In a shop, the ratio of the number of pens to the number of pencils is 3 : 2. The average (arithmetic mean) of the total number of pens and pencils together is 180. How many pencils are there in the shop?

Introduction / Context:
This problem combines ratios with the concept of average. We know the ratio between pens and pencils and the average of their total count. Using this information, we must determine how many pencils are in the shop. It is a straightforward application of ratio and basic arithmetic involving sums and averages.

Given Data / Assumptions:
• Ratio of pens : pencils = 3 : 2. • Average of pens and pencils together = 180. • Average of two numbers equals (sum of the numbers) / 2.

Concept / Approach:
If the number of pens is P and the number of pencils is Q, and the average of these two counts (P and Q) is 180, then (P + Q) / 2 = 180, giving P + Q = 360. From the ratio P : Q = 3 : 2, we write P = 3k and Q = 2k for some positive k. Substituting into the sum equation allows us to solve for k and hence obtain Q, the number of pencils.

Step-by-Step Solution:
Step 1: Let the number of pens = 3k and the number of pencils = 2k. Step 2: Given that the average of P and Q is 180, write (P + Q) / 2 = 180. Step 3: Substitute P and Q: (3k + 2k) / 2 = 180. Step 4: Simplify: (5k) / 2 = 180 ⇒ 5k = 360. Step 5: Solve for k: k = 360 / 5 = 72. Step 6: Therefore, number of pencils Q = 2k = 2 * 72 = 144.

Verification / Alternative check:
Number of pens P = 3k = 3 * 72 = 216. Check ratio: P : Q = 216 : 144 = 3 : 2 after dividing both numbers by 72. Check average: (P + Q) / 2 = (216 + 144) / 2 = 360 / 2 = 180, which matches the given average. All conditions are satisfied.

Why Other Options Are Wrong:
• 244, 344 and 444 either break the ratio 3 : 2 with the implied number of pens or do not yield an average of 180 when combined with the corresponding number of pens.

Common Pitfalls:
Some students misinterpret the statement and treat 180 as the total instead of the average, leading to P + Q = 180 instead of 360. Others may incorrectly apply the ratio to 180 directly. Always convert average information into a sum equation first, then apply the ratio.

Final Answer:
The number of pencils in the shop is 144.