Find the single discount (as percent) equivalent to successive discounts of :

(i) 80% and 80%

(ii) 60% and 60%

(iii) 60% and 80%

(i) Let M.P. be ₹ $100$.

1st discount % = 80%

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

And,

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

New M.P. = ₹ 20

2nd Discount % = 80%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 80 = \dfrac{\text{Discount}}{20} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{80 \times 20}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{1600}{100}\\[1em] \Rightarrow \text{Discount} = 16$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 20 - 16\\[1em] \Rightarrow \text{S.P.} = 4$

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

= 100 - 4

= 96

$\text{Discount\%} = \dfrac{\text{Single Discount}}{\text{Initial M.P.}} \times 100 \\[1em] \Rightarrow \text{Discount\%} = \dfrac{96}{100} \times 100 \%\\[1em] \Rightarrow \text{Discount\%} = \dfrac{96}{\cancel{100}} \times \cancel{100} \%\\[1em] \Rightarrow \text{Discount\%} = 96\%$

Hence, single equivalent discount = 96%.

(ii) Let M.P. be ₹ $100$.

1st discount % = 60%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 60 = \dfrac{\text{Discount}}{100} \times 100\\[1em] \Rightarrow 60 = \dfrac{\text{Discount}}{\cancel{100}} \times \cancel{100}\\[1em] \Rightarrow \text{Discount} = 60$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 100 - 60\\[1em] \Rightarrow \text{S.P.} = 40$

New M.P. = ₹ 40

2nd Discount % = 60%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 60 = \dfrac{\text{Discount}}{40} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{60 \times 40}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{2400}{100}\\[1em] \Rightarrow \text{Discount} = 24$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 40 - 24\\[1em] \Rightarrow \text{S.P.} = 16$

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

= 100 - 16

= 84

$\text{Discount\%} = \dfrac{\text{Single Discount}}{\text{Initial M.P.}} \times 100 \\[1em] \Rightarrow \text{Discount\%} = \dfrac{84}{100} \times 100 \%\\[1em] \Rightarrow \text{Discount\%} = \dfrac{84}{\cancel{100}} \times \cancel{100} \%\\[1em] \Rightarrow \text{Discount\%} = 84\%$

Hence, single equivalent discount = 84%.

(iii) Let M.P. be ₹ $100$.

1st discount % = 60%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 60 = \dfrac{\text{Discount}}{100} \times 100\\[1em] \Rightarrow 60 = \dfrac{\text{Discount}}{\cancel{100}} \times \cancel{100}\\[1em] \Rightarrow \text{Discount} = 60$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 100 - 60\\[1em] \Rightarrow \text{S.P.} = 40$

New M.P. = ₹ 40

2nd Discount % = 80%

$\text{Discount \%} = \dfrac{\text{Discount}}{\text{M.P.}} \times 100\\[1em] \Rightarrow 80 = \dfrac{\text{Discount}}{40} \times 100\\[1em] \Rightarrow \text{Discount} = \dfrac{80 \times 40}{100}\\[1em] \Rightarrow \text{Discount} = \dfrac{3200}{100}\\[1em] \Rightarrow \text{Discount} = 32$

And,

$\text{S.P. = M.P. - Discount}\\[1em] \Rightarrow \text{S.P.} = 40 - 32\\[1em] \Rightarrow \text{S.P.} = 8$

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

= 100 - 8

= 92

$\text{Discount\%} = \dfrac{\text{Single Discount}}{\text{Initial M.P.}} \times 100 \\[1em] \Rightarrow \text{Discount\%} = \dfrac{92}{100} \times 100 \%\\[1em] \Rightarrow \text{Discount\%} = \dfrac{92}{\cancel{100}} \times \cancel{100} \%\\[1em] \Rightarrow \text{Discount\%} = 92\%$

Hence, single equivalent discount = 92%.