If `40%` of a number is equal to two-third of another number, what is the ratio of first number to the second number?

Options

A. `2 : 5`
B. ` 3 :7`
C. `5 : 3`
D. ` 7 : 3`

Answer

To solve the problem step by step, we will first define the variables and then set up the equation based on the information given in the question.

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

### Step 2: Set Up the Equation
According to the problem, \( 40\% \) of the first number \( X \) is equal to \( \frac{2}{3} \) of the second number \( Y \). We can express this mathematically as:
\[ 0.4X = \frac{2}{3}Y \]

### Step 3: Eliminate the Decimal
To eliminate the decimal, we can multiply both sides of the equation by \( 10 \):
\[ 4X = \frac{20}{3}Y \]

### Step 4: Clear the Fraction
Next, we can multiply both sides by \( 3 \) to clear the fraction:
\[ 12X = 20Y \]

### Step 5: Rearranging the Equation
Now, we can rearrange the equation to find the ratio of \( X \) to \( Y \):
\[ \frac{X}{Y} = \frac{20}{12} \]

### Step 6: Simplify the Ratio
We can simplify \( \frac{20}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is \( 4 \):
\[ \frac{X}{Y} = \frac{5}{3} \]

### Step 7: State the Ratio
Thus, the ratio of the first number \( X \) to the second number \( Y \) is:
\[ X : Y = 5 : 3 \]