**Solution:**

Let's assume that the dealer mixes 'a' kg of tea costing $9 per kg and '6-a' kg of tea costing $12.6 per kg to make a mixture.

To find the proportion, we need to determine the values of 'a' and '6-a' such that the dealer realizes a profit of 20% by selling the mixture at $13.5 per kg.

**Step 1: Calculating the Cost Price (CP) of the Mixture:**

The cost price of the mixture will be the sum of the cost of the two types of tea used.

CP = (Cost of tea A) + (Cost of tea B)

CP = 9a + 12.6(6-a)
= 9a + 75.6 - 12.6a
= 75.6 - 3.6a

**Step 2: Calculating the Selling Price (SP) of the Mixture:**

The selling price of the mixture is given as $13.5 per kg.

SP = 13.5(6)
= 81

**Step 3: Calculating the Profit:**

Profit = SP - CP

Profit = 81 - (75.6 - 3.6a)
= 81 - 75.6 + 3.6a
= 5.4 + 3.6a

**Step 4: Calculating the Profit Percentage:**

Profit Percentage = (Profit / CP) * 100

20 = (5.4 + 3.6a) / (75.6 - 3.6a) * 100

Simplifying the equation further:

20 = (5.4 + 3.6a) * (100 / (75.6 - 3.6a))
2000 = 54 + 36a
1946 = 36a
a = 54.06 / 36
a = 1.5

Therefore, the dealer should mix 1.5 kg of tea costing $9 per kg and 4.5 kg of tea costing $12.6 per kg to realize a profit of 20%.

**Ratio of a:6:**

The ratio of a:6 represents the proportion of the two types of tea used in the mixture.

In this case, a = 1.5 kg and 6 = 6 kg.

The ratio of a:6 can be written as 1.5:6.

Simplifying the ratio by dividing both terms by the common factor of 1.5, we get:

1.5 / 1.5 : 6 / 1.5
1 : 4

Therefore, the ratio of a:6 is 1:4, indicating that for every 1 kg of tea costing $9 per kg, there should be 4 kg of tea costing $12.6 per kg in the mixture.