Question

parallelogram c is a scaled copy of parallelogram b.
diagrams: parallelogram b has top side ( 12\frac{1}{2} ), right side 35; parallelogram c has top side ( 8\frac{3}{4} ), right side ( 24\frac{1}{2} )
what scale factor takes parallelogram b to parallelogram c?

Explanation:

Step1: Convert mixed numbers to improper fractions

For the length of parallelogram B: \( 12\frac{1}{2}=\frac{12\times2 + 1}{2}=\frac{25}{2} \), \( 35=\frac{35}{1} \)
For the length of parallelogram C: \( 8\frac{3}{4}=\frac{8\times4+3}{4}=\frac{35}{4} \), \( 24\frac{1}{2}=\frac{24\times2 + 1}{2}=\frac{49}{2} \) (We can use either pair of corresponding sides to find the scale factor. Let's use the vertical sides: 35 (B) and \( 24\frac{1}{2} \) (C) or the horizontal sides: \( 12\frac{1}{2} \) (B) and \( 8\frac{3}{4} \) (C). Let's use the vertical sides first. Wait, actually, the scale factor is the ratio of corresponding sides of C to B. So take a side of C and divide by the corresponding side of B. Let's use the horizontal sides: \( 8\frac{3}{4}\div12\frac{1}{2} \) or vertical sides: \( 24\frac{1}{2}\div35 \). Let's compute \( 24\frac{1}{2}\div35 \). Convert \( 24\frac{1}{2} \) to improper fraction: \( \frac{49}{2} \), and 35 is \( \frac{35}{1} \). So \( \frac{49}{2}\div\frac{35}{1}=\frac{49}{2}\times\frac{1}{35}=\frac{49}{70}=\frac{7}{10} \)? Wait, no, wait. Wait, maybe I mixed up B and C. Wait, the problem is scale factor from B to C, so C is the image, B is the original. So scale factor is (length of C)/(length of B). Let's check the horizontal sides: \( 8\frac{3}{4} \) (C) and \( 12\frac{1}{2} \) (B). So \( 8\frac{3}{4}\div12\frac{1}{2} \). Convert to improper fractions: \( \frac{35}{4}\div\frac{25}{2}=\frac{35}{4}\times\frac{2}{25}=\frac{70}{100}=\frac{7}{10} \). Wait, or vertical sides: \( 24\frac{1}{2}\div35 \). \( 24\frac{1}{2}=\frac{49}{2} \), 35=\( \frac{35}{1} \). \( \frac{49}{2}\div\frac{35}{1}=\frac{49}{2}\times\frac{1}{35}=\frac{49}{70}=\frac{7}{10} \). So that's consistent. So the scale factor is \( \frac{7}{10} \) or 0.7. Wait, let's verify with the other side. \( 12\frac{1}{2}=\frac{25}{2} \), \( 8\frac{3}{4}=\frac{35}{4} \). \( \frac{35}{4}\div\frac{25}{2}=\frac{35}{4}\times\frac{2}{25}=\frac{70}{100}=\frac{7}{10} \). Yep, same result. So the scale factor is \( \frac{7}{10} \) or 0.7. Wait, but let's do it with mixed numbers. \( 24\frac{1}{2}\div35 \). \( 24\frac{1}{2} \) is 24.5, 35 is 35. 24.5 ÷ 35 = 0.7, which is \( \frac{7}{10} \). Similarly, \( 8\frac{3}{4} \) is 8.75, \( 12\frac{1}{2} \) is 12.5. 8.75 ÷ 12.5 = 0.7, which is \( \frac{7}{10} \). So that's correct.

Step2: Calculate the scale factor

To find the scale factor from B to C, we take a corresponding side length of C and divide it by the corresponding side length of B. Let's use the vertical sides:
Length of B (vertical): \( 35 \)
Length of C (vertical): \( 24\frac{1}{2}=\frac{49}{2} \)
Scale factor \( = \frac{24\frac{1}{2}}{35}=\frac{\frac{49}{2}}{35}=\frac{49}{2}\times\frac{1}{35}=\frac{49}{70}=\frac{7}{10} \)
Or using horizontal sides:
Length of B (horizontal): \( 12\frac{1}{2}=\frac{25}{2} \)
Length of C (horizontal): \( 8\frac{3}{4}=\frac{35}{4} \)
Scale factor \( = \frac{8\frac{3}{4}}{12\frac{1}{2}}=\frac{\frac{35}{4}}{\frac{25}{2}}=\frac{35}{4}\times\frac{2}{25}=\frac{70}{100}=\frac{7}{10} \)

Answer:

\( \frac{7}{10} \)