MBRA: Multiplying mixed numbers

• How to multiply Natural Numbers
• How to multiply fractions
• How to convert an Integer to a fraction
• That a Mixed Number is an Integer plus a fraction

Example: 7²⁄₇ × 6³⁄₄:

1. We start by drawing a picture of the multiplication. This is very similar to the picture(s) we would have drawn for Natural Number multiplication:

   	7 2⁄7
   6 3⁄4

2. Next, we break the mixed numbers up into Integer and fractional parts.
   • 7²⁄₇ = 7 + ²⁄₇, and
   • 6³⁄₄ = 6 + ³⁄₄

   Our picture then looks like this:

   	7	+ 2⁄7
   6
   + 3⁄4

3. We can then solve for the value of each piece. We can solve these in any order:
   • We'll solve the 6 × 7 piece first. 6 × 7 is 42, so:

     	7	+ 2⁄7
     6	42
     + 3⁄4

   • Next, 6 × ²⁄₇ = ⁶⁄₁ × ²⁄₇ = ¹²⁄₇:

     	7	+ 2⁄7
     6	42	12⁄7
     + 3⁄4

   • Third, ³⁄₄ × 7 = ³⁄₄ × ⁷⁄₁ = ²¹⁄₄:

     	7	+ 2⁄7
     6	42	12⁄7
     + 3⁄4	21⁄4

   • Finally, ³⁄₄ × ²⁄₇ = ⁶⁄₂₈:

     	7	+ 2⁄7
     6	42	12⁄7
     + 3⁄4	21⁄4	6⁄28

4. Now, we add the four pieces together. The pieces are:
   • 42,
   • ¹²⁄₇,
   • ²¹⁄₄, and
   • ⁶⁄₂₈
   Our final answer will be 42 + ¹²⁄₇ + ²¹⁄₄ + ⁶⁄₂₈.

   Which is: 49 ⁵⁄₂₈

5. We can check this by converting the two mixed numbers to improper fractions, multiplying the improper fractions and then converting back to a mixed number. So:
   • 6³⁄₄ = ²⁷⁄₄
   • 7²⁄₇ = ⁵¹⁄₇
   • ²⁷⁄₄ × ⁵¹⁄₇ = ¹³⁷⁷⁄₂₈
   • ¹³⁷⁷⁄₂₈ = 49 ⁵⁄₂₈

• The individual numbers can stay smaller, and
• The approach introduces (without naming it), the distributive property, which will appear when learning Algebra.