The average of three numbers is 42 the first is twice the second and the second is twice the third the difference between the largest and the smallest number is

Options

A. 18
B. 36
C. 54
D. 72

Answer

To solve the problem step by step, let's denote the three numbers as A, B, and C.

### Step 1: Understand the relationships between the numbers
- We know that the first number (A) is twice the second number (B), and the second number (B) is twice the third number (C).
- We can express these relationships mathematically:
- A = 2B
- B = 2C

### Step 2: Express all numbers in terms of the third number (C)
- Let's denote the third number (C) as X.
- Then, from the relationship B = 2C, we can express B as:
- B = 2X
- Now, substituting B into the equation for A:
- A = 2B = 2(2X) = 4X

### Step 3: Write the average of the three numbers
- The average of the three numbers A, B, and C is given as 42.
- The formula for the average is:
\[ \text{Average} = \frac{A + B + C}{3} \]
- Substituting the expressions for A, B, and C:
\[ \frac{4X + 2X + X}{3} = 42 \]

### Step 4: Simplify the equation
- Combine the terms in the numerator:
\[ \frac{7X}{3} = 42 \]

### Step 5: Solve for X
- To eliminate the fraction, multiply both sides by 3:
\[ 7X = 42 \times 3 \]
- Calculate the right side:
\[ 7X = 126 \]
- Now, divide both sides by 7 to find X:
\[ X = \frac{126}{7} = 18 \]

### Step 6: Find the values of A, B, and C
- Now that we have X, we can find the values of A, B, and C:
- C = X = 18
- B = 2X = 2(18) = 36
- A = 4X = 4(18) = 72

### Step 7: Calculate the difference between the largest and smallest numbers
- The largest number is A (72), and the smallest number is C (18).
- The difference is:
\[ A - C = 72 - 18 = 54 \]

### Final Answer
The difference between the largest and smallest number is **54**.

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