If s and t are positive integers such that s/t = 64.12, which of the following could be the remainder...

1. Translate the problem requirements: We need to find which remainder is possible when positive integer s is divided by positive integer t, given that \(\mathrm{s/t}\) equals the decimal 64.12
2. Convert decimal to quotient-remainder form: Express \(\mathrm{s/t = 64.12}\) as \(\mathrm{s = 64t + r}\) where r is the remainder, establishing the relationship \(\mathrm{r/t = 0.12}\)
3. Establish integer constraints: Since \(\mathrm{r/t = 0.12 = 3/25}\), derive that \(\mathrm{t = 25r/3}\), which requires r to be divisible by 3 for t to be an integer
4. Test answer choices against constraints: Check which remainder value makes t an integer and verify the solution works

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking for. We have two positive integers, s and t, and when we divide s by t, we get exactly 64.12. We need to figure out which of the given answer choices could be the remainder when s is divided by t.

Think of it this way: when you divide one number by another, you get a quotient (the whole number part) and a remainder (what's left over). For example, \(\mathrm{17 ÷ 5 = 3}\) remainder 2, which we could also write as \(\mathrm{17 ÷ 5 = 3.4}\).

Here, \(\mathrm{s ÷ t = 64.12}\), so the quotient is 64 and there's some remainder that creates the decimal part 0.12.

Process Skill: TRANSLATE - Converting the decimal division result into quotient-remainder language

2. Convert decimal to quotient-remainder form

When we divide s by t and get 64.12, this means:
\(\mathrm{s = 64 × t + remainder}\)

Let's call the remainder 'r'. So: \(\mathrm{s = 64t + r}\)

If we divide both sides by t:
\(\mathrm{s/t = 64 + r/t}\)

Since \(\mathrm{s/t = 64.12}\), we have:
\(\mathrm{64.12 = 64 + r/t}\)

This means: \(\mathrm{r/t = 0.12}\)

So the remainder divided by the divisor equals 0.12. This is the key relationship we need to work with.

3. Establish integer constraints

Now we need \(\mathrm{r/t = 0.12}\) where both r and t are positive integers.

Let's convert 0.12 to a fraction:
\(\mathrm{0.12 = 12/100 = 3/25}\) (dividing both numerator and denominator by 4)

So: \(\mathrm{r/t = 3/25}\)

This means: \(\mathrm{r × 25 = t × 3}\), or \(\mathrm{t = 25r/3}\)

For t to be a positive integer, the fraction \(\mathrm{25r/3}\) must equal a whole number. Since 25 and 3 don't share any common factors, r must be divisible by 3.

Process Skill: APPLY CONSTRAINTS - Using the requirement that t must be an integer to limit possible values of r

4. Test answer choices against constraints

Let's check each answer choice to see if it's divisible by 3:

1. \(\mathrm{r = 2}\): Not divisible by 3, so \(\mathrm{t = 25(2)/3 = 50/3}\) is not an integer ❌
2. \(\mathrm{r = 4}\): Not divisible by 3, so \(\mathrm{t = 25(4)/3 = 100/3}\) is not an integer ❌
3. \(\mathrm{r = 8}\): Not divisible by 3, so \(\mathrm{t = 25(8)/3 = 200/3}\) is not an integer ❌
4. \(\mathrm{r = 20}\): Not divisible by 3, so \(\mathrm{t = 25(20)/3 = 500/3}\) is not an integer ❌
5. \(\mathrm{r = 45}\): Divisible by 3! So \(\mathrm{t = 25(45)/3 = 1125/3 = 375}\), which is an integer ✓

Let's verify: If \(\mathrm{r = 45}\) and \(\mathrm{t = 375}\), then \(\mathrm{s = 64(375) + 45 = 24,000 + 45 = 24,045}\)
Check: \(\mathrm{s/t = 24,045/375 = 64.12}\) ✓

Final Answer

The answer is E. 45

Only when the remainder is 45 do we get integer values for both s and t that satisfy the condition \(\mathrm{s/t = 64.12}\).