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In our everyday lives, we often need to compare two quantities with different units. This comparison is known as a rate. For example, if a book sells for a total of \(\$6,000,000\) over 80 years, \(\$6,000,000\) and 80 years are the two quantities we are comparing. Rates help us understand the relationship between these quantities and can be applied in various contexts like speed (miles per hour), price (dollars per item), and productivity (units produced per hour).

Understanding rates is essential for making informed decisions, such as budgeting or evaluating the efficiency of different actions over time. Once you have a total value and a corresponding measurement, you can determine the rate by performing a division.

To calculate a unit rate, you must use division. This mathematical operation splits a number (the dividend) by another number (the divisor) to find how many times the divisor fits into the dividend. In this exercise, we need to determine how much money was earned per year from the total sales over a given period.

For example, to find out how much \(\$6,000,000\) sales are earned over 80 years, we perform the division:\[ \frac{6{,}000{,}000}{80} = 75{,}000 \]

This shows that every year, \(\$75,000\) was earned. Therefore, understanding and applying division is crucial for calculating rates and unit rates accurately.

A unit rate is a type of rate where the second quantity (denominator) is reduced to one, making it easy to understand the per unit amount. It's a standardized way to compare prices, speed, or any rates.

To find a unit rate, divide the numerator by the denominator. In the given exercise, we calculate the unit rate of book sales over 80 years by dividing the total sales \(\$6,000,000\) by 80 years:
\[ \frac{6{,}000{,}000}{80} = 75{,}000 \]

This tells us that the book earned \(\$75,000\) per year. Calculating unit rates helps you easily compare and analyze data. For example, when grocery shopping, knowing the unit price (like dollars per kilogram) instantly tells you which products are more cost-effective.

Sometimes, the resulting unit rate you calculate might not be a simple whole number. In such cases, rounding numbers helps you simplify and better communicate your results. Rounding involves adjusting the number to a specified degree of precision, like the nearest hundredth or tenth.

For instance, if a division yields \74.580357142\, you might round it to the nearest hundredth to get \74.58\. This makes the number easier to understand and use in real-world contexts. In the exercise, we ended up with \(\$75,000\), a whole number, so no further rounding was necessary.

Being proficient with rounding rules ensures that your calculations are both precise and practical. It’s a handy skill, especially when dealing with money, measurements, and statistics.