Two numbers are in the ratio 29: 20. On diminishing each of them by 420 the ratio becomes 17: 11. The difference between the number is:

Options

A. 1000
B. 1080
C. 960
D. 1440

Answer

To solve the problem, we need to find the difference between two numbers that are in the ratio 29:20. When each number is diminished by 420, the new ratio becomes 17:11. Let's break this down step by step.

### Step 1: Define the Numbers
Let the two numbers be represented as:
- First number = 29x
- Second number = 20x

### Step 2: Set Up the Equation After Diminishing
When we diminish each number by 420, the new numbers become:
- First number = 29x - 420
- Second number = 20x - 420

According to the problem, the new ratio of these numbers is 17:11. Therefore, we can set up the equation:
\[ \frac{29x - 420}{20x - 420} = \frac{17}{11} \]

### Step 3: Cross Multiply
Cross multiplying gives us:
\[ 11(29x - 420) = 17(20x - 420) \]

### Step 4: Expand Both Sides
Expanding both sides:
\[ 319x - 4620 = 340x - 7140 \]

### Step 5: Rearrange the Equation
Rearranging the equation to isolate x:
\[ 319x - 340x = -7140 + 4620 \]
\[ -21x = -2520 \]

### Step 6: Solve for x
Dividing both sides by -21:
\[ x = \frac{2520}{21} = 120 \]

### Step 7: Find the Original Numbers
Now that we have x, we can find the original numbers:
- First number = 29x = 29 * 120 = 3480
- Second number = 20x = 20 * 120 = 2400

### Step 8: Calculate the Difference
Now, we calculate the difference between the two numbers:
\[ \text{Difference} = 3480 - 2400 = 1080 \]

### Final Answer
The difference between the two numbers is **1080**.

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