Problem 647 In the following exercises, find... [FREE SOLUTION] (2024)

Short Answer

Rate

A rate is a way to compare two different kinds of quantities. For instance, in the exercise, we are comparing pounds to square inches. Understanding this comparison helps us see the relationship between the two quantities. Rates can be useful in everyday situations, such as determining the speed of a car (miles per hour) or the cost of groceries (dollars per item). When we say '90 pounds per 7.5 square inches,' the rate tells us how much 'pounds' change as 'square inches' change. Recognizing the relationship between the quantities is the foundation of comprehending rates.

Division

Division is the mathematical operation we use to find the unit rate. In the given exercise, we need to calculate how many pounds correspond to one square inch. To do this, we divide the total pounds by the total square inches. This is expressed as: \ Unit rate = \ \ \(\frac{90 \text{ pounds}}{7.5 \text{ square inches}}\). Practically, division breaks down a total amount into equal parts, enabling us to find the rate for one single part. For example, dividing 90 by 7.5 means distributing 90 pounds evenly across 7.5 square inches to find out what each single square inch gets.

Mathematical Calculation

Carrying out the correct mathematical calculation is crucial for determining the unit rate. First, we place the given values into our division expression. Here, it looks like this: \ \(\frac{90}{7.5}\). Next, we perform the actual division. This can be done manually or using a calculator. For manual calculation: \ \ 90 ÷ 7.5, think of it as: \ \ \ 90 divided by 7.5 simplifies directly to 12. So, the unit rate is 12 pounds per square inch. Correct calculations ensure that our interpretations and solutions are reliable and accurate.

Proportional Reasoning

Proportional reasoning lets us understand how quantities relate to one another. If one quantity changes, the other changes in proportion. For instance, if the area doubles, the weight would also double for the same rate. This concept helps us interpret the unit rate correctly: at ‘12 pounds per square inch,’ we recognize that each square inch holds 12 pounds. Hence, if we have 2 square inches, we’d have 24 pounds, and so forth. This proportional thinking aids in making sense of various problems by keeping the relationship between quantities consistent, regardless of their sizes.