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Mateo produces a street sign graphic on his computer that is 4 inches by 5 inches. He enlarges the graphic by a scale factor of 3 to print. Then Mateo enlarges the image by a scale factor of 4 before sending it to the machinist.



What are the dimensions of the sign? Write the smaller dimension first and the larger dimension second.

7 answers

First, we start with the original dimensions of the graphic, which are 4 inches by 5 inches.

  1. First Enlargement (Scale Factor of 3):

    • Smaller dimension: \( 4 \times 3 = 12 \) inches
    • Larger dimension: \( 5 \times 3 = 15 \) inches

    After the first enlargement, the dimensions are 12 inches by 15 inches.

  2. Second Enlargement (Scale Factor of 4):

    • Smaller dimension: \( 12 \times 4 = 48 \) inches
    • Larger dimension: \( 15 \times 4 = 60 \) inches

    After the second enlargement, the dimensions are 48 inches by 60 inches.

Finally, the dimensions of the sign, with the smaller dimension first and the larger dimension second, are 48 inches by 60 inches.

Are the enlarged graphics similar to the original

Yes, the enlarged graphics are similar to the original graphic.

Two shapes are considered similar if they have the same shape but may differ in size. This means that the ratios of corresponding dimensions are equal.

  1. Original Dimensions: 4 inches by 5 inches

    • The ratio of the widths (smaller dimension to larger dimension) is \( \frac{4}{5} \).

  2. After the First Enlargement: 12 inches by 15 inches

    • The ratio remains \( \frac{12}{15} = \frac{4}{5} \).

  3. After the Second Enlargement: 48 inches by 60 inches

    • The ratio remains \( \frac{48}{60} = \frac{4}{5} \).

Since the ratios of the dimensions are the same throughout the enlargements, we conclude that the enlarged graphics maintain the same shape as the original, confirming they are similar.

Pam takes a picture of a painting to use the image for a postcard. The painting is 16 inches by 40 inches. She reduces the image by a scale factor of 14
to view the image on her computer. Pam further reduces the image by a scale factor of 12
to match the dimensions of a postcard.



What are the dimensions of the postcard? Write the smaller dimension first and the larger dimension second.

Let's start with the original dimensions of the painting, which are 16 inches by 40 inches.

  1. First Reduction (Scale Factor of \( \frac{1}{14} \)):

    • Smaller dimension: \( 16 \times \frac{1}{14} = \frac{16}{14} = \frac{8}{7} \) inches
    • Larger dimension: \( 40 \times \frac{1}{14} = \frac{40}{14} = \frac{20}{7} \) inches

    After the first reduction, the dimensions are approximately \( 1.14 \) inches by \( 2.86 \) inches (or exactly \( \frac{8}{7} \) inches by \( \frac{20}{7} \) inches).

  2. Second Reduction (Scale Factor of \( \frac{1}{12} \)):

    • Smaller dimension: \( \frac{8}{7} \times \frac{1}{12} = \frac{8}{84} = \frac{2}{21} \) inches
    • Larger dimension: \( \frac{20}{7} \times \frac{1}{12} = \frac{20}{84} = \frac{5}{21} \) inches

    After the second reduction, the dimensions are approximately \( 0.10 \) inches by \( 0.24 \) inches (or exactly \( \frac{2}{21} \) inches by \( \frac{5}{21} \) inches).

Finally, writing the smaller dimension first and the larger dimension second, the dimensions of the postcard are:

\[ \frac{2}{21} \text{ inches by } \frac{5}{21} \text{ inches} \]

If we convert these to decimal form for easier understanding:

  • \( \frac{2}{21} \approx 0.095 \) inches
  • \( \frac{5}{21} \approx 0.238 \) inches

Thus, we can say the dimensions of the postcard are approximately 0.095 inches by 0.238 inches.

Are the reduced images similar to the original

Yes, the reduced images are similar to the original painting.

Two shapes are similar if they have the same shape but may differ in size, which means the ratios of corresponding dimensions should remain the same.

  1. Original Dimensions: 16 inches by 40 inches

    • The ratio of the original dimensions is \[ \frac{16}{40} = \frac{2}{5} \text{ (or 0.4)} \]

  2. After the First Reduction (Scale Factor of \( \frac{1}{14} \)):

    • Reduced dimensions:
      • Smaller dimension: \( 16 \times \frac{1}{14} = \frac{8}{7} \) inches
      • Larger dimension: \( 40 \times \frac{1}{14} = \frac{20}{7} \) inches
    • The ratio of these dimensions is \[ \frac{\frac{8}{7}}{\frac{20}{7}} = \frac{8}{20} = \frac{2}{5} \text{ (or 0.4)} \]

  3. After the Second Reduction (Scale Factor of \( \frac{1}{12} \)):

    • Further reduced dimensions:
      • Smaller dimension: \( \frac{8}{7} \times \frac{1}{12} = \frac{2}{21} \) inches
      • Larger dimension: \( \frac{20}{7} \times \frac{1}{12} = \frac{5}{21} \) inches
    • The ratio of these dimensions is \[ \frac{\frac{2}{21}}{\frac{5}{21}} = \frac{2}{5} \text{ (or 0.4)} \]

Since the ratios of the dimensions remain consistent throughout the reductions (\(\frac{2}{5}\)), we can conclude that the reduced images maintain the same shape as the original painting and are therefore similar.