Question

brody and his siblings found a rope in their garage that was $54\frac{1}{4}$ feet long. after getting permission from their dad, they cut off $7\frac{3}{4}$ feet of the rope to make a jump rope, $6\frac{1}{6}$ feet to use for playing tug-of-war, and $22\frac{1}{2}$ feet to make a swing. they left the rest of the rope in the garage. how many feet of rope are in the garage now?
\\(\frac{\square}{\square}\\) feet

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(54\frac{1}{4}\) to an improper fraction: \(54\frac{1}{4}=\frac{54\times4 + 1}{4}=\frac{217}{4}\)
Convert \(7\frac{3}{4}\) to an improper fraction: \(7\frac{3}{4}=\frac{7\times4+3}{4}=\frac{31}{4}\)
Convert \(6\frac{1}{6}\) to an improper fraction: \(6\frac{1}{6}=\frac{6\times6 + 1}{6}=\frac{37}{6}\)
Convert \(22\frac{1}{2}\) to an improper fraction: \(22\frac{1}{2}=\frac{22\times2+1}{2}=\frac{45}{2}\)

Step2: Find a common denominator for subtraction

The denominators are 4, 4, 6, and 2. The least common denominator (LCD) of 4, 6, and 2 is 12.
Convert each fraction to have a denominator of 12:
\(\frac{217}{4}=\frac{217\times3}{4\times3}=\frac{651}{12}\)
\(\frac{31}{4}=\frac{31\times3}{4\times3}=\frac{93}{12}\)
\(\frac{37}{6}=\frac{37\times2}{6\times2}=\frac{74}{12}\)
\(\frac{45}{2}=\frac{45\times6}{2\times6}=\frac{270}{12}\)

Step3: Subtract the lengths of the cut - off ropes from the original length

First, add the lengths of the cut - off ropes: \(\frac{93}{12}+\frac{74}{12}+\frac{270}{12}=\frac{93 + 74+270}{12}=\frac{437}{12}\)
Then, subtract this sum from the original length: \(\frac{651}{12}-\frac{437}{12}=\frac{651 - 437}{12}=\frac{214}{12}\)

Step4: Simplify the resulting fraction

Simplify \(\frac{214}{12}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 214 and 12 is 2.
\(\frac{214\div2}{12\div2}=\frac{107}{6}\)
Convert \(\frac{107}{6}\) back to a mixed number: \(107\div6 = 17\) with a remainder of \(5\), so \(\frac{107}{6}=17\frac{5}{6}\) (Wait, let's check the calculation again. Wait, maybe we made a mistake in step 3. Let's recalculate the sum of the cut - off ropes:

Wait, original length: \(54\frac{1}{4}\), cut - off lengths: \(7\frac{3}{4}\), \(6\frac{1}{6}\), \(22\frac{1}{2}\)

Another way: Work with mixed numbers directly.

\(54\frac{1}{4}-7\frac{3}{4}-6\frac{1}{6}-22\frac{1}{2}\)

First, \(54\frac{1}{4}-7\frac{3}{4}=53\frac{5}{4}-7\frac{3}{4}=(53 - 7)+(\frac{5}{4}-\frac{3}{4}) = 46+\frac{2}{4}=46\frac{1}{2}\)

Then, \(46\frac{1}{2}-22\frac{1}{2}=24\)

Then, \(24-6\frac{1}{6}=23\frac{6}{6}-6\frac{1}{6}=(23 - 6)+(\frac{6}{6}-\frac{1}{6})=17\frac{5}{6}\) Wait, no, wait:

Wait, \(54\frac{1}{4}-7\frac{3}{4}\): \(54\frac{1}{4}=53 + 1+\frac{1}{4}=53+\frac{5}{4}\), \(7\frac{3}{4}=7+\frac{3}{4}\), so \(53 - 7 = 46\), \(\frac{5}{4}-\frac{3}{4}=\frac{2}{4}=\frac{1}{2}\), so \(54\frac{1}{4}-7\frac{3}{4}=46\frac{1}{2}\)

Then \(46\frac{1}{2}-22\frac{1}{2}=(46 - 22)+(\frac{1}{2}-\frac{1}{2}) = 24\)

Then \(24-6\frac{1}{6}=23\frac{6}{6}-6\frac{1}{6}=17\frac{5}{6}\)? Wait, no, \(24 = 23 + 1=23+\frac{6}{6}\), \(6\frac{1}{6}=6+\frac{1}{6}\), so \(23-6 = 17\), \(\frac{6}{6}-\frac{1}{6}=\frac{5}{6}\), so the result is \(17\frac{5}{6}\)? Wait, but let's check with the first method.

Wait, original length: \(54\frac{1}{4}=\frac{217}{4}\), sum of cut - off: \(7\frac{3}{4}+6\frac{1}{6}+22\frac{1}{2}=\frac{31}{4}+\frac{37}{6}+\frac{45}{2}\)

\(\frac{31}{4}+\frac{45}{2}=\frac{31 + 90}{4}=\frac{121}{4}\), then \(\frac{121}{4}+\frac{37}{6}=\frac{363+74}{12}=\frac{437}{12}\)

\(\frac{217}{4}-\frac{437}{12}=\frac{651 - 437}{12}=\frac{214}{12}=\frac{107}{6}=17\frac{5}{6}\)

Wait, but let's do the calculation step by step again:

Original length: \(54\frac{1}{4}\) feet.

Cut - off lengths: \(7\frac{3}{4}\), \(6\frac{1}{6}\), \(22\frac{1}{2}\)

First, add \(7\frac{3}{4}\) and \(22\frac{1}{2}\):

\(7\frac{3}{4}+22\frac{1}{2}=7\frac{3}{4}+22\frac{2}{4}=29\frac{5}{4}=30\frac{1}{4}\)

Then add \(6\frac{1}{6}\) to \(30\frac{1}{4}\):

\(30\frac{1}{4}+6\frac{1}{6}=30\frac{3}{12}+6\frac{2}{12}=36\frac{5}{12}\)

Now, subtract \(36\frac{5}{12}\) from \(54\frac{1}{4}\):

\(54\frac{1}{4}-36\frac{5}{12}=54\frac{3}{12}-36\frac{5}{12}=53\frac{15}{12}-36\frac{5}{12}=(53 - 36)+(\frac{15}{12}-\frac{5}{12}) = 17\frac{10}{12}=17\frac{5}{6}\)

Yes, so the remaining length is \(17\frac{5}{6}\) feet.

Answer:

\(17\frac{5}{6}\)