An artist is hired to create an art display for the interior of a city building. The display is to span a total width of 10 yd. The artist has equally sized portraits placed side-by-side in a horizontal line with no gaps. Each portrait has a width of 45 in. How many portraits are in the display?

First, fill in the blanks on the left side of the equation using three of the ratios shown. Then write your answer on the right side of the equation.

Ratios:
[tex]\[
\frac{1 \text{ portrait}}{45 \text{ in}}, \quad
\frac{1 \text{ ft}}{12 \text{ in}}, \quad
\frac{12 \text{ in}}{1 \text{ ft}}, \quad
\frac{3 \text{ ft}}{1 \text{ yd}}, \quad
\frac{1 \text{ yd}}{3 \text{ ft}}
\][/tex]

[tex]\[
\frac{1 \text{ yd}}{3 \text{ ft}} \times \frac{1 \text{ ft}}{12 \text{ in}} \times \frac{1 \text{ portrait}}{45 \text{ in}} \times 10 \text{ yd} = \square \text{ portraits}
\][/tex]



Answer :

Sure, let's solve this step-by-step.

1. Convert the total width from yards to inches:
- Given:
- Total width [tex]\(= 10\)[/tex] yards
- Each portrait width [tex]\(= 45\)[/tex] inches

To convert 10 yards to inches, we need to use the conversions between yards to feet and feet to inches.

- 1 yard = 3 feet
- 1 foot = 12 inches

Therefore, the total width in inches can be calculated as:
[tex]\[ \text{Total width} = 10 \text{ yd} \times 3 \text{ ft/yd} \times 12 \text{ in/ft} \][/tex]
This converts to:
[tex]\[ \text{Total width} = 10 \times 3 \times 12 = 360 \text{ inches} \][/tex]

2. Calculate the number of portraits that can fit in the total width:
- Each portrait is 45 inches wide.

To find how many portraits can fit in the total width, we divide the total width by the width of one portrait:
[tex]\[ \text{Number of portraits} = \frac{\text{Total width}}{\text{Width of one portrait}} = \frac{360 \text{ in}}{45 \text{ in/portrait}} \][/tex]

Performing the division:
[tex]\[ \text{Number of portraits} = \frac{360}{45} = 8 \][/tex]

3. Fill in the blanks for the ratios and the final answer:

First, line up the necessary ratios for converting yards to inches and then to the number of portraits:
[tex]\[ \frac{10 \text{ yd}}{1} \times \frac{3 \text{ ft}}{1 \text{ yd}} \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{1 \text{ portrait}}{45 \text{ in}} = 8 \text{ portraits} \][/tex]

So the detailed step-by-step solution is as follows:

- Starting with 10 yards:
[tex]\[ 10 \text{ yd} \rightarrow \][/tex]
- Convert to feet:
[tex]\[ 10 \text{ yd} \times 3 \text{ ft/yd} = 30 \text{ ft} \][/tex]
- Convert to inches:
[tex]\[ 30 \text{ ft} \times 12 \text{ in/ft} = 360 \text{ in} \][/tex]
- Convert inches to the number of portraits:
[tex]\[ 360 \text{ in} \times \frac{1 \text{ portrait}}{45 \text{ in}} = 8 \text{ portraits} \][/tex]

Therefore, the artist can fit 8 portraits in the display.