Convert the decimal number 0.0068 into a vulgar fraction and express your answer in lowest terms.

Introduction / Context:
Converting terminating decimals into vulgar fractions is a basic skill in arithmetic. A vulgar fraction is simply an ordinary fraction like a/b. In this problem, the decimal 0.0068 must be written as a fraction and simplified to lowest terms. This type of conversion is often used in problems involving precision, measurement and ratios.

Given Data / Assumptions:

• Decimal given: 0.0068.
• We assume it is exact and terminating.
• We must express it as a fraction in lowest terms.

Concept / Approach:
To convert a terminating decimal to a fraction:

• Count the number of digits after the decimal point.
• Write the decimal as an integer divided by a power of 10 (10, 100, 1000, etc.).
• Simplify the resulting fraction by cancelling common factors between numerator and denominator.

Here, 0.0068 has four digits after the decimal point, so we will use 10^4 = 10000 as the denominator initially.

Step-by-Step Solution:
Step 1: Recognize that 0.0068 has four decimal places. Step 2: Write 0.0068 as 68/10000, because moving the decimal point four places to the right gives 68. Step 3: Now simplify the fraction 68/10000. Step 4: Find the greatest common divisor of 68 and 10000. Both are divisible by 4. Step 5: Divide the numerator by 4: 68 ÷ 4 = 17. Step 6: Divide the denominator by 4: 10000 ÷ 4 = 2500. Step 7: So 68/10000 simplifies to 17/2500. Step 8: Check if 17 and 2500 have any further common factors. 17 is prime and does not divide 2500, so 17/2500 is already in lowest terms.

Verification / Alternative Check:
Convert 17/2500 back to a decimal. Since 2500 = 25 * 100, we can write 17/2500 = (17/25) / 100. Now 17/25 = 0.68. Dividing 0.68 by 100 shifts the decimal two more places to the left, giving 0.0068. This confirms our fraction is correct.

Why Other Options Are Wrong:
7/1250 ≈ 0.0056, which is less than 0.0068.
17/5000 ≈ 0.0034, which is half of 0.0068, so it is not correct.
7/1750 ≈ 0.004, also not equal to 0.0068. Only 17/2500 converts exactly to the given decimal.

Common Pitfalls:
Students sometimes miscount decimal places and write 0.0068 as 68/1000 instead of 68/10000. Another error is stopping simplification too early, for example leaving the answer as 68/10000. Always look for common divisors to reduce the fraction completely.

Final Answer:
The decimal 0.0068 expressed as a vulgar fraction in lowest terms is 17/2500.