Problem 18 - Algebra Form and Function

Question

A sample of \(n\) frogs has a total weight of \(W\) grams. (a) What is the mean weight of a frog in the sample? (b) The largest frog eats the smallest one. What is the mean weight now?

Step-by-Step Solution

Answer

What is the new mean weight of the frogs after the largest frog eats the smallest one? Answer: The mean weight of the frogs in the sample can be found by dividing the total weight (W) by the number of frogs (n) using the formula: Mean weight = \(\frac{W}{n}\) After the largest frog eats the smallest one, the new mean weight can be found by dividing the total weight (which remains unchanged) by the new number of frogs (n-1), using the formula: New mean weight = \(\frac{W}{n-1}\)

Step 1: Find the mean weight of the frogs in the sample

To find the mean weight of the frogs, we need to divide the total weight of the frogs by the number of frogs in the sample. The formula for the mean weight is: Mean weight = \(\frac{W}{n}\)

Step 2: Calculate the mean weight given the total weight and number of frogs

Plug the values of the total weight (W) and the number of frogs (n) into the formula to calculate the mean weight. Assuming we have these values, we can substitute them into the formula: Mean weight = \(\frac{W}{n}\)

Step 3: Calculate the number of frogs after the largest frog eats the smallest one

Since the largest frog eats the smallest one, the number of frogs in the sample decreases by 1. So, the new number of frogs is: New number of frogs = n - 1

Step 4: Find the new total weight after the largest frog eats the smallest one

Since the largest frog ate the smallest one, the total weight of the sample does not change. The new total weight is still W. This is because the smallest frog's weight is added to the largest frog's weight, and the total weight remains the same. New total weight = W

Step 5: Calculate the new mean weight after the largest frog eats the smallest one

To find the new mean weight, we need to divide the new total weight by the new number of frogs. We can use the same formula for the mean weight as before, but with the new number of frogs from Step 3: New mean weight = \(\frac{New\; total\; weight}{New\; number\; of\; frogs}\) = \(\frac{W}{n - 1}\)

Key Concepts

Total Weight

The total weight is a critical starting point for understanding how to calculate the mean weight. Imagine you have a group of frogs, and you want to know their combined weight. This combined weight is what we refer to as the *total weight*.

• In mathematical terms, if our group of frogs weighs a combined total of \( W \) grams, then \( W \) represents the total weight.
• This encompasses the weight of every frog in the group.

Knowing the total weight helps us determine average or mean values later, as it represents the sum of all individual weights in a single number. This simplifies calculations and gives a clear understanding of the entire group's heft.

Number of Frogs

The number of frogs in your sample, denoted by \( n \), is essential. It tells us how many individual components we divide the total weight by to find the average.

• This number can change, like when a frog eats another.
• Initially, if you have a sample of \( n \) frogs, every calculation depends on this original count.

When something happens, like the largest frog eating the smallest one, the number changes. The new count of frogs becomes \( n - 1 \).
Recognizing how many frogs there are gives context to your mean calculations, telling you precisely how you distribute the total weight.

New Mean Weight

The concept of mean weight is all about averages. To find the mean weight of the frogs, you divide the total weight \( W \) by the number of frogs \( n \). But what happens when one frog consumes another?

Impact on Mean Weight

After the smallest frog gets eaten by the biggest, the total weight still remains \( W \), because the biggest frog takes on the smallest frog's weight. However, the number of frogs decreases by one, leading to a new mean calculation:

• New number of frogs: \( n - 1 \)
• New mean weight: \[ \frac{W}{n - 1} \]

By dividing the unchanged total weight by the new, smaller number of frogs, you see how the average weight per frog increases when the group size decreases. That's the charm of the mean—it adjusts dynamically with changes in the group!