Three numbers are in the ratio `5 : 6 : 7`. If the product of the numbers is 5670, then the greatest number is

Options

A. a) 15
B. b) 18
C. c) 21
D. d) 28

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the numbers based on the ratio
The three numbers are in the ratio 5:6:7. We can express these numbers in terms of a variable \( x \):
- First number = \( 5x \)
- Second number = \( 6x \)
- Third number = \( 7x \)

### Step 2: Write the equation for the product of the numbers
According to the problem, the product of these three numbers is given as 5670. Therefore, we can write the equation:
\[ (5x) \times (6x) \times (7x) = 5670 \]

### Step 3: Simplify the equation
Now, we can simplify the left side of the equation:
\[ 5 \times 6 \times 7 \times x^3 = 5670 \]
Calculating \( 5 \times 6 \times 7 \):
\[ 5 \times 6 = 30 \]
\[ 30 \times 7 = 210 \]
So, we have:
\[ 210x^3 = 5670 \]

### Step 4: Solve for \( x^3 \)
Next, we will isolate \( x^3 \) by dividing both sides of the equation by 210:
\[ x^3 = \frac{5670}{210} \]

### Step 5: Calculate \( \frac{5670}{210} \)
Now, we perform the division:
\[ \frac{5670}{210} = 27 \]
So, we have:
\[ x^3 = 27 \]

### Step 6: Find the value of \( x \)
To find \( x \), we take the cube root of both sides:
\[ x = \sqrt[3]{27} = 3 \]

### Step 7: Calculate the actual numbers
Now that we have \( x \), we can find the actual numbers:
- First number = \( 5x = 5 \times 3 = 15 \)
- Second number = \( 6x = 6 \times 3 = 18 \)
- Third number = \( 7x = 7 \times 3 = 21 \)

### Step 8: Identify the greatest number
Among the numbers 15, 18, and 21, the greatest number is:
\[ \text{Greatest number} = 21 \]

Thus, the greatest number is **21**.

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