Question

avery is hanging three picture frames above her couch. each picture frame measures ( 9\frac{1}{5} ) inches by ( 11\frac{2}{3} ) inches. what is the amount of area that the picture frames will take up?

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(9\frac{1}{5}\) to an improper fraction. \(9\frac{1}{5}=\frac{9\times5 + 1}{5}=\frac{46}{5}\). Then, convert \(11\frac{2}{3}\) to an improper fraction. \(11\frac{2}{3}=\frac{11\times3+2}{3}=\frac{35}{3}\).

Step2: Calculate the area of one picture frame

The area of a rectangle is length times width. So the area of one picture frame is \(\frac{46}{5}\times\frac{35}{3}\). Simplify this: \(\frac{46\times35}{5\times3}=\frac{46\times7}{3}=\frac{322}{3}\) square inches.

Step3: Calculate the total area for three picture frames

Multiply the area of one frame by 3: \(3\times\frac{322}{3}= 322\) square inches. Wait, that seems off. Wait, no, let's recalculate step 2. Wait, \(\frac{46}{5}\times\frac{35}{3}\): 35 divided by 5 is 7, so \(46\times7 = 322\), then over 3? Wait, no, \(\frac{46}{5}\times\frac{35}{3}=\frac{46\times35}{15}=\frac{46\times7}{3}=\frac{322}{3}\approx107.33\) per frame. Then three frames: \(3\times\frac{322}{3}=322\)? Wait, no, that can't be. Wait, no, I made a mistake. Wait, \(9\frac{1}{5}\) is \(9.2\) and \(11\frac{2}{3}\) is approximately \(11.666...\). So area of one frame is \(9.2\times11.666...\). Let's calculate that: \(9.2\times11.666... = 9.2\times\frac{35}{3}=\frac{9.2\times35}{3}=\frac{322}{3}\approx107.333\). Then three frames: \(3\times\frac{322}{3}=322\). Wait, that's correct because the 3s cancel. Wait, but let's do it with fractions properly.

Wait, \(9\frac{1}{5}=\frac{46}{5}\), \(11\frac{2}{3}=\frac{35}{3}\). Area of one frame: \(\frac{46}{5}\times\frac{35}{3}=\frac{46\times35}{15}\). 35 and 15 have a common factor of 5: 35÷5 = 7, 15÷5 = 3. So \(\frac{46\times7}{3}=\frac{322}{3}\). Then three frames: \(3\times\frac{322}{3}=322\). Oh, right, because multiplying by 3 and dividing by 3 cancels out. So the total area is 322 square inches.

Wait, but let's check with decimals. \(9\frac{1}{5}=9.2\), \(11\frac{2}{3}\approx11.6667\). Area of one frame: \(9.2\times11.6667\approx107.333\). Three frames: \(107.333\times3 = 322\). Yep, that matches. So the calculation is correct.

Answer:

322 square inches