In a society `16(2)/(3)%` people were interested in music only, `33(1)/(3)%` were interested in Yoga only, 25% were interested in cycling only and rest were interested in other activities. If no. of people who are interested in other activities is 450 then what is the no. of people who are interested in music.

Options

A. 400
B. 300
C. 250
D. 350

Answer

To solve the problem step by step, we will follow the given information and perform the necessary calculations.

### Step 1: Define the total population
Let's assume the total number of people in the society is \( 100x \).

### Step 2: Calculate the number of people interested in music
The percentage of people interested in music is given as \( 16 \frac{2}{3} \% \).
This can be converted to an improper fraction:
\[ 16 \frac{2}{3} = \frac{50}{3} \% \]
Thus, the number of people interested in music is:
\[ \text{People interested in music} = \frac{50}{3} \% \text{ of } 100x = \frac{50}{3} \times \frac{100}{100}x = \frac{50x}{3} \]

### Step 3: Calculate the number of people interested in yoga
The percentage of people interested in yoga is \( 33 \frac{1}{3} \% \), which is:
\[ 33 \frac{1}{3} = \frac{100}{3} \% \]
Thus, the number of people interested in yoga is:
\[ \text{People interested in yoga} = \frac{100}{3} \% \text{ of } 100x = \frac{100}{3} \times \frac{100}{100}x = \frac{100x}{3} \]

### Step 4: Calculate the number of people interested in cycling
The percentage of people interested in cycling is \( 25 \% \). Thus, the number of people interested in cycling is:
\[ \text{People interested in cycling} = 25 \% \text{ of } 100x = \frac{25}{100} \times 100x = 25x \]

### Step 5: Calculate the total number of people interested in music, yoga, and cycling
Now, we can find the total number of people interested in music, yoga, and cycling:
\[ \text{Total} = \frac{50x}{3} + \frac{100x}{3} + 25x \]
To combine these, we need a common denominator:
\[ \text{Total} = \frac{50x + 100x + 75x}{3} = \frac{225x}{3} = 75x \]

### Step 6: Calculate the number of people interested in other activities
The total number of people in the society is \( 100x \). Therefore, the number of people interested in other activities is:
\[ \text{People interested in other activities} = 100x - 75x = 25x \]

### Step 7: Set up the equation based on the given information
We know that the number of people interested in other activities is given as 450:
\[ 25x = 450 \]

### Step 8: Solve for \( x \)
To find \( x \):
\[ x = \frac{450}{25} = 18 \]

### Step 9: Calculate the number of people interested in music
Now we can find the number of people interested in music using the value of \( x \):
\[ \text{People interested in music} = \frac{50x}{3} = \frac{50 \times 18}{3} = \frac{900}{3} = 300 \]

### Final Answer
The number of people who are interested in music is **300**.

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