Beth is designing a card. She brings to a print shop a drawing that is 6 inches by 3 inches and asks them to print the card after applying a scale factor of [tex]\frac{2}{3}[/tex]. When she returns to the print shop, they hand her a box of cards, each of which is 9 inches by [tex]13 \frac{1}{2}[/tex] inches, which is incorrect.

What should be the correct dimensions of the card?

A. The scale factor is a reduction, and the new card should be 4 in [tex]\times[/tex] 2 in.
B. The scale factor is a reduction, and the new card should be 4 in [tex]\times[/tex] 6 in.
C. The scale factor is for an enlargement, and the print shop created the right size card.
D. The scale factor is for an enlargement, but the print shop should have made the card 9 in by 6 in.



Answer :

Let's walk through the problem step by step to determine what should be the correct dimensions of Beth's card after applying the scale factor.

1. Initial dimensions of the card:
- Length: 6 inches
- Width: 3 inches

2. Scale factor:
- The provided scale factor is [tex]\(\frac{2}{3}\)[/tex].

3. Applying the scale factor:
- The scale factor is a reduction since [tex]\(\frac{2}{3}\)[/tex] is less than 1.
- To find the new dimensions, we multiply the initial dimensions by the scale factor.

[tex]\[ \text{New Length} = 6 \, \text{inches} \times \frac{2}{3} = 4 \, \text{inches} \][/tex]

[tex]\[ \text{New Width} = 3 \, \text{inches} \times \frac{2}{3} = 2 \, \text{inches} \][/tex]

4. Verification of current card dimensions:
- The print shop gave Beth a card measuring 9 inches by 13.5 inches.
- Comparing this to the initial dimensions multiplied by the scale factor, the print shop's dimensions are incorrect.

5. Correct dimensions of the card:
- The calculations show that after applying the scale factor of [tex]\(\frac{2}{3}\)[/tex], the correct dimensions of the card should be:

[tex]\[ \text{Length} = 4 \, \text{inches} \][/tex]

[tex]\[ \text{Width} = 2 \, \text{inches} \][/tex]

Thus, the correct dimensions of the card should be 4 inches by 2 inches, confirming that the first option is correct:

The correct answer is:
The scale factor is a reduction, and the new card should be 4 inches by 2 inches.