If decreasing `120` by `x%` gives the same result as increasing `40` by `x%`, then `x%` of `210` is what per cent less than `(x+ 20)%` of `180`?

Options

A. A)33 `1/3`
B. B)18
C. C)16 `2/3`
D. D)20

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Set up the equation based on the problem statement
We know that decreasing `120` by `x%` gives the same result as increasing `40` by `x%`. This can be expressed mathematically as:

\[ 120 - \left(\frac{x}{100} \times 120\right) = 40 + \left(\frac{x}{100} \times 40\right) \]

### Step 2: Simplify the equation
We can rewrite the equation:

\[ 120 - \frac{120x}{100} = 40 + \frac{40x}{100} \]

This simplifies to:

\[ 120 - 1.2x = 40 + 0.4x \]

### Step 3: Rearranging the equation
Now, let's move all terms involving `x` to one side and constant terms to the other side:

\[ 120 - 40 = 1.2x + 0.4x \]

This simplifies to:

\[ 80 = 1.6x \]

### Step 4: Solve for `x`
Now, divide both sides by `1.6` to find `x`:

\[ x = \frac{80}{1.6} = 50 \]

So, we have found that \( x = 50\% \).

### Step 5: Calculate \( x\% \) of `210`
Now we need to calculate \( 50\% \) of `210`:

\[ \frac{50}{100} \times 210 = 105 \]

### Step 6: Calculate \( (x + 20)\% \) of `180`
Next, we calculate \( (50 + 20)\% \) of `180`, which is \( 70\% \):

\[ \frac{70}{100} \times 180 = 126 \]

### Step 7: Find the difference between the two results
Now we need to find out how much less \( 105 \) is than \( 126 \):

\[ 126 - 105 = 21 \]

### Step 8: Calculate the percentage difference
To find out what percentage \( 21 \) is of \( 126 \):

\[ \text{Percentage less} = \left(\frac{21}{126}\right) \times 100 \]

Calculating this gives:

\[ \frac{21}{126} = \frac{1}{6} \approx 0.1667 \]

So,

\[ \left(0.1667 \times 100\right) \approx 16.67\% \]

### Final Answer
Thus, \( x\% \) of \( 210 \) is approximately \( 16.67\% \) less than \( (x + 20)\% \) of \( 180 \).

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