Directions (67-69):The radar graph given below shows the number of buckets (plastic + wooden) sold by five different shops (P , Q, R, S and T) in two different months (May and June). Read the data carefully and answer the questions.

In May, if total number of plastic buckets sold by T are four times of the difference between total number of buckets sold by Q in both months, then find the minimum possible average of total wooden buckets sold by T in May and total plastic buckets sold by T in June (Given, number of wooden buckets sold by T in June > total wooden buckets sold by T in May and it also a perfect square of even number).

Step 1: Extract the relevant data from the radar graph
By analyzing the radar graph, we can find the total number of buckets (plastic + wooden) sold by shops Q and T in the months of May and June:

• For shop Q:
  • Total buckets sold in May (blue line, square marker) = 120
  • Total buckets sold in June (orange line, circle marker) = 100
• For shop T:
  • Total buckets sold in May (blue line, labeled value) = 168
  • Total buckets sold in June (orange line, circle marker) = 176

Step 2: Calculate the number of wooden buckets sold by T in May
The difference between the total number of buckets sold by shop Q in both months is:
$|120-100|=20$
According to the question, the total number of plastic buckets sold by T in May is four times this difference:
${\text{Plastic buckets sold by T in May}}_{\mathrm{}}=4\times 20=80$
Using the total buckets sold by T in May (168), we can find the wooden buckets sold by T in May:
${\text{Wooden buckets sold by T in May (W}}_{\text{T, May}})=168-80=88$

Step 3: Apply the constraints for T's sales in June
Let $W_{\text{T, June}}$ be the number of wooden buckets sold by T in June, and $P_{\text{T, June}}$ be the number of plastic buckets sold by T in June.
We are given two conditions for $W_{\text{T, June}}$:

1. It is greater than the total wooden buckets sold by T in May:
   $W_{\text{T, June}}>88$
   
2. It is a perfect square of an even number:
   Possible values of perfect squares of even numbers greater than 88 are:
   ${10}^2=100$, ${12}^2=144$, ${14}^2=196$, etc.

Since the total number of buckets sold by T in June is 176, the wooden buckets sold cannot exceed 176:
$W_{\text{T, June}}<176$
Therefore, the possible values for $W_{\text{T, June}}$ are 100 or 144.

Step 4: Minimize the average
We want to find the minimum possible average of $W_{\text{T, May}}$ and $P_{\text{T, June}}$:
$\text{Average}=\frac{88+P_{\text{T, June}}}{2}$
To minimize this average, we need to minimize the number of plastic buckets sold in June ($P_{\text{T, June}}$). Given that:
$P_{\text{T, June}}=176-W_{\text{T, June}}$
To minimize $P_{\text{T, June}}$, we must maximize $W_{\text{T, June}}$ to its largest possible value (144):
${\text{Minimum P}}_{\text{T, June}}=176-144=32$
Now, calculate the minimum average:
$\text{Minimum Average}=\frac{88+32}{2}=\frac{120}{2}=60$