Find a single discount (as percent) equivalent to following successive discounts:

(i) 20% and 12%

(ii) 10%, 20% and 20%

(iii) 20%, 10% and 5%

(i) Let M.P. be ₹ $100$.

1st discount % = 20%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 20 = \dfrac{\text{Discount}}{100} \times 100\\[1em] \Rightarrow 20 = \dfrac{\text{Discount}}{\cancel{100}} \times \cancel{100}\\[1em] \Rightarrow \text{Discount} = 20$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 100 - 20\\[1em] \Rightarrow \text{S.P.} = 80$

New M.P. = ₹ 80

2nd Discount % = 12%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 12 = \dfrac{\text{Discount}}{80} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{12 \times 80}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{960}{100}\\[1em] \Rightarrow \text{Discount} = 9.6$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 80 - 9.6\\[1em] \Rightarrow \text{S.P.} = 70.4$

Single equivalent discount = Initial M.P. - Final S.P.

= 100 - 70.4

= 29.6

$\text{Discount\%} = \dfrac{\text{Single Discount}}{\text{Initial M.P.}} \times 100 \\[1em] \Rightarrow \text{Discount\%} = \dfrac{29.6}{100} \times 100 \%\\[1em] \Rightarrow \text{Discount\%} = \dfrac{29.6}{\cancel{100}} \times \cancel{100} \%\\[1em] \Rightarrow \text{Discount\%} = 29.6\%$

Hence, single equivalent discount = 29.6%.

(ii) Let M.P. be ₹ $100$.

1st discount % = 10%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 10 = \dfrac{\text{Discount}}{100} \times 100\\[1em] \Rightarrow 10 = \dfrac{\text{Discount}}{\cancel{100}} \times \cancel{100}\\[1em] \Rightarrow \text{Discount} = 10$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 100 - 10\\[1em] \Rightarrow \text{S.P.} = 90$

New M.P. = ₹ 90

2nd Discount % = 20%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 20 = \dfrac{\text{Discount}}{90} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{20 \times 90}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{1800}{100}\\[1em] \Rightarrow \text{Discount} = 18$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 90 - 18\\[1em] \Rightarrow \text{S.P.} = 72$

New M.P. = ₹ 72

2nd Discount % = 20%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 20 = \dfrac{\text{Discount}}{72} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{20 \times 72}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{1440}{100}\\[1em] \Rightarrow \text{Discount} = 14.4$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 72 - 14.4\\[1em] \Rightarrow \text{S.P.} = 57.6$

Single equivalent discount = Initial M.P. - Final S.P.

= 100 - 57.6

= 42.4

$\text{Discount\%} = \dfrac{\text{Single Discount}}{\text{Initial M.P.}} \times 100 \\[1em] \Rightarrow \text{Discount\%} = \dfrac{42.4}{100} \times 100 \%\\[1em] \Rightarrow \text{Discount\%} = \dfrac{42.4}{\cancel{100}} \times \cancel{100} \%\\[1em] \Rightarrow \text{Discount\%} = 42.4\%$

Hence, single equivalent discount = 42.4%.

(iii) Let M.P. be ₹ $100$.

1st discount % = 20%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 20 = \dfrac{\text{Discount}}{100} \times 100\\[1em] \Rightarrow 20 = \dfrac{\text{Discount}}{\cancel{100}} \times \cancel{100}\\[1em] \Rightarrow \text{Discount} = 20$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 100 - 20\\[1em] \Rightarrow \text{S.P.} = 80$

New M.P. = ₹ 80

2nd Discount % = 10%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 10 = \dfrac{\text{Discount}}{80} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{10 \times 80}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{800}{100}\\[1em] \Rightarrow \text{Discount} = 8$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 80 - 8\\[1em] \Rightarrow \text{S.P.} = 72$

New M.P. = ₹ 72

3rd Discount % = 5%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 5 = \dfrac{\text{Discount}}{72} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{5 \times 72}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{360}{100}\\[1em] \Rightarrow \text{Discount} = 3.6$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 72 - 3.6\\[1em] \Rightarrow \text{S.P.} = 68.4$

Single equivalent discount = Initial M.P. - Final S.P.

= 100 - 68.4

= 31.6

$\text{Discount\%} = \dfrac{\text{Single Discount}}{\text{Initial M.P.}} \times 100 \\[1em] \Rightarrow \text{Discount\%} = \dfrac{31.6}{100} \times 100 \%\\[1em] \Rightarrow \text{Discount\%} = \dfrac{31.6}{\cancel{100}} \times \cancel{100} \%\\[1em] \Rightarrow \text{Discount\%} = 31.6\%$

Hence, single equivalent discount = 31.6%.