The Computational Cost of
Mixed Numbers

1. Unconscious Lack of Optimization

In elementary arithmetic, when dealing with calculations involving mixed numbers, we are often taught the following rule: "Convert them to improper fractions first, then calculate."

I had accepted this teaching without much doubt. However, when I reconsidered this procedure from an engineering perspective (computational complexity and memory efficiency), I felt a sense of unease.

"Isn't this sacrificing computational efficiency (performance)?"

The inefficiency becomes particularly pronounced in "addition and subtraction where denominators are different (requiring common denominators)."

2. Case Study: Addition with Common Denominators

Let's look at a concrete example. Consider the following calculation:

$$3 \frac{5}{12} + 2 \frac{7}{18}$$

A. The "School" Way: Improper Fractions

First, convert the mixed numbers to improper fractions, then find a common denominator.

1. Conversion (Multiplication cost incurred):
   \(3 \frac{5}{12} \rightarrow \frac{3 \times 12 + 5}{12} = \frac{41}{12}\)
   \(2 \frac{7}{18} \rightarrow \frac{2 \times 18 + 7}{18} = \frac{43}{18}\)
   (The numerators are already getting large)
2. Common Denominator (LCM 36):
   \(\frac{41 \times 3}{36} + \frac{43 \times 2}{36} = \frac{123}{36} + \frac{86}{36}\)
   (3-digit numbers appear. Mental arithmetic becomes difficult)
3. Addition and Reconversion:
   \(\frac{209}{36} \rightarrow 209 \div 36 = 5 \dots 29 \rightarrow \mathbf{5 \frac{29}{36}}\)

Evaluation: Intermediate values swelled to "123", "86", and "209".

B. The "Engineer" Way: Divide and Conquer

Separate (Divide) the integer part and the fractional part for calculation.

1. Separation and Common Denominator:
   Integer Part: \(3 + 2 = \mathbf{5}\)
   Fractional Part: \(\frac{5}{12} + \frac{7}{18} = \frac{15}{36} + \frac{14}{36}\)
2. Addition:
   Fractional Part: \(\frac{15 + 14}{36} = \frac{29}{36}\)
3. Combination:
   \(\mathbf{5 \frac{29}{36}}\)

Evaluation: The maximum value handled is only "36". It can be solved instantly with mental arithmetic.

3. Why is "Improper Fraction Conversion" Recommended?

If Method B is clearly more efficient, why do schools teach Method A? While we cannot know the exact intent, it is likely that they prioritize "algorithmic generality."

While the separation method works for addition and subtraction, it does not apply to multiplication and division (e.g., \(1 \frac{1}{2} \times 1 \frac{1}{2}\)). Establishing conversion to improper fractions as a standard procedure that works for all four arithmetic operations is very rational in terms of avoiding confusion.

4. Dr.WataWata Insight