QuestionDetermine the number of students in a classical music lecture class who like different composers using a Venn diagram. Let M = Mozart, B = Beethoven, H = Haydn. Find the number of students who like: (a) exactly 2 composers, (b) exactly 1 composer, (c) none of the composers, (d) M but not B or H.

Studdy Solution

STEP 1

1. The sets M, B, and H represent the students who like Mozart, Beethoven, and Haydn respectively.
2. The numbers given represent the total number of students who like each individual composer and combinations of composers.
3. The total number of students in the class is 68.
4. The Venn diagram can be used to represent the overlaps and unique elements in each set.

STEP 2

1. Determine the number of students who like exactly one composer.
2. Determine the number of students who like exactly two composers.
3. Determine the number of students who like all three composers.
4. Determine the number of students who like none of the composers.
5. Determine the number of students who like Mozart, but neither Beethoven nor Haydn.

STEP 3

Calculate the number of students who like exactly one composer by subtracting the students who like exactly two composers and all three composers from the total who like each individual composer.

STEP 4

For Mozart (M), subtract the number of students who like both Mozart and Beethoven (MB), both Mozart and Haydn (MH), and all three composers (MBH) from the total number who like Mozart.
$|M|_{\text{only}} = |M| - |MB| - |MH| + |MBH|$

STEP 5

For Beethoven (B), subtract the number of students who like both Beethoven and Mozart (BM), both Beethoven and Haydn (BH), and all three composers (MBH) from the total number who like Beethoven.
$|B|_{\text{only}} = |B| - |BM| - |BH| + |MBH|$

STEP 6

For Haydn (H), subtract the number of students who like both Haydn and Mozart (HM), both Haydn and Beethoven (HB), and all three composers (MBH) from the total number who like Haydn.
$|H|_{\text{only}} = |H| - |HM| - |HB| + |MBH|$

STEP 7

Calculate the number of students who like exactly two composers by subtracting the students who like all three composers from the total number who like each pair of composers.

STEP 8

For students who like Mozart and Beethoven (MB) but not Haydn, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Beethoven.
$|MB|_{\text{only}} = |MB| - |MBH|$

STEP 9

For students who like Mozart and Haydn (MH) but not Beethoven, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Haydn.
$|MH|_{\text{only}} = |MH| - |MBH|$

STEP 10

For students who like Beethoven and Haydn (BH) but not Mozart, subtract the number of students who like all three composers (MBH) from the total number who like both Beethoven and Haydn.
$|BH|_{\text{only}} = |BH| - |MBH|$

STEP 11

The number of students who like all three composers has already been given.
$|MBH| = 9$

STEP 12

Calculate the number of students who like none of the composers by subtracting the number of students who like at least one composer from the total number of students in the class.

STEP 13

Add the number of students who like exactly one composer, exactly two composers, and all three composers. Subtract this sum from the total number of students to find the number who like none of the composers.
$\text{None} = \text{Total students} - (|M|_{\text{only}} + |B|_{\text{only}} + |H|_{\text{only}} + |MB|_{\text{only}} + |MH|_{\text{only}} + |BH|_{\text{only}} + |MBH|)$

STEP 14

To find the number of students who like Mozart but neither Beethoven nor Haydn, use the number of students who like exactly one composer from Mozart's set.
$|M|_{\text{only}} = \text{Number of students who like Mozart but neither Beethoven nor Haydn}$
Now let's perform the calculations:
STEP_2: $|M|_{\text{only}} = 38 - 15 - 21 + 9 = 11$
STEP_3: $|B|_{\text{only}} = 38 - 15 - 16 + 9 = 16$
STEP_4: $|H|_{\text{only}} = 33 - 21 - 16 + 9 = 5$
STEP_6: $|MB|_{\text{only}} = 15 - 9 = 6$
STEP_7: $|MH|_{\text{only}} = 21 - 9 = 12$
STEP_8: $|BH|_{\text{only}} = 16 - 9 = 7$
STEP_11: $\text{None} = 68 - (11 + 16 + 5 + 6 + 12 + 7 + 9) = 68 - 66 = 2$
The answers to the questions are: (a) The number of students who like exactly two composers is $|MB|_{\text{only}} + |MH|_{\text{only}} + |BH|_{\text{only}} = 6 + 12 + 7 = 25$ . (b) The number of students who like exactly one composer is $|M|_{\text{only}} + |B|_{\text{only}} + |H|_{\text{only}} = 11 + 16 + 5 = 32$ . (c) The number of students who like none of the composers is $2$ . (d) The number of students who like Mozart, but neither Beethoven nor Haydn is $|M|_{\text{only}} = 11$ .

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Studdy Solution

STEP 1

STEP 2

STEP 3

Calculate the number of students who like exactly one composer by subtracting the students who like exactly two composers and all three composers from the total who like each individual composer.

STEP 4

For Mozart (M), subtract the number of students who like both Mozart and Beethoven (MB), both Mozart and Haydn (MH), and all three composers (MBH) from the total number who like Mozart.
$|M|_{\text{only}} = |M| - |MB| - |MH| + |MBH|$

STEP 5

For Beethoven (B), subtract the number of students who like both Beethoven and Mozart (BM), both Beethoven and Haydn (BH), and all three composers (MBH) from the total number who like Beethoven.
$|B|_{\text{only}} = |B| - |BM| - |BH| + |MBH|$

STEP 6

For Haydn (H), subtract the number of students who like both Haydn and Mozart (HM), both Haydn and Beethoven (HB), and all three composers (MBH) from the total number who like Haydn.
$|H|_{\text{only}} = |H| - |HM| - |HB| + |MBH|$

STEP 7

Calculate the number of students who like exactly two composers by subtracting the students who like all three composers from the total number who like each pair of composers.

STEP 8

For students who like Mozart and Beethoven (MB) but not Haydn, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Beethoven.
$|MB|_{\text{only}} = |MB| - |MBH|$

STEP 9

For students who like Mozart and Haydn (MH) but not Beethoven, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Haydn.
$|MH|_{\text{only}} = |MH| - |MBH|$

STEP 10

For students who like Beethoven and Haydn (BH) but not Mozart, subtract the number of students who like all three composers (MBH) from the total number who like both Beethoven and Haydn.
$|BH|_{\text{only}} = |BH| - |MBH|$

STEP 11

The number of students who like all three composers has already been given.
$|MBH| = 9$

STEP 12

Calculate the number of students who like none of the composers by subtracting the number of students who like at least one composer from the total number of students in the class.

STEP 13

STEP 14

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Studdy Solution

STEP 1

STEP 2

STEP 3

Calculate the number of students who like exactly one composer by subtracting the students who like exactly two composers and all three composers from the total who like each individual composer.

STEP 4

STEP 5

STEP 6

STEP 7

Calculate the number of students who like exactly two composers by subtracting the students who like all three composers from the total number who like each pair of composers.

STEP 8

For students who like Mozart and Beethoven (MB) but not Haydn, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Beethoven.∣MB∣only=∣MB∣−∣MBH∣ |MB|_{\text{only}} = |MB| - |MBH| ∣MB∣only​=∣MB∣−∣MBH∣

STEP 9

For students who like Mozart and Haydn (MH) but not Beethoven, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Haydn.∣MH∣only=∣MH∣−∣MBH∣ |MH|_{\text{only}} = |MH| - |MBH| ∣MH∣only​=∣MH∣−∣MBH∣

STEP 10

For students who like Beethoven and Haydn (BH) but not Mozart, subtract the number of students who like all three composers (MBH) from the total number who like both Beethoven and Haydn.∣BH∣only=∣BH∣−∣MBH∣ |BH|_{\text{only}} = |BH| - |MBH| ∣BH∣only​=∣BH∣−∣MBH∣

STEP 11

The number of students who like all three composers has already been given.∣MBH∣=9 |MBH| = 9 ∣MBH∣=9

STEP 12

Calculate the number of students who like none of the composers by subtracting the number of students who like at least one composer from the total number of students in the class.

STEP 13

STEP 14

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For students who like Mozart and Beethoven (MB) but not Haydn, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Beethoven.
$|MB|_{\text{only}} = |MB| - |MBH|$

For students who like Mozart and Haydn (MH) but not Beethoven, subtract the number of students who like all three composers (MBH) from the total number who like both Mozart and Haydn.
$|MH|_{\text{only}} = |MH| - |MBH|$

For students who like Beethoven and Haydn (BH) but not Mozart, subtract the number of students who like all three composers (MBH) from the total number who like both Beethoven and Haydn.
$|BH|_{\text{only}} = |BH| - |MBH|$

The number of students who like all three composers has already been given.
$|MBH| = 9$