Two-third of one-fourth of a number is equal to `40%` of another number. Which one of the following statements is true about the numbers?

Options

A. Second number is `2cdot 4` times of the first number
B. First number is `2cdot 4` times of the second number
C. First number is more than the second number by `40%`
D. First number is more than the second number by `60%`

Answer

To solve the problem step by step, let's denote the first number as \( x \) and the second number as \( y \).

### Step 1: Set up the equation based on the problem statement.
According to the problem, two-thirds of one-fourth of the first number \( x \) is equal to 40% of the second number \( y \).

This can be expressed mathematically as:
\[ \frac{2}{3} \times \frac{1}{4} \times x = 0.4 \times y \]

### Step 2: Simplify the left side of the equation.
Calculating the left side:
\[ \frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6} \]
So, the equation now becomes:
\[ \frac{1}{6} x = 0.4 y \]

### Step 3: Convert 40% to a fraction.
We know that 40% can be expressed as:
\[ 0.4 = \frac{40}{100} = \frac{2}{5} \]
Now we can rewrite the equation:
\[ \frac{1}{6} x = \frac{2}{5} y \]

### Step 4: Cross-multiply to eliminate the fractions.
Cross-multiplying gives us:
\[ 5x = 12y \]

### Step 5: Solve for \( x \) in terms of \( y \).
Rearranging the equation to express \( x \) in terms of \( y \):
\[ x = \frac{12}{5} y \]

### Step 6: Convert \( \frac{12}{5} \) to a decimal.
Calculating \( \frac{12}{5} \):
\[ \frac{12}{5} = 2.4 \]
Thus, we have:
\[ x = 2.4y \]

### Step 7: Interpret the result.
This means that the first number \( x \) is 2.4 times the second number \( y \).

### Step 8: Analyze the options given in the question.
1. The first option states that the second number is 2.4 times the first number, which is incorrect.
2. The second option states that the first number is 2.4 times the second number, which is correct.
3. The third option states that the first number is more than the second number by 40%, which is incorrect.
4. The fourth option states that the first number is more than the second number by 60%, which is also incorrect.

### Conclusion:
The only true statement about the numbers is that the first number is 2.4 times the second number.