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# When it rains, every student in Margo's school brings a raincoat

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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.

May 1, 2023

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71.4% of the students bring an umbrella only.

What is the percentage?

Percentage is a way of expressing a number as a fraction of 100. It is represented by the symbol %. It is a useful way of comparing values and calculating changes over time.

Let's denote the total number of students in Margo's school by the letter N. We can use a Venn diagram to represent the number of students who bring a raincoat, an umbrella, or both, as shown below:

RAINCOAT         UMBRELLA

________         ________

|                  |       |                 |

|                  |       |                 |

|                  |       |                 |

|      A         |       |        C       |

|                  |       |                 |

|                  |       |                 |

|________|       |________|

BOTH (B)         ONLY (D)

We know from the problem statement that 50% of the students bring both a raincoat and an umbrella. Therefore, the number of students who bring both is equal to 0.5N.

We also know that the total number of umbrellas brought is twice the number of raincoats. Let's denote the number of raincoats brought by the letter A. Then the number of umbrellas brought is equal to 2A.

We can use these values to fill in the Venn diagram. The total number of students is N, so the number of students who bring only a raincoat is (A - 0.5N) and the number of students who bring only an umbrella is (2A - 0.5N). The number of students who bring both is 0.5N.

To find the percentage of students who bring an umbrella only, we need to divide the number of students who bring only an umbrella by the total number of students and multiply by 100:

(2A - 0.5N) / N * 100

We can simplify this expression using the fact that the number of umbrellas brought is twice the number of raincoats:

(2A - 0.5N) / N * 100

= (2 * A - 0.5N) / N * 100

= (3A - 0.5N) / N * 100

We can't directly solve for the value of A, but we can use the fact that the sum of the numbers in the Venn diagram adds up to N:

A - 0.5N + 0.5N + 2A - 0.5N = N

3.5A - N = N

A = 2N / 7

Substituting this value of A into the expression for the percentage of students who bring an umbrella only, we get:

(3A - 0.5N) / N * 100

= (3 * 2N / 7 - 0.5N) / N * 100

= 71.4%

Therefore, 71.4% of the students bring an umbrella only.

May 1, 2023