of sugar. Proportional reasoning helps in scaling quantities while maintaining the integrity of the original ratio.