the distance from new York to Chicago is about 700 miles.
a- if this represents 9 1/3 inches on the map, what is the scale used?
b- calculate the number of inches on the map if the scale is inches/miles = 1/50 .
c- what is the difference in miles traveled over 6 inches using the scales in parts a and b
a- To find the scale, we can set up a proportion:
(700 miles)/(9 1/3 inches) = x miles/1 inch
To simplify, we can convert the mixed number to an improper fraction:
(700 miles)/(28/3 inches) = x miles/1 inch
Now we can multiply both sides by (1 inch)(28/3 inches):
700 miles = (x miles)(28/3 inches)
Simplifying the right side:
700 miles = (28/3)x miles
Now we can solve for x by multiplying both sides by 3/28:
x = (700 miles)(3/28) = 75 miles
Therefore, the scale used is 1 inch represents 75 miles.
b- If the scale is 1 inch represents 1/50 miles, we can calculate the number of inches on the map by setting up a proportion:
(1 inch)/(1/50 miles) = x inches/1 mile
To simplify, we can invert the fraction on the right side:
(1 inch)/(1/50 miles) = x inches/(1 mile/1)
Now we can multiply both sides by (1/50 miles)(1 mile/1):
1 inch = x inches/(1/50)
Simplifying the right side:
1 inch = x inches * (50/1)
Now we solve for x by dividing both sides by 50:
x = 1 inch / 50 = 0.02 inches
Therefore, the number of inches on the map using a scale of 1 inch represents 1/50 miles is 0.02 inches.
c- To find the difference in miles traveled over 6 inches using the scales in parts a and b, we can set up proportions using the scales:
For scale a: (9 1/3 inches)/(1 inch) = (700 miles)/x miles
To solve for x, we multiply both sides by x:
9 1/3 inches = (700 miles)/(1 inch) * x miles
Now we solve for x by dividing both sides by (700 miles)/(1 inch):
x = (9 1/3 inches) * (1 inch) / (700 miles) = 28/3 miles
Therefore, using scale a, 6 inches represents (28/3 miles) * 6 inches = 56 miles.
For scale b: (0.02 inches)/(1 inch) = (1/50 miles)/x miles
To solve for x, we multiply both sides by x:
0.02 inches = (1/50 miles)/(1 inch) * x miles
Now we solve for x by dividing both sides by (1/50 miles)/(1 inch):
x = (0.02 inches) * (1 inch) / (1/50 miles) = 1 mile
Therefore, using scale b, 6 inches represents 6 inches * 1 mile = 6 miles.
The difference in miles traveled over 6 inches using the scales in parts a and b is 56 miles - 6 miles = 50 miles.