Question

a particular dress requires $4\frac{1}{4}$ yards of fabric for manufacturing. if the matching jacket requires $\frac{2}{3}$ yards less fabric, how much fabric is needed for both pieces? (enter your answer as a simplified mixed number.) \underline{\quad\quad} yd

Explanation:

Step1: Simplify the dress's fabric amount

First, simplify \(4\frac{1}{4}\). \(4\frac{1}{4}=\frac{17}{4}\) (since \(4\times4 + 1=17\)).

Step2: Find the jacket's fabric amount

The jacket requires \(\frac{2}{3}\) yards less than the dress. So we subtract \(\frac{2}{3}\) from \(\frac{17}{4}\). To subtract these fractions, we need a common denominator, which is \(12\).
\(\frac{17}{4}-\frac{2}{3}=\frac{17\times3}{4\times3}-\frac{2\times4}{3\times4}=\frac{51}{12}-\frac{8}{12}=\frac{43}{12}\). Convert \(\frac{43}{12}\) to a mixed number: \(3\frac{7}{12}\).

Step3: Find total fabric for both pieces

Now, add the fabric for the dress (\(\frac{17}{4}\)) and the jacket (\(\frac{43}{12}\)). Convert \(\frac{17}{4}\) to twelfths: \(\frac{17\times3}{4\times3}=\frac{51}{12}\). Then add \(\frac{51}{12}+\frac{43}{12}=\frac{94}{12}\). Simplify \(\frac{94}{12}\): divide numerator and denominator by \(2\) to get \(\frac{47}{6}\). Convert \(\frac{47}{6}\) to a mixed number: \(7\frac{5}{6}\)? Wait, no, let's check again. Wait, maybe I made a mistake in step 2. Wait, the dress is \(4\frac{1}{4}\) which is \(\frac{17}{4}\), the jacket is dress minus \(\frac{2}{3}\), so jacket is \(\frac{17}{4}-\frac{2}{3}\). Let's recalculate that: \(\frac{17}{4}=\frac{51}{12}\), \(\frac{2}{3}=\frac{8}{12}\), so \(\frac{51 - 8}{12}=\frac{43}{12}=3\frac{7}{12}\). Then total is dress (\(4\frac{1}{4}=\frac{17}{4}=\frac{51}{12}\)) plus jacket (\(\frac{43}{12}\)): \(\frac{51 + 43}{12}=\frac{94}{12}=\frac{47}{6}\). Wait, \(\frac{47}{6}\) is \(7\frac{5}{6}\)? Wait, no, \(6\times7 = 42\), \(47-42 = 5\), so \(7\frac{5}{6}\)? Wait, but let's check the initial numbers again. Wait, \(4\frac{1}{4}\) is \(4.25\), \(\frac{2}{3}\) is approximately \(0.666\), so jacket is \(4.25 - 0.666\approx3.583\). Then total is \(4.25+3.583\approx7.833\). Now, \(\frac{47}{6}\approx7.833\), and \(7\frac{5}{6}=\frac{47}{6}\), which matches. Wait, but let's do the steps again carefully.

Alternative approach:
Dress: \(4\frac{1}{4}\) yards.
Jacket: \(4\frac{1}{4}-\frac{2}{3}\) yards.
Total: \(4\frac{1}{4}+(4\frac{1}{4}-\frac{2}{3}) = 2\times4\frac{1}{4}-\frac{2}{3}\).
Calculate \(2\times4\frac{1}{4}\): \(2\times4 + 2\times\frac{1}{4}=8+\frac{1}{2}=8\frac{1}{2}=\frac{17}{2}\).
Now, \(\frac{17}{2}-\frac{2}{3}\). Common denominator is \(6\). \(\frac{17\times3}{2\times3}-\frac{2\times2}{3\times2}=\frac{51}{6}-\frac{4}{6}=\frac{47}{6}=7\frac{5}{6}\).

Answer:

\(7\frac{5}{6}\)