91Ó°ÊÓ

In algebra, a proportion is an equation that shows two ratios are equivalent. It's like a balance scale—what you do to one side must also be done to the other to maintain the equation's equality. With proportions, if you know three of the four values, you can always find the missing one by using cross-multiplication, a straightforward yet powerful technique.

Consider the proportion \( \frac{65}{20} = \frac{14}{n} \). In this scenario, the known ratio of 65 to 20 is said to be equivalent to the ratio of 14 to the unknown variable \(n\). Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other and setting the products equal to each other—this is rooted in the concept that two fractions are equal only if the cross products are equal. This method leads us directly to an equation that we can solve for the missing value.

Isolating a variable means rearranging an algebraic equation so the variable of interest is alone on one side of the equation. This is the pivotal step in solving for that variable. Once you've cross multiplied, as in the example with proportions, you'll usually get an equation that still needs to be simplified to isolate the variable.

In our case, after cross-multiplying, we obtained the equation \(65n = 280\). To isolate the variable \(n\), we need to get \(n\) alone on one side of the equal sign. Here, \(n\) is being multiplied by 65, so we do the opposite operation—division. Dividing both sides of the equation by 65 gives us \(n = \frac{280}{65}\), effectively isolating \(n\) and allowing us to find its value.

After solving for an unknown in algebra, we're often asked to round the answer to a specific place value. Rounding makes numbers easier to work with, especially if they're not whole numbers. In our problem, the solution involves rounding the value of \(n\) to the nearest hundredth.

Once we divided \(280\) by \(65\) to isolate and solve for \(n\), we got a precise answer that may not be a whole number. In this case, the unrounded answer is approximately \(4.30769...\). Rounding to the hundredth place means looking at the thousandths place (third decimal place). If it is 5 or greater, we round the hundredths place up by one. If it's less than 5, we leave the hundredths place as is. Thus, \(4.30769...\) rounds to \(4.31\), because the digit in the thousandths place is a 7, which is greater than 5. Remembering this rule ensures accuracy when rounding numbers in algebraic problems.