When positive integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12, what is...

1. Translate the problem requirements: We need to understand that \(\mathrm{x/y = 96.12}\) means x divided by y gives quotient 96 with remainder 9, and find the value of y
2. Connect the decimal to the remainder: Recognize that the decimal part 0.12 represents the remainder 9 divided by y
3. Set up the remainder equation: Since \(\mathrm{0.12 = 9/y}\), we can solve directly for y
4. Verify with the division algorithm: Check that our answer satisfies the original conditions

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what the problem is telling us in everyday terms.

We have two positive integers: x and y. When we divide x by y, we get a remainder of 9. We also know that when we perform this division on a calculator, we get \(\mathrm{x ÷ y = 96.12}\).

Think of this like dividing cookies among friends. If you have x cookies and y friends, each friend gets 96 cookies, but there are 9 cookies left over (the remainder). The decimal 0.12 in our result represents those leftover cookies as a fraction of the total group size.

Process Skill: TRANSLATE - Converting the problem's language about division and remainders into a clear mathematical relationship

2. Connect the decimal to the remainder

Now here's the key insight: that decimal part 0.12 is directly related to our remainder of 9.

When we divide and get 96.12, we can think of this as:

• The whole number part (96) tells us how many complete times y goes into x
• The decimal part (0.12) represents the remainder (9) expressed as a fraction of y

In simple terms: if 9 leftover items represent 0.12 of the group size, then we can figure out what the group size is!

This means: \(\mathrm{0.12 = 9/y}\)

3. Set up the remainder equation

Now we can solve this directly. We know that:

\(\mathrm{0.12 = 9/y}\)

To find y, we can rearrange this equation:

\(\mathrm{y = 9 ÷ 0.12}\)

Let's convert 0.12 to a fraction to make this easier:

\(\mathrm{0.12 = 12/100 = 3/25}\)

So: \(\mathrm{y = 9 ÷ (3/25) = 9 × (25/3) = (9 × 25)/3 = 225/3 = 75}\)

Therefore, \(\mathrm{y = 75}\).

4. Verify with the division algorithm

Let's check our answer makes sense.

If \(\mathrm{y = 75}\), then from \(\mathrm{x/y = 96.12}\), we get:

\(\mathrm{x = 96.12 × 75 = 7,209}\)

Now let's verify using the division algorithm. When we divide 7,209 by 75:

• 75 goes into 7,209 exactly 96 times: \(\mathrm{96 × 75 = 7,200}\)
• Remainder: \(\mathrm{7,209 - 7,200 = 9}\) ✓
• Decimal check: \(\mathrm{9/75 = 0.12}\) ✓

Perfect! Our answer checks out completely.

Final Answer

The value of y is 75, which corresponds to answer choice (B).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the relationship between decimal and remainder

Many students fail to recognize that when \(\mathrm{x/y = 96.12}\), the decimal part 0.12 directly represents the remainder as a fraction of the divisor. They might try to work with 96.12 as a simple decimal without connecting it to the remainder of 9, missing the key insight that \(\mathrm{0.12 = 9/y}\).

2. Confusion about division algorithm components

Students often struggle to properly set up the division algorithm equation \(\mathrm{x = qy + r}\), where q is the quotient, y is the divisor, and r is the remainder. They might incorrectly think that 96.12 means 96 remainder 12, rather than understanding that 96 is the whole number quotient and the remainder is 9.

3. Attempting to use trial and error with answer choices

Instead of establishing the mathematical relationship, some students might immediately start plugging in answer choices without understanding the underlying connection between the decimal representation and the remainder, leading to inefficient problem-solving.

Errors while executing the approach

1. Decimal to fraction conversion errors

When converting 0.12 to a fraction, students commonly make arithmetic mistakes. They might incorrectly convert 0.12 to fractions like \(\mathrm{12/10}\) instead of \(\mathrm{12/100}\), or fail to simplify \(\mathrm{12/100}\) to \(\mathrm{3/25}\), leading to calculation errors.

2. Division of fractions mistakes

When solving \(\mathrm{y = 9 ÷ 0.12}\), students often struggle with dividing by a decimal or fraction. They might forget to multiply by the reciprocal when computing \(\mathrm{9 ÷ (3/25)}\), or make arithmetic errors in calculating \(\mathrm{(9 × 25)/3 = 225/3 = 75}\).

3. Verification calculation errors

During the checking phase, students might make computational mistakes when verifying that \(\mathrm{x = 96.12 × 75 = 7,209}\), or when confirming that \(\mathrm{7,209 ÷ 75}\) gives quotient 96 and remainder 9.

Errors while selecting the answer

1. Selecting a related but incorrect value

Students might calculate correctly but then select 96 (choice A) thinking it's the answer because it's the quotient from the division, rather than recognizing that \(\mathrm{y = 75}\) is what the question asks for.

2. Misreading the question requirement

Some students might solve for x instead of y, or confuse which variable the question is asking for, especially after working through multiple calculations involving both x and y values.