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Probability

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When it rains, every student in Margo's school brings a raincoat, an umbrella, or both.  And of the students bring both.  The total number of umbrellas brought is equal to twice the number of raincoats.

What percent of the students bring an umbrella only?  Explain your solution in complete sentences.

Mar 13, 2023

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To solve the problem, we can use the following steps:

1. Let R be the number of students who bring raincoats, U be the number of students who bring umbrellas, and B be the number of students who bring both.

2. We know that U = 2R, since the total number of umbrellas brought is equal to twice the number of raincoats brought.

3. We also know that R + U - B = N, where N is the total number of students in Margo's school. This equation represents the fact that every student brings either a raincoat, an umbrella, or both.

4. We want to find the percentage of students who bring an umbrella only. This is given by:

(U-B)/N * 100%

5. Substituting U=2R and B in terms of R, we get:

(U-B)/N * 100% = ((U/N) - (1/3)) * 100% = ((2R/N) - (1/3)) * 100%

6. To find R in terms of N, we use the fact that 3R - B = N. Substituting B=U/2 and U=2R, we get:

3R - (U/2) = N --> R = (2/5)N and U = 2R = (4/5)N

7. Substituting R=(2/5)N and U=(4/5)N in the expression for the percentage of students who bring an umbrella only, we get:

((2R/N) - (1/3)) * 100% = ((4/5) - (1/3)) * 100% = 46.67%

Mar 13, 2023