in a survey of 260 dancers:
172 knew the ballroom dances
145 knew latin dances
102 knew swing dances
of these, 65 knew the ballroom and swing dances.
93 knew ballroom and latin dances
56 knew latin and swing dances
31 dancers indicated they knew all 3 styles of dance
1. how many knew only ballroom dances
2. how many knew only swing dances
3. how many knew exactly 2 of the 3 dances
4. how many knew latin and swing dances, but not ballroom dances
5. how many knew none of these dances
1. 172 - 65 - 93 = ?
2. same process, except with swing.
3. 65 + 93 + 56 = ?
4. 56
5. 260 - 172 - 56 = ?
To answer these questions, we can use a technique called Venn diagrams, which can help us visualize and organize the different combinations of dancers.
1. To find the number of dancers who knew only ballroom dances, we need to subtract the number of dancers who knew ballroom dances and other styles from the total number of dancers who knew ballroom dances. From the information given, we know that 172 dancers knew ballroom dances, and 65 of those also knew swing dances. Therefore, the number of dancers who knew only ballroom dances is 172 - 65 = 107.
2. Similarly, to find the number of dancers who knew only swing dances, we need to subtract the number of dancers who knew swing dances and other styles from the total number of dancers who knew swing dances. From the information given, we know that 102 dancers knew swing dances, and 65 of those also knew ballroom dances. Therefore, the number of dancers who knew only swing dances is 102 - 65 = 37.
3. To find the number of dancers who knew exactly 2 out of the 3 dances, we add up the number of dancers who knew each combination of two dances. From the information given, we know that 65 dancers knew ballroom and swing dances, 93 dancers knew ballroom and latin dances, and 56 dancers knew latin and swing dances. Adding these up gives us 65 + 93 + 56 = 214 dancers who knew exactly 2 dances.
4. To find the number of dancers who knew latin and swing dances, but not ballroom dances, we need to subtract the number of dancers who knew all 3 dances from the number of dancers who knew latin and swing dances. From the information given, we know that 31 dancers knew all 3 dances and 56 dancers knew latin and swing dances. Therefore, the number of dancers who knew latin and swing dances, but not ballroom dances, is 56 - 31 = 25.
5. To find the number of dancers who knew none of these dances, we need to subtract the total number of dancers who knew at least one of the dances from the total number of dancers surveyed. From the information given, we know that 31 dancers knew all 3 dances. Therefore, the number of dancers who knew none of these dances is 260 - 31 = 229.
So, the answers are:
1. 107 dancers knew only ballroom dances.
2. 37 dancers knew only swing dances.
3. 214 dancers knew exactly 2 of the 3 dances.
4. 25 dancers knew latin and swing dances, but not ballroom dances.
5. 229 dancers knew none of these dances.