Three times the cube of a number is seven times the other number. What is the ratio of the first number to the second number ?

Options

A. 6
B. 49
C. 144
D. can't be determined

Answer

To solve the problem, we need to set up an equation based on the information given in the question. Let's break it down step by step.

### Step 1: Define the Variables
Let the first number be \( N_1 \) and the second number be \( N_2 \).

### Step 2: Set Up the Equation
According to the problem, three times the cube of the first number is equal to seven times the second number. This can be expressed mathematically as:
\[ 3 \times N_1^3 = 7 \times N_2 \]

### Step 3: Rearranging the Equation
We can rearrange the equation to express \( N_1^3 \) in terms of \( N_2 \):
\[ N_1^3 = \frac{7}{3} N_2 \]

### Step 4: Assume a Value for \( N_2 \)
To find the ratio, we can assume a value for \( N_2 \). Let's assume:
\[ N_2 = 3 \times 7^2 \]
This assumption is made to simplify calculations later.

### Step 5: Substitute \( N_2 \) Back into the Equation
Now, substituting \( N_2 \) into the equation for \( N_1^3 \):
\[ N_1^3 = \frac{7}{3} \times (3 \times 7^2) \]
The \( 3 \) in the numerator and denominator cancels out:
\[ N_1^3 = 7 \times 7^2 = 7^3 \]

### Step 6: Solve for \( N_1 \)
Taking the cube root of both sides gives us:
\[ N_1 = 7 \]

### Step 7: Calculate the Ratio
Now, we can calculate the ratio of \( N_1 \) to \( N_2 \):
\[ \text{Ratio} = \frac{N_1}{N_2} = \frac{7}{3 \times 7^2} \]
This simplifies to:
\[ \frac{7}{3 \times 49} = \frac{7}{147} = \frac{1}{21} \]

Thus, the ratio of the first number to the second number is:
\[ \text{Ratio} = 1 : 21 \]

### Final Answer
The ratio of the first number to the second number is \( 1 : 21 \).

---