Question

completing a two-way table
softball\tno softball\ttotal
swim\ta\tb\t22
do not swim\tc\t5\td
total\t20\te\t32
a summer camp has 32 campers. 22 of them swim, 20 play softball, and 5 do not play softball or swim. which values correctly complete the table?
○ a = 15, b = 10, c = 7, d = 5, e = 12
○ a = 15, b = 7, c = 5, d = 10, e = 12
○ a = 14, b = 7, c = 5, d = 12, e = 10
○ a = 14, b = 12, c = 7, d = 5, e = 10

Explanation:

Step1: Find \( d \) (Do Not Swim Total)

Total campers = 32, Swim total = 22. So \( d = 32 - 22 = 10 \)? Wait, no—wait, 5 do not play softball or swim. Wait, let's re-express. The "Do Not Swim" row: the "No Softball" column is 5 (since 5 do not play softball or swim, so that's the intersection of Do Not Swim and No Softball). Wait, the table:

• Swim row: Softball (\( a \)), No Softball (\( b \)), Total 22.
• Do Not Swim row: Softball (\( c \)), No Softball (5), Total \( d \).
• Total column: Softball (20), No Softball (\( e \)), Total 32.

First, total No Softball: \( e = 32 - 20 = 12 \)? Wait, no—wait, the "No Softball" column total is \( e \), and total campers is 32, so Softball total (20) + No Softball total (\( e \)) = 32 ⇒ \( e = 32 - 20 = 12 \)? Wait, but the Do Not Swim row, No Softball is 5, so Swim row No Softball (\( b \)) = \( e - 5 \). Let's check the options.

Wait, another approach: The number of campers who do not play softball or swim is 5 (Do Not Swim and No Softball). So in the Do Not Swim row, No Softball is 5. Then, Do Not Swim total (\( d \)): since total campers is 32, Swim total is 22, so \( d = 32 - 22 = 10 \)? Wait, no—32 total, 22 swim, so 10 do not swim. But the Do Not Swim row has No Softball = 5, so Softball in Do Not Swim (\( c \)) is \( d - 5 = 10 - 5 = 5 \)? Wait, no, let's use the Softball total. Softball total is 20, which is \( a + c = 20 \). Swim total is 22, so \( a + b = 22 \). No Softball total is \( b + 5 = e \), and total campers \( 22 + d = 32 \) ⇒ \( d = 10 \). Also, \( e = 32 - 20 = 12 \), so \( b + 5 = 12 \) ⇒ \( b = 7 \). Then, \( a + b = 22 \) ⇒ \( a = 22 - 7 = 15 \). Then, \( a + c = 20 \) ⇒ \( c = 20 - 15 = 5 \). Then, \( d = c + 5 = 5 + 5 = 10 \)? Wait, no—\( d = c + 5 \), and \( d = 32 - 22 = 10 \), which matches. So let's check:

• \( a = 15 \), \( b = 7 \) (since \( b = e - 5 = 12 - 5 = 7 \), and \( e = 32 - 20 = 12 \)), \( c = 5 \), \( d = 10 \), \( e = 12 \). Looking at the options, the second option is \( a = 15, b = 7, c = 5, d = 10, e = 12 \). Wait, but let's verify again:

Total Softball: \( a + c = 15 + 5 = 20 \) (correct). Total Swim: \( a + b = 15 + 7 = 22 \) (correct). Do Not Swim total: \( c + 5 = 5 + 5 = 10 \) (correct, since 32 - 22 = 10). No Softball total: \( b + 5 = 7 + 5 = 12 \) (correct, 32 - 20 = 12). So the second option is \( a = 15, b = 7, c = 5, d = 10, e = 12 \). Wait, but in the options, the second option is "a = 15, b = 7, c = 5, d = 10, e = 12". Wait, let's check the options again:

First option: \( a = 15, b = 10, c = 7, d = 5, e = 12 \) – no, \( d \) should be 10, not 5.

Second option: \( a = 15, b = 7, c = 5, d = 10, e = 12 \) – matches.

Third option: \( a = 14, b = 7, c = 5, d = 12, e = 10 \) – \( d \) wrong.

Fourth option: \( a = 14, b = 12, c = 7, d = 5, e = 10 \) – wrong.

So the correct option is the second one.

Step2: Verify each variable

• \( a \): Softball and Swim. Softball total is 20, so \( a = 20 - c \). \( c \) is Softball and Do Not Swim. Do Not Swim total is \( d = 10 \), and Do Not Swim and No Softball is 5, so \( c = 10 - 5 = 5 \). Thus, \( a = 20 - 5 = 15 \).
• \( b \): Swim and No Softball. No Softball total is \( e = 12 \), so \( b = 12 - 5 = 7 \).
• \( c \): Do Not Swim and Softball. Do Not Swim total \( d = 10 \), so \( c = 10 - 5 = 5 \).
• \( d \): Do Not Swim total. \( 32 - 22 = 10 \).
• \( e \): No Softball total. \( 32 - 20 = 12 \).

All variables match the second option: \( a = 15, b = 7, c = 5, d = 10, e = 12 \).

Answer:

The correct option is: \( a = 15, b = 7, c = 5, d = 10, e = 12 \) (the second option in the list).