The average height of 40 sticks is 154 cm. Some sticks of average height 160 cm are taken out. If the new average height is 150 cm, then how many sticks are remaining?

Options

A. 12
B. 16
C. 24
D. 28

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Calculate the total height of the 40 sticks.
Given that the average height of 40 sticks is 154 cm, we can calculate the total height (L40) of these sticks using the formula:

\[ L_{40} = \text{Average Height} \times \text{Number of Sticks} \]
\[ L_{40} = 154 \, \text{cm} \times 40 = 6160 \, \text{cm} \]

### Step 2: Let x be the number of sticks taken out.
Let \( x \) be the number of sticks that are removed, and we know that the average height of these sticks is 160 cm. Therefore, the total height of the sticks taken out (Lx) can be expressed as:

\[ L_{x} = \text{Average Height of x sticks} \times \text{Number of Sticks Taken Out} \]
\[ L_{x} = 160 \, \text{cm} \times x = 160x \, \text{cm} \]

### Step 3: Calculate the total height of the remaining sticks.
After removing \( x \) sticks, the number of remaining sticks is \( 40 - x \). The new average height of the remaining sticks is given as 150 cm. Thus, the total height of the remaining sticks (L40-x) can be calculated as:

\[ L_{40-x} = \text{Average Height of Remaining Sticks} \times \text{Number of Remaining Sticks} \]
\[ L_{40-x} = 150 \, \text{cm} \times (40 - x) = 150(40 - x) = 6000 - 150x \, \text{cm} \]

### Step 4: Set up the equation.
According to the problem, the total height of the original sticks is equal to the sum of the heights of the sticks taken out and the remaining sticks:

\[ L_{40} = L_{x} + L_{40-x} \]
Substituting the values we have:

\[ 6160 = 160x + (6000 - 150x) \]

### Step 5: Simplify the equation.
Now, let's simplify the equation:

\[ 6160 = 160x + 6000 - 150x \]
\[ 6160 = 10x + 6000 \]

### Step 6: Solve for x.
Now, isolate \( x \):

\[ 6160 - 6000 = 10x \]
\[ 160 = 10x \]
\[ x = \frac{160}{10} = 16 \]

### Step 7: Calculate the number of remaining sticks.
Now that we have \( x \), we can find the number of remaining sticks:

\[ \text{Remaining Sticks} = 40 - x = 40 - 16 = 24 \]

Thus, the number of sticks remaining is **24**.

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