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When we talk about solving proportions, we are dealing with two ratios that are set equal to each other. A ratio is a comparison of two quantities, and a proportion states that two ratios are equivalent.

To solve a proportion, we often use a method known as cross multiplication. This involves multiplying the numerator of one ratio by the denominator of the other ratio, and doing the same with the remaining numerator and denominator. For example, given the proportion \( \frac{a}{b} = \frac{c}{d} \) we would cross multiply to get \( a \times d = b \times c \). After cross multiplication, we can solve for the unknown variable by isolating it on one side of the equation. This might involve additional steps like simplifying the equation, which is our next topic.

Simplifying equations is a critical step in solving math problems, as it allows us to more easily see the solution. Simplifying typically involves reducing the equation to its most basic form. This can include combining like terms, multiplying out parentheses, or dividing both sides of an equation by the same number to isolate the variable.

In our example, once we cross multiplied \(0.7\cdot n = 1.2\cdot 6.4\), we completed the multiplication to simplify it to \(0.7\cdot n = 7.68\). To solve for \(n\), we then needed to isolate the variable \(n\) by dividing both sides by 0.7. Simplifying equations allows us to pinpoint the precise value of our unknown variable, streamlining the path to our solution.

After solving an equation, we often have to present our answer in a specific way, which can include rounding numbers to a certain decimal place. Rounding numbers is the process of adjusting a number to make it simpler or to fit a certain level of precision.

For instance, the final calculation in our problem was to divide 7.68 by 0.7, resulting in a long decimal. The instructions specified to round to the nearest hundredth, which means we need to look at the third decimal place. If it's 5 or above, we round up; if it's 4 or below, we round down. Thus, \( \frac{7.68}{0.7} \) rounded to the nearest hundredth is \( 10.97 \). Rounding numbers is a fundamental concept, especially when dealing with money, measurements, and other real-world applications where extreme precision is not necessary or not practicable.