If first number is `8(1)/(3)%` less than third number and ratio of second and third number is 15:16 then average of first and third number is how much percent less/more then second number?

Options

A. `2.66%`
B. `2.22%`
C. `3.333%`
D. `2.45%`

Answer

To solve the problem step by step, we will follow the given information and calculations from the video transcript.

### Step 1: Understand the given percentages and ratios
The first number is \(8 \frac{1}{3}\% \) less than the third number. We convert this percentage into a fraction:
\[ 8 \frac{1}{3}\% = \frac{25}{3}\% \]

### Step 2: Define the variables
Let:
- The third number be \(16x\) (based on the ratio of the second and third numbers being \(15:16\)).
- The second number be \(15x\).

### Step 3: Calculate the first number
Since the first number is \( \frac{25}{3}\% \) less than the third number, we can express the first number as:
\[ \text{First Number} = \text{Third Number} - \left(\frac{25}{3}\% \text{ of Third Number}\right) \]
\[ = 16x - \left(\frac{25}{3} \times \frac{16x}{100}\right) \]
\[ = 16x - \left(\frac{25 \times 16x}{300}\right) \]
\[ = 16x - \frac{400x}{300} \]
\[ = 16x - \frac{4x}{3} \]
To combine these, we convert \(16x\) into a fraction with a denominator of 3:
\[ = \frac{48x}{3} - \frac{4x}{3} = \frac{44x}{3} \]
Thus, the first number is:
\[ \text{First Number} = \frac{44x}{3} \]

### Step 4: Calculate the average of the first and third numbers
The average of the first and third numbers is given by:
\[ \text{Average} = \frac{\text{First Number} + \text{Third Number}}{2} \]
\[ = \frac{\frac{44x}{3} + 16x}{2} \]
Convert \(16x\) into a fraction with a denominator of 3:
\[ = \frac{\frac{44x}{3} + \frac{48x}{3}}{2} = \frac{\frac{92x}{3}}{2} = \frac{92x}{6} = \frac{46x}{3} \]

### Step 5: Find the difference between the average and the second number
Now, we need to find how much the average is more or less than the second number:
\[ \text{Second Number} = 15x \]
The difference is:
\[ \text{Difference} = \frac{46x}{3} - 15x \]
Convert \(15x\) into a fraction with a denominator of 3:
\[ = \frac{46x}{3} - \frac{45x}{3} = \frac{x}{3} \]

### Step 6: Calculate the percentage difference
To find how much more the average is than the second number in percentage terms:
\[ \text{Percentage} = \left(\frac{\text{Difference}}{\text{Second Number}}\right) \times 100 \]
\[ = \left(\frac{\frac{x}{3}}{15x}\right) \times 100 = \left(\frac{1}{45}\right) \times 100 = \frac{100}{45} \approx 2.22\% \]

### Final Answer
The average of the first and third numbers is approximately \(2.22\%\) more than the second number.

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