Answer :
Certainly! Let's work through the problem step-by-step.
### Step 1: Determine the total number of people in the group.
The total number of people is given as 120.
### Step 2: Calculate the number of people who like tea only.
A fraction \(\frac{3}{10}\) of the total number of people like tea only. Therefore, the number of people who like tea only is:
[tex]\[ \frac{3}{10} \times 120 = 36 \][/tex]
### Step 3: Calculate the number of people who like coffee only.
A fraction \(\frac{5}{12}\) of the total number of people like coffee only. Therefore, the number of people who like coffee only is:
[tex]\[ \frac{5}{12} \times 120 = 50 \][/tex]
### Step 4: Calculate the number of people who like both tea and coffee.
The remaining people in the group like both tea and coffee. To find this number, we subtract the number of people who like tea only and coffee only from the total number:
[tex]\[ 120 - (36 + 50) = 34 \][/tex]
So, 34 people in the group like both tea and coffee.
### Step 5: Determine the fraction of people who like both tea and coffee.
To find the fraction of the total number of people who like both tea and coffee, we divide the number of people who like both by the total number of people:
[tex]\[ \frac{34}{120} \][/tex]
Simplify this fraction by dividing the numerator and the denominator by their greatest common divisor:
[tex]\[ \frac{34}{120} \approx 0.2833 \][/tex]
So, the fraction is approximately:
[tex]\[ \frac{34}{120} \approx 0.2833 \approx \frac{2833}{10000} \][/tex]
### Final Results
- i. The number of people who like tea and coffee both is \(34\).
- ii. The fraction of the total number of people who like tea and coffee both is approximately \(0.2833\).
These steps provide the detailed solution to the given problem.
### Step 1: Determine the total number of people in the group.
The total number of people is given as 120.
### Step 2: Calculate the number of people who like tea only.
A fraction \(\frac{3}{10}\) of the total number of people like tea only. Therefore, the number of people who like tea only is:
[tex]\[ \frac{3}{10} \times 120 = 36 \][/tex]
### Step 3: Calculate the number of people who like coffee only.
A fraction \(\frac{5}{12}\) of the total number of people like coffee only. Therefore, the number of people who like coffee only is:
[tex]\[ \frac{5}{12} \times 120 = 50 \][/tex]
### Step 4: Calculate the number of people who like both tea and coffee.
The remaining people in the group like both tea and coffee. To find this number, we subtract the number of people who like tea only and coffee only from the total number:
[tex]\[ 120 - (36 + 50) = 34 \][/tex]
So, 34 people in the group like both tea and coffee.
### Step 5: Determine the fraction of people who like both tea and coffee.
To find the fraction of the total number of people who like both tea and coffee, we divide the number of people who like both by the total number of people:
[tex]\[ \frac{34}{120} \][/tex]
Simplify this fraction by dividing the numerator and the denominator by their greatest common divisor:
[tex]\[ \frac{34}{120} \approx 0.2833 \][/tex]
So, the fraction is approximately:
[tex]\[ \frac{34}{120} \approx 0.2833 \approx \frac{2833}{10000} \][/tex]
### Final Results
- i. The number of people who like tea and coffee both is \(34\).
- ii. The fraction of the total number of people who like tea and coffee both is approximately \(0.2833\).
These steps provide the detailed solution to the given problem.
