A man piles 150 tookpicks in layers so that each layer has one less tookpick than the below. If the top layer has three tookpicks , how many layers are there ?

Options

A. 15
B. 17
C. 20
D. 148

Explanation

To solve the problem of how many layers of toothpicks are stacked, we can follow these steps:

### Step 1: Understand the Problem
The problem states that a man piles 150 toothpicks in layers, with each layer having one less toothpick than the layer below it. The top layer has 3 toothpicks.

### Step 2: Define the Variables
Let:
- \( n \) = number of layers
- The number of toothpicks in the top layer (1st layer) = 3
- The number of toothpicks in the 2nd layer = 4
- The number of toothpicks in the 3rd layer = 5
- Continuing this way, the number of toothpicks in the \( n \)-th layer = \( 3 + (n - 1) \)

### Step 3: Write the Sum of the Toothpicks
The total number of toothpicks in \( n \) layers can be expressed as:
\[ \text{Total Toothpicks} = 3 + 4 + 5 + \ldots + (3 + (n - 1)) \]
This can be rewritten as:
\[ \text{Total Toothpicks} = \sum_{k=0}^{n-1} (3 + k) = 3n + \sum_{k=0}^{n-1} k \]
Using the formula for the sum of the first \( n-1 \) integers, \( \sum_{k=0}^{n-1} k = \frac{(n-1)n}{2} \), we can substitute:
\[ \text{Total Toothpicks} = 3n + \frac{(n-1)n}{2} \]

### Step 4: Set Up the Equation
We know the total number of toothpicks is 150:
\[ 3n + \frac{(n-1)n}{2} = 150 \]

### Step 5: Clear the Fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 2(3n) + (n-1)n = 300 \]
This simplifies to:
\[ 6n + n^2 - n = 300 \]
or:
\[ n^2 + 5n - 300 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we need to factor the quadratic equation \( n^2 + 5n - 300 = 0 \). We look for two numbers that multiply to -300 and add to 5. These numbers are 20 and -15:
\[ (n + 20)(n - 15) = 0 \]

### Step 7: Solve for \( n \)
Setting each factor to zero gives:
1. \( n + 20 = 0 \) → \( n = -20 \) (not valid since \( n \) must be positive)
2. \( n - 15 = 0 \) → \( n = 15 \)

### Conclusion
Thus, the number of layers is \( n = 15 \).