For the fractions 5/8, 7/12, 3/4 and 13/16, which of the following options shows them correctly arranged in ascending order of their numerical value?

Introduction / Context:
Comparing and ordering fractions is a fundamental arithmetic skill that is very important in aptitude exams. In this problem, we are given four fractions and asked to choose the option that arranges them in ascending order, that is, from the smallest value to the largest value. Being comfortable with fraction comparison helps in topics like ratios, percentages and data interpretation.

Given Data / Assumptions:

• Fractions given: 5/8, 7/12, 3/4 and 13/16.
• We must arrange them in increasing order (ascending order).
• All fractions are positive proper fractions (less than 1).

Concept / Approach:
There are several ways to compare fractions:

• Convert each fraction to an equivalent fraction with a common denominator.
• Convert each fraction to a decimal value and then compare the decimals.
• Use cross-multiplication to compare pairs of fractions.

For explanation clarity, converting to decimal or using approximate values works very well in exam conditions.

Step-by-Step Solution:
Step 1: Convert 7/12 to a decimal: 7 ÷ 12 ≈ 0.5833. Step 2: Convert 5/8 to a decimal: 5 ÷ 8 = 0.625. Step 3: Convert 3/4 to a decimal: 3 ÷ 4 = 0.75. Step 4: Convert 13/16 to a decimal: 13 ÷ 16 = 0.8125. Step 5: Now compare these decimal values: 0.5833, 0.625, 0.75, 0.8125. Step 6: Ordering them from smallest to largest gives 0.5833 < 0.625 < 0.75 < 0.8125. Step 7: Translating back to fractions, this order is 7/12 < 5/8 < 3/4 < 13/16.

Verification / Alternative Check:
We can also check using cross-multiplication. For example, to compare 7/12 and 5/8, compare 7 * 8 = 56 and 5 * 12 = 60. Since 56 < 60, we have 7/12 < 5/8. Similar pairwise checks will confirm the order 7/12 < 5/8 < 3/4 < 13/16.

Why Other Options Are Wrong:
Option A: 5/8 < 7/12 is incorrect because 7/12 is actually smaller than 5/8.
Option B: Again starts with 5/8 and 7/12 in the wrong order and also misplaces 13/16 and 3/4.
Option C: Puts 7/12 last, which is wrong since 7/12 is the smallest fraction.
Only option D shows the correct ascending order.

Common Pitfalls:
A common error is to compare only numerators or only denominators without considering the fraction as a whole. Another mistake is to think that the larger denominator always gives a smaller fraction, which is not correct unless numerators are equal. Always compare fractions logically, either by using a common denominator, decimals, or cross-multiplication.

Final Answer:
The correct ascending order is 7/12 < 5/8 < 3/4 < 13/16.