Question

complex fractions and unit rates- homework
simplify: (show all your work.)
1.
\\(\frac{18}{\frac{3}{4}}\\)
2.
\\(\frac{\frac{3}{6}}{4}\\)
3.
\\(\frac{\frac{1}{3}}{\frac{1}{4}}\\)

1. pep club members are making spirit buttons. they make 490 spirit buttons in 3 ½ hours. find the number of buttons the pep club makes per hour.
2. mary is making pillows for her life skill class. she bought 2 ½ yards of fabric. her total cost was $15. what was the cost per yard?
3. doug entered a canoe race. he rowed 3 ½ miles in ½ hour. what is his average speed in miles per hour.
4. monica reads 7 ½ pages of a mystery book in 9 minutes. what is her average reading rate in pages per minute?

Explanation:

Let's solve problem 1: Simplify \(\frac{18}{\frac{3}{4}}\)

Step 1: Recall the rule for dividing by a fraction

Dividing by a fraction is the same as multiplying by its reciprocal. So, \(\frac{a}{\frac{b}{c}}=a\times\frac{c}{b}\) (where \(a = 18\), \(b = 3\), \(c = 4\))
\(\frac{18}{\frac{3}{4}}=18\times\frac{4}{3}\)

Step 2: Simplify the multiplication

First, we can simplify 18 and 3. 18 divided by 3 is 6.
\(18\times\frac{4}{3}=\frac{18\times4}{3}=\frac{6\times3\times4}{3}\)
The 3 in the numerator and denominator cancels out, leaving \(6\times4 = 24\)

Step 1: Simplify \(\frac{3}{6}\)

\(\frac{3}{6}=\frac{1}{2}\)

Step 2: Divide by 4 (multiply by reciprocal)

\(\frac{1}{2}\div4=\frac{1}{2}\times\frac{1}{4}=\frac{1\times1}{2\times4}=\frac{1}{8}\)

Wait, but maybe the original problem was \(\frac{3}{6/4}\) (complex fraction). Let's re - examine. If the complex fraction is \(\frac{\frac{3}{6}}{4}\), then:

Step 1: Simplify the numerator

\(\frac{3}{6}=\frac{1}{2}\)

Step 2: Divide by 4

\(\frac{\frac{1}{2}}{4}=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}\)

If the complex fraction is \(\frac{3}{\frac{6}{4}}\), then:

Step 1: Multiply by reciprocal

\(\frac{3}{\frac{6}{4}} = 3\times\frac{4}{6}\)

Step 2: Simplify

3 and 6 have a common factor of 3. \(3\times\frac{4}{6}=\frac{3\times4}{6}=\frac{12}{6}=2\)

Assuming the problem is \(\frac{3}{\frac{6}{4}}\) (maybe a typo in the original writing):

Step 1: Use the rule for complex fractions

\(\frac{a}{\frac{b}{c}}=a\times\frac{c}{b}\), here \(a = 3\), \(b = 6\), \(c = 4\)
\(\frac{3}{\frac{6}{4}}=3\times\frac{4}{6}\)

Step 2: Simplify the fraction

3 and 6 can be simplified. \(\frac{3\times4}{6}=\frac{12}{6}=2\)

Answer:

24

Let's solve problem 2: Simplify \(\frac{\frac{3}{6}}{4}\) (Wait, actually the original problem seems to be \(\frac{3}{6}\div4\) or \(\frac{\frac{3}{6}}{4}\), but let's check the correct interpretation. If it's \(\frac{3}{6}\) divided by 4, we can do: