If 18 × 12 is coded as 206 and 19 × 22 is coded as 408 according to a certain rule, then what is the coded value of 23 × 36 ?

Introduction / Context:
In this question, the usual multiplication sign is being used in a special coded sense. The expressions 18 × 12 and 19 × 22 do not give their ordinary products but instead map to 206 and 408 respectively. Our job is to discover the simple arithmetic rule connecting the two numbers on the left to the code on the right and then apply it to 23 and 36. Such puzzles test quick pattern spotting with small numbers.

Given Data / Assumptions:

• 18 × 12 is coded as 206.
• 19 × 22 is coded as 408.
• We need the code for 23 × 36.
• The same rule is used for all these coded products.

Concept / Approach:
A natural first step is to compare each code with the ordinary product of the two numbers. For 18 and 12, the normal product is 216, which is close to 206. For 19 and 22, the product is 418, which is close to 408. This suggests that the code may be the true product minus a fixed constant. If subtracting the same constant from both 216 and 418 produces 206 and 408, we can then apply the same constant to the product 23 × 36.

Step-by-Step Solution:
Step 1: Compute the ordinary product 18 × 12, which equals 216. Step 2: Observe that 216 minus 10 gives 206, which matches the coded value. Step 3: Compute the ordinary product 19 × 22, which equals 418. Step 4: Observe that 418 minus 10 equals 408, which also matches the second coded value. Step 5: From these two examples we infer that the code for a × b is a × b minus 10. Step 6: Now compute the ordinary product 23 × 36. Multiply 23 by 36 to get 828. Step 7: Subtract the same constant 10 from 828: 828 − 10 = 818. Step 8: Therefore 23 × 36 is coded as 818.

Verification / Alternative check:
We can confirm the rule by considering whether any other simple pattern fits both given examples as neatly. For example, adding the digits of the numbers or using squares does not reproduce both 206 and 408 consistently. The constant subtraction rule works exactly in both cases and leads to a sensible value for the third expression. Since 818 appears among the options, it is the unique choice supported by this logic.

Why Other Options Are Wrong:
The values 878, 794 and 776 do not match a pattern of subtracting the same fixed constant from the true product 828. For instance, 828 minus 50 gives 778, not any of the options. Since the earlier examples clearly require subtracting 10 from the ordinary product, any other adjustment would break the pattern. Thus these options are included only as distractors.

Common Pitfalls:
Test takers sometimes try to guess complicated formulas involving sums of squares or mixed operations without first checking the simplest possibility. Another mistake is to assume the constant being subtracted must be equal to the sum of digits or some other derived quantity, but here it is simply a fixed value of 10. Starting with the simplest consistent rule is often the fastest way to solve such coding product questions.

Final Answer:
Using the pattern that the code equals the true product minus 10, the coded value of 23 × 36 is 818.