In a group of children, one-fifth play football, one-fourth play basketball, two-fifths play cricket, and the remaining 30 play other games. Find the total number of children in the group.

Answer

To solve the problem step by step, we will first define the total number of children in the group and then set up an equation based on the information given in the question.

### Step 1: Define the total number of children
Let the total number of children in the group be \( x \).

### Step 2: Set up the equation based on the information given
According to the problem:
- One-fifth of the children play football: \( \frac{x}{5} \)
- One-fourth of the children play basketball: \( \frac{x}{4} \)
- Two-fifths of the children play cricket: \( \frac{2x}{5} \)
- The remaining children play other games, which is given as 30.

Now, we can express the total number of children in terms of these fractions:
\[ \frac{x}{5} + \frac{x}{4} + \frac{2x}{5} + 30 = x \]

### Step 3: Combine the fractions
To combine the fractions, we need a common denominator. The least common multiple of 5 and 4 is 20. So, we convert each fraction:
- \( \frac{x}{5} = \frac{4x}{20} \)
- \( \frac{x}{4} = \frac{5x}{20} \)
- \( \frac{2x}{5} = \frac{8x}{20} \)

Now substituting back into the equation:
\[ \frac{4x}{20} + \frac{5x}{20} + \frac{8x}{20} + 30 = x \]

### Step 4: Combine the fractions on the left side
Adding the fractions together:
\[ \frac{4x + 5x + 8x}{20} + 30 = x \]
This simplifies to:
\[ \frac{17x}{20} + 30 = x \]

### Step 5: Isolate \( x \)
Now, we will isolate \( x \) by subtracting \( \frac{17x}{20} \) from both sides:
\[ 30 = x - \frac{17x}{20} \]
To simplify the right side, we can express \( x \) as \( \frac{20x}{20} \):
\[ 30 = \frac{20x - 17x}{20} \]
This simplifies to:
\[ 30 = \frac{3x}{20} \]

### Step 6: Solve for \( x \)
Now, multiply both sides by 20 to eliminate the fraction:
\[ 600 = 3x \]
Now, divide both sides by 3:
\[ x = 200 \]

### Conclusion
The total number of children in the group is \( 200 \).

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