Question

An ant stands at one vertex of a rectangular sheet of paper. The ant then proceeds to walk all the way around the perimeter of the paper. The dimensions of the sheet of paper are 3 5/8 inches by 2 7/8 inches. How many inches does the ant travel? Show your work. Suppose that each dimension of the sheet of paper described in question 1 is increased by one inch. How many inches will the ant travel in this case? Explain your thinking.

Answer 1

To find the distance the ant travels around the perimeter of the rectangular sheet of paper, we first need to calculate the perimeter. The perimeter \( P \) of a rectangle is given by the formula:

\[ P = 2 \times (\text{length} + \text{width}) \]

Given the dimensions of the sheet of paper:

• Length = \( 3 \frac{5}{8} \) inches
• Width = \( 2 \frac{7}{8} \) inches

We first convert these mixed numbers into improper fractions.

For the length:

\[ 3 \frac{5}{8} = 3 + \frac{5}{8} = \frac{3 \times 8}{8} + \frac{5}{8} = \frac{24}{8} + \frac{5}{8} = \frac{29}{8} \text{ inches} \]

For the width:

\[ 2 \frac{7}{8} = 2 + \frac{7}{8} = \frac{2 \times 8}{8} + \frac{7}{8} = \frac{16}{8} + \frac{7}{8} = \frac{23}{8} \text{ inches} \]

Now we can plug these values into the perimeter formula:

\[ P = 2 \times \left(\frac{29}{8} + \frac{23}{8}\right) \]

Calculating \( \frac{29}{8} + \frac{23}{8} \):

\[ \frac{29}{8} + \frac{23}{8} = \frac{29 + 23}{8} = \frac{52}{8} = \frac{13}{2} \]

Now substituting back into the perimeter formula:

\[ P = 2 \times \frac{13}{2} = 13 \text{ inches} \]

Thus, the ant travels 13 inches around the perimeter of the rectangular sheet of paper.

Next, let's increase each dimension by one inch:

New length = \( 3 \frac{5}{8} + 1 = 4 \frac{5}{8} \) inches New width = \( 2 \frac{7}{8} + 1 = 3 \frac{7}{8} \) inches

Converting these to improper fractions:

New length:

\[ 4 \frac{5}{8} = 4 + \frac{5}{8} = \frac{4 \times 8}{8} + \frac{5}{8} = \frac{32}{8} + \frac{5}{8} = \frac{37}{8} \text{ inches} \]

New width:

\[ 3 \frac{7}{8} = 3 + \frac{7}{8} = \frac{3 \times 8}{8} + \frac{7}{8} = \frac{24}{8} + \frac{7}{8} = \frac{31}{8} \text{ inches} \]

Now we calculate the new perimeter:

\[ P' = 2 \times \left(\frac{37}{8} + \frac{31}{8}\right) \]

Calculating \( \frac{37}{8} + \frac{31}{8} \):

\[ \frac{37}{8} + \frac{31}{8} = \frac{37 + 31}{8} = \frac{68}{8} = \frac{17}{2} \]

Now substituting back into the new perimeter formula:

\[ P' = 2 \times \frac{17}{2} = 17 \text{ inches} \]

Thus, when each dimension is increased by one inch, the ant travels 17 inches around the new perimeter of the rectangular sheet of paper.

Final Answers:

• Original perimeter: 13 inches
• Increased dimensions perimeter: 17 inches