When 50 % of one number is added to the second number . The second number increased to its four thirds . What the ratio between the first and the second number ?

Options

A. `3:2`
B. `3:4`
C. `2:3`
D. `2:5`

Answer

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

### Step 1: Understand the problem
According to the problem, when 50% of the first number \( X \) is added to the second number \( Y \), the result is equal to four-thirds of the second number \( Y \).

### Step 2: Set up the equation
50% of \( X \) can be expressed as:
\[ \frac{50}{100} \times X = \frac{1}{2}X \]
When this is added to \( Y \), we have:
\[ Y + \frac{1}{2}X = \frac{4}{3}Y \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( X \) and \( Y \):
\[ \frac{1}{2}X + Y = \frac{4}{3}Y \]
Subtract \( Y \) from both sides:
\[ \frac{1}{2}X = \frac{4}{3}Y - Y \]
This simplifies to:
\[ \frac{1}{2}X = \frac{4}{3}Y - \frac{3}{3}Y \]
\[ \frac{1}{2}X = \frac{1}{3}Y \]

### Step 4: Solve for the ratio \( \frac{X}{Y} \)
To find the ratio \( \frac{X}{Y} \), we can multiply both sides of the equation by 2:
\[ X = \frac{2}{3}Y \]
Now, we can express the ratio \( \frac{X}{Y} \):
\[ \frac{X}{Y} = \frac{2}{3} \]

### Step 5: Write the final answer
Thus, the ratio between the first number \( X \) and the second number \( Y \) is:
\[ \text{Ratio } (X:Y) = 2:3 \]