Question

a group of 75 math students were asked whether they like algebra and whether they like geometry. a total of 45 students like algebra, 53 like geometry, and 6 do not like either subject. algebra vs. geometry | | likes geometry | does not like geometry | total | likes algebra | a | b | 45 | does not like algebra | c | 6 | d | total | 53 | e | 75 | what are the correct values of a, b, c, d, and e? a = 16, b = 29, c = 22, d = 30, e = 24 a = 29, b = 16, c = 30, d = 22, e = 24 a = 16, b = 29, c = 24, d = 22, e = 30 a = 29, b = 16, c = 24, d = 30, e = 22

Explanation:

Step1: Find e (Total Does Not Like Geometry)

Total students: 75, Total Likes Geometry: 53. So \( e = 75 - 53 = 22 \)? Wait, no, wait: Total "Does Not Like Geometry" is \( e \), and total is 75. Also, "Does Not Like Geometry" column: \( b + 6 = e \), and total row: \( 45 + d = 75 \), so \( d = 75 - 45 = 30 \). Then, "Does Not Like Algebra" row: \( c + 6 = d = 30 \), so \( c = 30 - 6 = 24 \)? Wait, no, let's start over.

Total students: 75. 6 do not like either, so number of students who like at least one: \( 75 - 6 = 69 \).

Number who like algebra: 45, like geometry: 53. Using principle of inclusion-exclusion: \( |A \cup G| = |A| + |G| - |A \cap G| \). So \( 69 = 45 + 53 - a \) (since \( a \) is \( |A \cap G| \)). So \( 69 = 98 - a \) → \( a = 98 - 69 = 29 \)? Wait, no, wait: \( |A \cup G| = 75 - 6 = 69 \). \( |A| = 45 \), \( |G| = 53 \). So \( |A \cap G| = |A| + |G| - |A \cup G| = 45 + 53 - 69 = 29 \). So \( a = 29 \).

Then, for "Likes Algebra" row: \( a + b = 45 \) → \( 29 + b = 45 \) → \( b = 16 \).

For "Total" column "Does Not Like Geometry": \( e = b + 6 = 16 + 6 = 22 \)? Wait, no, total "Does Not Like Geometry" is \( e \), and total students is 75, so \( 53 + e = 75 \) → \( e = 75 - 53 = 22 \). Then, "Does Not Like Geometry" column: \( b + 6 = e \) → \( 16 + 6 = 22 \), which matches.

For "Does Not Like Algebra" row: \( c + 6 = d \), and \( a + c = 53 \) (since "Likes Geometry" column total is 53) → \( 29 + c = 53 \) → \( c = 24 \). Then, \( d = c + 6 = 24 + 6 = 30 \).

So let's check:

• \( a = 29 \), \( b = 16 \), \( c = 24 \), \( d = 30 \), \( e = 22 \). Which matches the last option: \( a = 29, b = 16, c = 24, d = 30, e = 22 \).

Wait, let's verify each step:

1. Total students: 75. 6 do not like either, so students who like at least one: 75 - 6 = 69.

1. Number who like algebra (A): 45, like geometry (G): 53. So \( |A \cap G| = a = |A| + |G| - |A \cup G| = 45 + 53 - 69 = 29 \). So \( a = 29 \).

1. Likes Algebra row: \( a + b = 45 \) → \( b = 45 - 29 = 16 \).

1. Total column "Does Not Like Geometry" (e): \( 75 - 53 = 22 \). So \( e = 22 \).

1. Does Not Like Geometry column: \( b + 6 = e \) → \( 16 + 6 = 22 \), correct.

1. Likes Geometry column: \( a + c = 53 \) → \( c = 53 - 29 = 24 \).

1. Does Not Like Algebra row: \( c + 6 = d \) → \( 24 + 6 = 30 \), so \( d = 30 \).

So the values are \( a = 29, b = 16, c = 24, d = 30, e = 22 \), which is the last option.

Step1: Find e (Total Does Not Like Geometry)

Total students: 75, Likes Geometry total: 53. So \( e = 75 - 53 = 22 \).

Step2: Find a (Likes Both Algebra and Geometry)

Students who like at least one: \( 75 - 6 = 69 \). Using inclusion - exclusion: \( |A \cup G|=|A| + |G|-|A \cap G| \), so \( 69 = 45 + 53 - a \), solving gives \( a = 29 \).

Step3: Find b (Likes Algebra, Does Not Like Geometry)

Likes Algebra row: \( a + b = 45 \), so \( 29 + b = 45 \), \( b = 16 \).

Step4: Find c (Does Not Like Algebra, Likes Geometry)

Likes Geometry column: \( a + c = 53 \), so \( 29 + c = 53 \), \( c = 24 \).

Step5: Find d (Does Not Like Algebra, Total)

Does Not Like Algebra row: \( c + 6 = d \), so \( 24 + 6 = 30 \), \( d = 30 \).

Answer:

a = 29, b = 16, c = 24, d = 30, e = 22 (the last option: \( a = 29, b = 16, c = 24, d = 30, e = 22 \))