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Out 2000 staff 48% preferred coffee 54% tea and 64% cocoa. Of the total 28% used coffee and tea 32% tea and cocoa and 30% coffee and cocoa. Only 6% did none of these. Find the number having all the three.
- a)360
- b)280
- c)160
- d)None
Correct answer is option 'A'. Can you explain this answer?
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Out 2000 staff 48% preferred coffee 54% tea and 64% cocoa. Of the tota...
Given information:
- Out of 2000 staff, 48% preferred coffee, 54% preferred tea and 64% preferred cocoa.
- 28% of the staff used both coffee and tea, 32% used both tea and cocoa and 30% used both coffee and cocoa.
- 6% of the staff did not use any of these.
To find: The number of staff who used all three - coffee, tea and cocoa.
Solution:
Let's assume the total number of staff who preferred coffee, tea and cocoa as x, y and z respectively. Then we can write the following equations based on the given information:
x + y + z = 2000 ---(1) (Total number of staff)
0.48*2000 = x + 0.28y + 0.3z ---(2) (48% preferred coffee, 28% used both coffee and tea, 30% used both coffee and cocoa)
0.54*2000 = y + 0.28x + 0.32z ---(3) (54% preferred tea, 28% used both coffee and tea, 32% used both tea and cocoa)
0.64*2000 = z + 0.3x + 0.32y ---(4) (64% preferred cocoa, 30% used both coffee and cocoa, 32% used both tea and cocoa)
We can solve these equations to find the values of x, y and z. One way to do this is to use matrix equations. Writing the above equations in matrix form, we get:
| 1 1 1 | | x | | 2000 |
| 1 0.28 0.3 | | y | | 0.48*2000 |
| 0.28 1 0.32 | x | z | = | 0.54*2000 |
| 0.3 0.32 1 | | y | | 0.64*2000 |
Using any matrix calculator, we can solve for x, y and z. (Note that the determinant of the matrix is non-zero, so the equations have a unique solution.)
We get x = 600, y = 840 and z = 960.
Now we need to find the number of staff who used all three - coffee, tea and cocoa. Let's assume this number is k. Then we can write:
k + 0.28k + 0.3k - 0.28*0.3*2000 = 0.06*2000 (From the inclusion-exclusion principle, where we subtract the staff who used exactly two of the drinks.)
Solving for k, we get k = 360.
Therefore, the number of staff who used all three - coffee, tea and cocoa - is 360.
Answer: (a) 360.
- Out of 2000 staff, 48% preferred coffee, 54% preferred tea and 64% preferred cocoa.
- 28% of the staff used both coffee and tea, 32% used both tea and cocoa and 30% used both coffee and cocoa.
- 6% of the staff did not use any of these.
To find: The number of staff who used all three - coffee, tea and cocoa.
Solution:
Let's assume the total number of staff who preferred coffee, tea and cocoa as x, y and z respectively. Then we can write the following equations based on the given information:
x + y + z = 2000 ---(1) (Total number of staff)
0.48*2000 = x + 0.28y + 0.3z ---(2) (48% preferred coffee, 28% used both coffee and tea, 30% used both coffee and cocoa)
0.54*2000 = y + 0.28x + 0.32z ---(3) (54% preferred tea, 28% used both coffee and tea, 32% used both tea and cocoa)
0.64*2000 = z + 0.3x + 0.32y ---(4) (64% preferred cocoa, 30% used both coffee and cocoa, 32% used both tea and cocoa)
We can solve these equations to find the values of x, y and z. One way to do this is to use matrix equations. Writing the above equations in matrix form, we get:
| 1 1 1 | | x | | 2000 |
| 1 0.28 0.3 | | y | | 0.48*2000 |
| 0.28 1 0.32 | x | z | = | 0.54*2000 |
| 0.3 0.32 1 | | y | | 0.64*2000 |
Using any matrix calculator, we can solve for x, y and z. (Note that the determinant of the matrix is non-zero, so the equations have a unique solution.)
We get x = 600, y = 840 and z = 960.
Now we need to find the number of staff who used all three - coffee, tea and cocoa. Let's assume this number is k. Then we can write:
k + 0.28k + 0.3k - 0.28*0.3*2000 = 0.06*2000 (From the inclusion-exclusion principle, where we subtract the staff who used exactly two of the drinks.)
Solving for k, we get k = 360.
Therefore, the number of staff who used all three - coffee, tea and cocoa - is 360.
Answer: (a) 360.
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Out 2000 staff 48% preferred coffee 54% tea and 64% cocoa. Of the total 28% used coffee and tea 32% tea and cocoa and 30% coffee and cocoa. Only 6% did none of these. Find the number having all the three.a)360b)280c)160d)NoneCorrect answer is option 'A'. Can you explain this answer?
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