Making a new scale drawing of the sandbox using a scale from the playground to the new drawing of 20 inches to 1 inch requires the proportional conversion of the length of the sandbox in the original drawing to the length in the new drawing.
The justification of the accuracy of this drawing is based on the fact that the scale is proportional (equal ratios are used).
How the new scale drawing is developed:
The scale from the playground to the original drawing = 12 inches to 1 inch.
Based on the above, if the length of the sandbox in the original drawing is x inches, then the length of the sandbox in real life is 12x inches.
The scale from the playground to the new drawing = 20 inches to 1 inch.
If the length of the sandbox in the new drawing is y inches, then the length of the sandbox in real life is 20y inches.
Set up a proportion to find the length of the sandbox in the new drawing:
12x / 1 = 20y / 1
Simplifying this equation:
12x = 20y
y = (12/20)x
y = (3/5)x
The length of the sandbox in the new drawing = (3/5) times the length of the sandbox in the original drawing.
Thus, using the correct proportions, the new drawing will be an accurate representation of the sandbox.