Question

A carpenter used 9 1/2 ft of cedar for every 7 2/3 ft of redwood for a construction project. If the carpenter uses 4 3/4 ft of cedar, how much redwood does he need?

Answer 1

We can make a proportion to represent the woods used. On the top  is cedar, and on the bottom is the redwood. On the left is what was used the first time, on the right is the current problem.

Before making the proportion, let's do a preliminary step and write all the fractions as improper fractions. We do this because we can't directly multiply or divide mixed numbers, but we can with fractions.

$9\frac{1}{2} = \frac{2 * 9 + 1}{2} =\frac{19}{2}$  <-- Multiply denominator times whole number and add the leftover fraction.

$7\frac{2}{3} = \frac{3 * 7 + 2}{3} =\frac{23}{3}$  

$4\frac{3}{4} = \frac{4 * 4 + 3}{4} =\frac{19}{4}$  

Now we set up the proportion. We solve it by cross multiplying.

$\frac{Cedar}{Redwood} =\frac{Cedar}{Redwood}$

$\frac{19/2}{23/3} =\frac{19/4}{R}$

$\frac{19}{2} R = \frac{23}{3} *\frac{19}{4}$

It may seem right to simplify the right side, but let's leave it alone. We divide both sides by 19/2.

$R = (\frac{23}{3} *\frac{19}{4} ) / \frac{19}{2}$

Dividing by a fraction means multiplying by its reciprocal. Thus,

$R = \frac{23}{3} *\frac{19}{4} * \frac{2}{19}$

Here's where not simplifying really pays off. We can simplify NOW and the terms go away more easily. The 19s go away and simplify to 1, and the 2 and 4 simplify to 1 and 2.  All this happens by dividing out common numbers.

$R = \frac{23}{3} *\frac{1}{2} * \frac{1}{1}$

Now we multiply all the tops and all the bottoms.

R = 23 * 1 * 1 / 3 * 2 * 1

R = 23 / 6

While R is 23/6, we need to make it a mixed number. The way we made mixed numbers as improper fractions happens here, except in reverse.

23 ÷ 6 = 3 with a remainder of 5, as 3 × 6 + 5 = 23. Thus, $\frac{23}{6} = 3\frac{5}{6}$.

As a result, $3\frac{5}{6}$ feet of redwood is needed.