In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbox. If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox? (A) 5 (B) 7 (C) 10 (D) 17 (E) 20 (ebasznan) m donoubraglion 2.32 25 LB
In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbox. If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox? (A) 5 (B) 7 (C) 10 (D) 17 (E) 20 (ebasznan) m donoubraglion 2.32 25 LB
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbox. If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox?
(A) 5
(B) 7
(C) 10
(D) 17
(E) 20
To solve this, we use the principle of inclusion-exclusion:
Let:
- \( A \) be the number of students with backpacks,
- \( B \) be the number of students with lunchboxes,
- \( X \) be the number of students with both items.
According to the problem:
- \( A + B - X = 43 \),
- \( A = 23 \),
- \( B = 25 \).
Substituting \( A \) and \( B \):
\[ 23 + 25 - X = 43 \]
Simplifying:
\[ 48 - X = 43 \]
\[ X = 5 \]
Thus, the number of students who brought both a backpack and a lunchbox is \( 5 \) (Option A).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F455248bd-ea7a-4e00-86ee-f35f2059a6c7%2F6xys4ze_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In a school picnic, a total of 43 students brought a backpack, a lunchbox, or both a backpack and a lunchbox. If there are a total of 23 backpacks and 25 lunchboxes, how many students brought both a backpack and a lunchbox?
(A) 5
(B) 7
(C) 10
(D) 17
(E) 20
To solve this, we use the principle of inclusion-exclusion:
Let:
- \( A \) be the number of students with backpacks,
- \( B \) be the number of students with lunchboxes,
- \( X \) be the number of students with both items.
According to the problem:
- \( A + B - X = 43 \),
- \( A = 23 \),
- \( B = 25 \).
Substituting \( A \) and \( B \):
\[ 23 + 25 - X = 43 \]
Simplifying:
\[ 48 - X = 43 \]
\[ X = 5 \]
Thus, the number of students who brought both a backpack and a lunchbox is \( 5 \) (Option A).
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