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Given: A circle is divided into five sections--one purple, one yellow, one red, one blue, and one green.
ResponderExcluirGiven: The purple section’s area is half the combined area of the other sections.
Given: The yellow section’s area is one-third the combined area of the other sections.
Given: The blue section’s area is one-fourth the combined area of the other sections.
Given: The red section’s area is one-fifth the combined area of the other sections.
Question: What fraction of the circle’s area is green?
Before we start, let’s define some variables:
a = area of the purple section
b = area of the yellow section
c = area of the red section
d = area of the blue section
f = area of the green section
We will also define the area of the whole circle to be 1.
Since the circle is divided into five sections, this means that the combined area of the sections is equal to the area of the circle. In other words:
a + b + c + d + f = 1 (Equation 1)
The other given information can similarly be expressed by equations as follows:
a = (b + c + d + f)/2 (Equation 2)
b = (a + c + d + f)/3 (Equation 3)
c = (a + b + d + f)/4 (Equation 4)
d = (a + b + c + f)/5 (Equation 5)
Multiplying Equations 2, 3, 4, and 5 by appropriate amounts to eliminate fractions, we get:
2a = b + c + d + f (Equation 6)
3b = a + c + d + f (Equation 7)
4c = a + b + d + f (Equation 8)
5d = a + b + c + f (Equation 9)
Substituting Equations 6, 7, 8, and 9 in turn into Equation 1, we obtain the following:
3a = 1, 4b = 1, 5c = 1, 6d = 1
In other words:
a = 1/3, b = 1/4, c = 1/5, d = 1/6
Substituting these values into Equation 1 and subtracting, we get:
f = 1 – (1/3 + 1/4 + 1/5 + 1/6) = (60 – 20 – 15 – 12 – 10)/60 = 3/60 = 1/20
Thus, the area of the green section is 1/20 the area of the circle.
My answer: 1/20
Well done, Jake. I liked your practical approach.
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