Question: A palynologist analyzing pollen counts observes that the average of three counts — $ 3u+2 $, $ 5u+7 $, and $ 4u+1 $ — equals 62. What is the value of $ u $?

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When studying ancient or modern pollen samples, palynologists rely heavily on accurate measurements to reconstruct past climates, track seasonal changes, and support forensic or archaeological investigations. A key analytical step involves calculating the average of multiple pollen counts. Recently, a palynologist encountered a real-world math problem: analyzing three pollen count measurements — $ 3u+2 $, $ 5u+7 $, and $ 4u+1 $ — where the average equals 62. Solving for $ u $ reveals critical data about environmental conditions.

Understanding the Problem: Middle School Math Meets Palynology

Understanding the Context

The average of three numbers is found by summing them and dividing by 3. In this case, the three pollen counts are expressions involving a variable $ u $. The average is given as:

$$
rac{(3u+2) + (5u+7) + (4u+1)}{3} = 62
$$

This equation is central to translating numerical data into ecological insights — a core task in palynology. By solving for $ u $, scientists uncover patterns in pollen distribution linked to seasonal shifts, vegetation types, or environmental stressors.

Step-by-Step Calculation

Question: A palynologist analyzing pollen counts observes that the average of three counts — $ 3u+2 $, $ 5u+7 $, and $ 4u+1 $ — equals 62. What is the value of $ u $?

Key Insights

Let’s break down the equation:

Add the expressions:

$$
(3u + 2) + (5u + 7) + (4u + 1) = (3u + 5u + 4u) + (2 + 7 + 1) = 12u + 10
$$

Set up the average equation:

$$
rac{12u + 10}{3} = 62
$$

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Final Thoughts

Multiply both sides by 3 to eliminate the denominator:

$$
12u + 10 = 186
$$

Subtract 10 from both sides:

$$
12u = 176
$$

Divide by 12:

$$
u = rac{176}{12} = rac{44}{3}
$$

Interpreting the Result in Context

While $ u = rac{44}{3} $ is a fractional value, it may represent a scaled or averaged metric in palynological data — such as adjusted counts from sediment layers or time-weighted pollen accumulations. In real research, such values help calibrate models predicting pollen deposition rates, identifying diseased plant activity, or mapping climate shifts over millennia.

Why This Matters for Palynologists

Accurate algebraic modeling of pollen counts enables scientists to: