Discount

(a) Discount $= 25\% \times 24 = 0.25 \times 24 = 6$, so $6. Selling price $= 24 - 6 = 18$.

(b) 25% off means paying $100\% - 25\% = 75\%$ of the marked price: $24 \times 0.75 = 18$.

Discount $= 12\% \times 32.50 = 0.12 \times 32.50 = 3.9$, so $3.90.

Selling price $= 32.50 - 3.90 = 28.60$, so $28.60.

Check in one step: 12% off means paying 88%, and $32.50 \times 0.88 = 28.6$.

Discount $= 160 - 128 = 32$, so $32.

As a percentage of the marked price: $\frac{32}{160} \times 100\% = 20\%$

Check: $160 \times 0.8 = 128$

Discount $= 28 - 19.90 = 8.10$, so $8.10.

As a percentage of the marked price: $\frac{8.10}{28} \times 100\% = 28.928\ldots\% \approx 28.9\%$

(a) The discount is $389 - 299 = 90$, so $90, which Ravi has right. But a percentage discount is a percentage of the marked price, $389, the price the discount is taken off. Ravi has divided by the selling price, $299.

(b) $\frac{90}{389} \times 100\% = 23.136\ldots\% \approx 23.1\%$

Ravi's way gives $\frac{90}{299} \times 100\% \approx 30.1\%$, which is too large.

(a) After 20% off: $1200 \times 0.8 = 960$, so $960.

After a further 5% off the $960: $960 \times 0.95 = 912$, so $912.

(b) The buyer pays $0.8 \times 0.95 = 0.76$ of the marked price, which is 76%. The single discount is $100\% - 76\% = 24\%$.

Check: $1200 - 912 = 288$, and $\frac{288}{1200} \times 100\% = 24\%$.

After 10% off: $250 \times 0.9 = 225$, so $225. After a further 10% off: $225 \times 0.9 = 202.5$, so $202.50.

She pays $0.9 \times 0.9 = 0.81$ of the marked price, which is 81%, so the single discount is $100\% - 81\% = 19\%$.

Check: $250 - 202.50 = 47.50$, and $\frac{47.50}{250} \times 100\% = 19\%$.

20% off would give $250 \times 0.8 = 200$, which is $200. The second 10% is taken of $225, not of $250, so it is only $22.50.

Shop A: $86 \times 0.75 = 64.5$, so $64.50.

Shop B: $86 \times 0.85 \times 0.9 = 65.79$, so $65.79.

Difference: $65.79 - 64.50 = 1.29$

At Shop B the buyer pays $0.85 \times 0.9 = 0.765$ of the marked price, which is 76.5%, so the single discount is $100\% - 76.5\% = 23.5\%$, less than 25%.

(a) 12% off: $120 \times 0.88 = 105.6$, so $105.60. $18 off: $120 - 18 = 102$, so $102. "$18 off" is better, by $105.60 - 102 = 3.60$, which is $3.60.

(b) 12% off: $200 \times 0.88 = 176$, so $176. $18 off: $200 - 18 = 182$, so $182. "12% off" is better, by $182 - 176 = 6$, which is $6.

(c) The same price when 12% of the marked price is $18. 1% is $18 \div 12 = 1.5$, which is $1.50, so 100% is $1.5 \times 100 = 150$, which is $150.

Check: 12% of $150 is $0.12 \times 150 = 18$. Below $150 the $18 voucher takes off more; above $150 the 12% voucher does, as (a) and (b) show.

(a) After 10% off: $400 \times 0.9 = 360$. After a further 20% off: $360 \times 0.8 = 288$. After a further 5% off: $288 \times 0.95 = 273.6$, so $273.60.

In one line: $400 \times 0.9 \times 0.8 \times 0.95 = 273.6$

(b) He pays $0.9 \times 0.8 \times 0.95 = 0.684$ of the marked price, which is 68.4%, so the single discount is $100\% - 68.4\% = 31.6\%$.

Check: $400 - 273.60 = 126.40$, and $\frac{126.40}{400} \times 100\% = 31.6\%$. Adding the three discounts would give 35%, which is too much.

Sale price: $80 \times 0.8 = 64$, so $64.

Price on the last day: 40% off the marked price is $80 \times 0.6 = 48$, so $48.

The further discount is $64 - 48 = 16$, so $16, and it is taken off the sale price: $\frac{16}{64} \times 100\% = 25\%$.

Check with the multipliers: $0.8 \times 0.75 = 0.6$, which is 60% of the marked price, so 40% off altogether. The further discount is not $40\% - 20\% = 20\%$: that would give $64 \times 0.8 = 51.2$, which is $51.20, not $48.