Find a single discount (as percent) equivalent to following successive discounts:
`20%, 10% and 5%`

Answer

To find a single discount equivalent to the successive discounts of 20%, 10%, and 5%, we can use the formula for combining two successive discounts. The formula for combining two discounts \(D_1\) and \(D_2\) is:

\[ D = D_1 + D_2 - \frac{D_1 \times D_2}{100} \]

Since we have three discounts, we will first combine the first two discounts (20% and 10%) and then combine the result with the third discount (5%).

### Step 1: Combine the first two discounts (20% and 10%)

Let:
- \(D_1 = 20\%\)
- \(D_2 = 10\%\)

Using the formula:

\[ D' = D_1 + D_2 - \frac{D_1 \times D_2}{100} \]

Substituting the values:

\[ D' = 20 + 10 - \frac{20 \times 10}{100} \]

Calculating the product:

\[ D' = 20 + 10 - \frac{200}{100} \]

\[ D' = 20 + 10 - 2 \]

\[ D' = 28\% \]

### Step 2: Combine the result with the third discount (5%)

Now we have a new discount \(D' = 28\%\) and we need to combine it with \(D_3 = 5\%\).

Using the formula again:

\[ D = D' + D_3 - \frac{D' \times D_3}{100} \]

Substituting the values:

\[ D = 28 + 5 - \frac{28 \times 5}{100} \]

Calculating the product:

\[ D = 28 + 5 - \frac{140}{100} \]

\[ D = 28 + 5 - 1.4 \]

\[ D = 33 - 1.4 \]

\[ D = 31.6\% \]

### Final Result

The single discount equivalent to the successive discounts of 20%, 10%, and 5% is **31.6%**.

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