Numerical Structure Math Example 4

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Example 4

medium
Show that 1a+1b=a+bab\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{a+b}{ab} using the structure of fraction arithmetic, and verify with a=3a=3, b=4b=4.

Solution

  1. 1
    Common denominator of 1a\frac{1}{a} and 1b\frac{1}{b} is abab: 1a=bab\dfrac{1}{a} = \dfrac{b}{ab} and 1b=aab\dfrac{1}{b} = \dfrac{a}{ab}.
  2. 2
    Add: bab+aab=a+bab\dfrac{b}{ab} + \dfrac{a}{ab} = \dfrac{a+b}{ab}.
  3. 3
    Verify with a=3,b=4a=3, b=4: 13+14=4+312=712\dfrac{1}{3} + \dfrac{1}{4} = \dfrac{4+3}{12} = \dfrac{7}{12}. ✓

Answer

1a+1b=a+bab\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{a+b}{ab}; for a=3,b=4a=3, b=4: 712\dfrac{7}{12}.
The algebraic identity for adding unit fractions follows directly from the structure of fraction addition: find a common denominator, rewrite each fraction, then add numerators. The formula a+bab\frac{a+b}{ab} is a compact structural result used in harmonic series, circuit theory, and lens equations.

About Numerical Structure

The underlying patterns, relationships, and algebraic properties—like commutativity and distributivity—that organize numbers into coherent systems.

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More Numerical Structure Examples

Example 1 medium

Identify which properties (commutative, associative, distributive, identity, inverse) are illustrate

Example 2 hard

Use the distributive property to compute [formula] mentally, and explain the structural reasoning.

Example 3 easy

Simplify using the appropriate property: (a) [formula], (b) [formula].

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