Surfboard Designs

Explanation

Analyze

• The surfboard is to have a \(\frac38\)-inch-thick fiberglass finish (Given)
• Each layer of fiberglass is \(\frac{1}{16}\) of an inch thick. (Given)
• How many layers of fiberglass need to be applied? (Find)

Form

Think of the \(\frac38\)-inch-thick finish separated into an unknown number of equally thick layers of fiberglass.  This indicates division.  We translate the words of the problem to numbers and symbols

[The number of layers of fiberglass that are needed] is equal to [the thickness of the finish] divided by [the thickness of 1 layer of the fiberglass]

The number of layers of fiberglass that are needed = \(\frac38 ÷ \frac{1}{16}\)

Solve

To find the quotient, we will use the rule for dividing two fractions

• Multiply \(\frac38\) by the reciprocal of \(\frac{1}{16}\), which is \(\frac{16}{1}\)
  • \(\frac38 \color{red}{÷ \frac{1}{16}} = \frac38 \color{red}{\cdot \frac{16}{1}}\)
• Multiply the numerators.  Multiply the denominators
  • = \(\frac{3 \cdot 16}{8 \cdot 1}\)
• To simplify, factor 16 as 2 • 8.  Then remove the common factor of 8 from the numerator and denominator.
  • = \(\require{cancel} \frac{3 \cdot 2 \cdot \cancel{8}^1}{\cancel{8}_1 \cdot 1}\)
• Multiply the remaining factors in the numerators.  Multiply the remaining factors in the denominator.
  • = \(\frac61\)
• Any whole number divided by 1 is the same number
  • = 6

State The number of layers of fiberglass needed is 6

Check 

If 6 layers of fiberglass, each \(\frac{1}{16}\) of an inch thick, are used, the finished thickness will be \(\frac{6}{16}\) of an inch.  If we simplify \(\frac{6}{16}\), we see that it is equivalent to the desired finish thickness

$\require{cancel} \frac{6}{16} = \frac{\cancel{2}^1 \cdot 3}{\cancel{2}_1} \cdot 8 = \frac38$

The result checks