A single discount equivalent to three successive discounts of `22 (2)/(9)%, 14 (2)/(7)% and 6.25%` is?

Options

A. `35%`
B. `33.33%`
C. `31.25%`
D. `37.5%`

Answer

To find the single discount equivalent to three successive discounts of \(22 \frac{2}{9}\%\), \(14 \frac{2}{7}\%\), and \(6.25\%\), we can follow these steps:

### Step 1: Convert the mixed fractions to improper fractions
1. Convert \(22 \frac{2}{9}\%\) to an improper fraction:
\[ 22 \frac{2}{9} = \frac{22 \times 9 + 2}{9} = \frac{198 + 2}{9} = \frac{200}{9} \]

2. Convert \(14 \frac{2}{7}\%\) to an improper fraction:
\[ 14 \frac{2}{7} = \frac{14 \times 7 + 2}{7} = \frac{98 + 2}{7} = \frac{100}{7} \]

### Step 2: Use the formula for successive discounts
The formula for calculating the equivalent single discount \(D\) from successive discounts \(A\), \(B\), and \(C\) is:
\[ D = A + B + C - \frac{AB}{100} - \frac{BC}{100} - \frac{CA}{100} \]

Here, we'll first calculate the equivalent of the first two discounts.

### Step 3: Calculate the first two discounts
Let:
- \(A = \frac{200}{9}\)
- \(B = \frac{100}{7}\)

Using the formula:
\[ D_{1} = A + B - \frac{AB}{100} \]
Calculating \(AB\):
\[ AB = \frac{200}{9} \times \frac{100}{7} = \frac{20000}{63} \]

Now substituting back into the formula:
\[ D_{1} = \frac{200}{9} + \frac{100}{7} - \frac{20000}{6300} \]

Finding a common denominator (which is 63):
\[ D_{1} = \frac{200 \times 7}{63} + \frac{100 \times 9}{63} - \frac{20000}{6300} \]
\[ = \frac{1400 + 900}{63} - \frac{20000}{6300} \]
\[ = \frac{2300}{63} - \frac{20000}{6300} \]

### Step 4: Simplify the expression
Convert \(\frac{20000}{6300}\) to have the same denominator:
\[ \frac{20000}{6300} = \frac{20000 \div 100}{6300 \div 100} = \frac{200}{63} \]
Now substituting back:
\[ D_{1} = \frac{2300 - 200}{63} = \frac{2100}{63} \]

### Step 5: Include the third discount
Now we need to include the third discount \(C = 6.25\%\) which can be expressed as:
\[ C = \frac{625}{100} = 6.25 \]

Now we calculate the final single discount:
\[ D = D_{1} + C - \frac{D_{1} \times C}{100} \]
Substituting the values:
\[ D = \frac{2100}{63} + 6.25 - \frac{\frac{2100}{63} \times 6.25}{100} \]

Calculating \(\frac{2100 \times 6.25}{6300}\):
\[ = \frac{13125}{6300} \]

Now substituting back:
\[ D = \frac{2100}{63} + \frac{625}{100} - \frac{13125}{6300} \]

### Step 6: Final Calculation
Finding a common denominator and simplifying gives us the final equivalent single discount.

### Final Result
After performing all calculations, we find that the equivalent single discount is approximately \(37.5\%\).