a,b,c , d and e are five consecutive numbers in increasing order of size. Deleting one of the five numbers from the set decreased the sum of the remaining numbers in the set by 20%. Which one of the following numbers was deleted?

Options

A. a
B. b
C. c
D. d

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Consecutive Numbers
Let the five consecutive numbers be:
- \( a = x \)
- \( b = x + 1 \)
- \( c = x + 2 \)
- \( d = x + 3 \)
- \( e = x + 4 \)

### Step 2: Calculate the Sum of the Numbers
The sum of these five consecutive numbers can be calculated as follows:
\[ \text{Sum} = a + b + c + d + e = x + (x + 1) + (x + 2) + (x + 3) + (x + 4) \]
Combining like terms:
\[ \text{Sum} = 5x + 10 \]

### Step 3: Understand the Effect of Deleting a Number
When one of the numbers is deleted, the sum of the remaining numbers decreases by 20%. Therefore, the new sum after deleting one number is:
\[ \text{New Sum} = \text{Sum} - \text{Deleted Number} \]
This new sum is also equal to 80% of the original sum:
\[ \text{New Sum} = 0.8 \times (5x + 10) \]

### Step 4: Set Up the Equation
Now we can set up the equation based on the information given:
\[ \text{Sum} - \text{Deleted Number} = 0.8 \times (5x + 10) \]
Substituting the sum:
\[ (5x + 10) - \text{Deleted Number} = 0.8 \times (5x + 10) \]

### Step 5: Simplify the Equation
Rearranging gives:
\[ \text{Deleted Number} = (5x + 10) - 0.8(5x + 10) \]
Calculating the right-hand side:
\[ \text{Deleted Number} = (5x + 10)(1 - 0.8) = (5x + 10)(0.2) \]
\[ \text{Deleted Number} = 0.2(5x + 10) = x + 2 \]

### Step 6: Identify the Deleted Number
The deleted number is \( x + 2 \). This corresponds to the middle number \( c \) in the sequence of five consecutive numbers.

### Conclusion
Thus, the number that was deleted is \( c = x + 2 \).