A choir is singing at a festival. On the first night, 12 choir members were absent, so the choir stood in 5 equal rows. On the second night, only 1 member was absent, so the choir stood in 6 equal rows. The same member of people stood in each row each night. How many members are in the choir ? 

Options

A. 65
B. 67
C. 70
D. 64

Answer

To solve the problem step-by-step, let's define the variables and set up the equations based on the information given.

### Step 1: Define the variable
Let the total number of choir members be \( X \).

### Step 2: Set up the equation for the first night
On the first night, 12 members were absent. Therefore, the number of members present that night is:
\[ X - 12 \]
These members stood in 5 equal rows, so we can express this as:
\[ \frac{X - 12}{5} \]
This represents the number of members in each row on the first night.

### Step 3: Set up the equation for the second night
On the second night, only 1 member was absent. Therefore, the number of members present that night is:
\[ X - 1 \]
These members stood in 6 equal rows, so we can express this as:
\[ \frac{X - 1}{6} \]
This represents the number of members in each row on the second night.

### Step 4: Set the two expressions equal
Since the number of members in each row is the same on both nights, we can set the two expressions equal to each other:
\[ \frac{X - 12}{5} = \frac{X - 1}{6} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 6(X - 12) = 5(X - 1) \]

### Step 6: Distribute both sides
Distributing the terms on both sides results in:
\[ 6X - 72 = 5X - 5 \]

### Step 7: Isolate the variable
Now, we will isolate \( X \) by subtracting \( 5X \) from both sides:
\[ 6X - 5X - 72 = -5 \]
This simplifies to:
\[ X - 72 = -5 \]

### Step 8: Solve for \( X \)
Now, add 72 to both sides:
\[ X = 72 - 5 \]
Thus,
\[ X = 67 \]

### Conclusion
The total number of members in the choir is \( 67 \).

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