Arrange the fractions 3/4, 5/12, 13/16, 16/29 and 3/8 in ascending order of their magnitude (from smallest to largest).

Introduction / Context:
Ordering fractions from smallest to largest is a fundamental skill that helps in understanding ratios, proportions and comparisons in many quantitative problems. Here, five different fractions are given, and you must identify the option that lists them in ascending order of their magnitude.

Given Data / Assumptions:

• Fractions: 3/4, 5/12, 13/16, 16/29 and 3/8.
• We must order them from smallest to largest.
• All fractions are positive and less than 1, except 13/16 and 3/4 which are close to 1.

Concept / Approach:
We can compare fractions using:

• Decimal conversion (approximate values).
• Cross-multiplication between pairs.
• Converting all fractions to a common denominator (though this may be more tedious here).
In exam situations, converting to approximate decimals is often the fastest method when denominators are not too large.




Step-by-Step Solution:
Step 1: Convert each fraction into a decimal approximation. 3/8 = 0.375. 5/12 ≈ 5 ÷ 12 ≈ 0.4167. 16/29 ≈ 16 ÷ 29 ≈ 0.5517 (since 0.55 * 29 ≈ 15.95). 3/4 = 0.75 exactly. 13/16 = 0.8125. Step 2: List these decimal values: 0.375, 0.4167, 0.5517, 0.75, 0.8125. Step 3: Order them from smallest to largest: 0.375 < 0.4167 < 0.5517 < 0.75 < 0.8125. Step 4: Convert the ordered decimals back to fractions: 3/8 < 5/12 < 16/29 < 3/4 < 13/16.



Verification / Alternative Check:
We can verify individual pairwise comparisons using cross-multiplication. For example, to compare 3/8 and 5/12, check 3 * 12 = 36 and 5 * 8 = 40. Since 36 < 40, 3/8 < 5/12. Similar checks confirm the rest of the ordering, reinforcing our answer.




Why Other Options Are Wrong:
Option A: Places 3/4 before 3/8, which is wrong since 3/4 is much larger than 3/8.
Option C: Incorrectly places 13/16 before 3/4, even though 13/16 is greater than 3/4 in actual value.
Option D: Places 13/16 before 16/29 and 3/4, which contradicts the decimal comparisons. Only option B matches the correct ascending order.




Common Pitfalls:
A frequent error is to compare numerators or denominators separately without considering the fraction as a whole. Another mistake is miscalculating decimal approximations. Always check your arithmetic carefully, especially when dealing with non-terminating decimals, and be sure to maintain the correct order based on accurate comparisons.




Final Answer:
The correct ascending order is 3/8 < 5/12 < 16/29 < 3/4 < 13/16.