Numerical Structure Math Example 2

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Example 2

hard
Use the distributive property to compute 47×9847 \times 98 mentally, and explain the structural reasoning.

Solution

  1. 1
    Rewrite 98=100−298 = 100 - 2.
  2. 2
    Apply distributive property: 47×98=47×(100−2)=47×100−47×247 \times 98 = 47 \times (100 - 2) = 47 \times 100 - 47 \times 2.
  3. 3
    47×100=4,70047 \times 100 = 4{,}700 and 47×2=9447 \times 2 = 94.
  4. 4
    47×98=4,700−94=4,60647 \times 98 = 4{,}700 - 94 = 4{,}606.

Answer

47×98=4,60647 \times 98 = 4{,}606
The distributive property a(b+c)=ab+aca(b+c) = ab + ac is the engine of mental multiplication. By decomposing a difficult factor into a nearby round number plus a small adjustment, we replace one hard multiplication with two easy ones. This strategy underpins algebraic factoring and FOIL.

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 3 easy

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

Example 4 medium

Show that [formula] using the structure of fraction arithmetic, and verify with [formula], [formula]

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