Quick Revision

CP, SP and the Multiplier

Profit = SP − CP and loss = CP − SP. Profit % and loss % are on the CP unless the question says otherwise.
Multiplier form: $\text{SP} = \text{CP} \times \frac{100 + p}{100}$ , and $\frac{100 - l}{100}$ for a loss. CP ₹720 with a $12\frac{1}{2}$ % gain sells for 720 × $\frac{9}{8}$ = ₹810.
Going back: $\text{CP} = \text{SP} \times \frac{100}{100 + p}$ . Sold for ₹1,035 at a 15% gain means CP = 1,035 ÷ 1.15 = ₹900.
Trap: never take p% off the SP to get the CP; divide.

Profit of r% on the SP: SP = 100 and CP = 100 − r, so profit on CP = $\frac{r}{100 - r} \times 100$ . So 40% on the SP is $66\frac{2}{3}$ % on the CP.
p% on the CP is $\frac{p}{100 + p} \times 100$ on the SP. So 50% on the CP is $33\frac{1}{3}$ % on the SP.
Profit of $\frac{1}{6}$ of the SP is $\frac{1}{5}$ of the CP, which is 20%.
CP : SP = 9 : 11 is a profit of 2 parts on 9 = $22\frac{2}{9}$ %.

Gain at one price equals the loss at another (in rupees or %): CP is halfway. Gain at ₹1,100 equals loss at ₹860, so CP = (1,100 + 860) ÷ 2 = ₹980.
A change of ₹d in the SP that moves one result to another: CP = d ÷ gap %.
Loss to gain: add. A 4% loss to a 9% gain is a 13% gap, so ₹299 more means CP = 299 ÷ 0.13 = ₹2,300.
Two gains or two losses: subtract (16% and 10% leave 6%).

Repairs, transport, packing, loading, tax paid and labour all go into the CP before you work out the profit.
Bought for ₹7,600 with ₹400 transport: CP = ₹8,000. Sold for ₹9,000, the profit is ₹1,000, which is 12.5%.
Trap: dividing the profit by the bare price of ₹7,600 gives the wrong percent.
Several pieces bought together: total profit = profit per piece × number of pieces.
After a first sale at a loss, the price needed for the rest = total target − money already received.

Bring both sides to one article or to the LCM of the counts. Buy 8 for ₹56 (₹7 each) and sell 7 for ₹56 (₹8 each): profit = $\frac{1}{7}$ = $14\frac{2}{7}$ %.
Same money on both sides: profit % = $\frac{\text{bought} - \text{sold}}{\text{sold}} \times 100$ .
CP of x = SP of y: profit % = $\frac{x - y}{y} \times 100$ . CP of 33 = SP of 30 gives 10%.
Selling n and gaining the SP of z: $\frac{z}{n - z} \times 100$ . Gaining the CP of z: $\frac{z}{n} \times 100$ .

Same SP, x% gain on one and x% loss on the other: always a loss of $\frac{x^2}{100}$ %. At 8% each, the loss is 0.64%.
The loss comes from the dearer item, the one sold at a loss.
Different rates: find each CP and add. At ₹1,500 each, a 50% gain and a 25% loss mean CPs of ₹1,000 and ₹2,000. ₹3,000 goes in and ₹3,000 comes out: no profit, no loss.
Same CP instead of same SP: the overall % is the plain average.

The overall % is the weighted average of the rates, with each part weighted by its cost.
$\frac{2}{3}$ at a 15% gain and $\frac{1}{3}$ at a 6% loss: 10 − 2 = 8% gain.
Take the stock as ₹100, ₹300 or the LCM of the fractions and work in rupees. That also finds a missing rate for the last part.
Trap: the plain average of the rates is right only when the parts are equal.
Finding the sizes of the parts from the overall % is alligation (Chapter 6).

Take the old CP as ₹100, write both CPs and SPs, and compare.
Profit 30%, CP up 4%, SP the same: 130 against 104 is a profit of 25%.
Bought x% cheaper and sold ₹d more or less: find the gap between the two SPs for a CP of ₹100, then scale it to ₹d.
Higher price, fewer units: compare units × profit per unit before and after.
Material 70% and labour 30% of the cost, rising 10% and 20%: cost rises by 7 + 6 = 13%.

The discount is always on the MP: SP = MP × $\frac{100 - d}{100}$ .
Markup m% and discount d%: profit % = $m - d - \frac{md}{100}$ . Markup 60% and discount 15%: 60 − 15 − 9 = 36%.
MP : CP = 4 : 3 with 10% off gives SP = 3.6, a profit of 0.6 on 3 = 20%.
The same m% for markup and discount always gives a loss of $\frac{m^2}{100}$ %: 1% for 10% each.
Discount equal to profit: CP = SP − discount.

$\text{MP} = \text{CP} \times \frac{100 + p}{100 - d}$ , with p the profit wanted and d the discount.
CP ₹1,240, 10% profit after 12% off: SP must be ₹1,364, so MP = 1,364 ÷ 0.88 = ₹1,550.
Markup % needed = $\frac{100 + p}{100 - d}$ − 1. With 25% off and an 8% profit: $\frac{108}{75}$ = 1.44, so mark 44% above CP.
Traps: adding d% to the SP, or p + d to the CP. The SP is (100 − d)% of the MP, so divide.

Two discounts: single equivalent = $d_1 + d_2 - \frac{d_1 d_2}{100}$ . Two of 25% make 43.75%, not 50%.
Three or more: multiply what is left. Three of 30% leave 0.7 × 0.7 × 0.7 = 0.343, a discount of 65.7%.
The order of the discounts does not change the answer.
A single discount of $d_1 + d_2$ always beats successive discounts of $d_1$ and $d_2$ .
Missing second discount: final price ÷ price after the first discount gives what is left.

Buy x get y free: the customer takes x + y and pays for x. Discount = $\frac{y}{x + y} \times 100$ . Buy 4 get 2 free is $33\frac{1}{3}$ %.
Trap: dividing by x instead of x + y.
Free items plus a d% discount on the bill: treat the free items as one discount and combine the two like successive discounts.
To compare schemes, turn each into a single discount. Buy 11 get 1 free ( $8\frac{1}{3}$ %) beats 5% then 3% (7.85%).

GST or sales tax is charged on the discounted price: pay = MP × (1 − d) × (1 + t). MP ₹2,400 with 10% off and 18% GST: 2,160 × 1.18 = ₹2,548.80.
Never net the two as (t − d)% of the MP.
A trade discount is given by a wholesaler to a retailer on the list price. A cash discount, for paying at once, comes after it. List ₹6,000 with 15% and 2%: ₹4,998.

Selling at CP with a short weight: gain % = $\frac{\text{error}}{\text{true} - \text{error}} \times 100$ . Divide by what is actually given.
A metre rod 4 cm short: $\frac{4}{96} \times 100 = 4\frac{1}{6}$ %.
Profit or discount plus a false weight: gain = $\frac{\text{money received}}{\text{cost of goods given}}$ − 1. They multiply, never add.
x% less weight and y% more price: $\frac{100 + y}{100 - x}$ − 1. Giving 10% less and charging 8% more gives $\frac{108}{90}$ = 1.2, a 20% gain.
Cheating while buying and selling: multiply both effects.

Chain of sales: last SP = first CP × each multiplier. A 25% profit and then a 4% loss make 1.25 × 0.96 = 1.2, a 20% gain overall.
Going back to the first CP: divide by each multiplier, never add or subtract the percentages.
Commission of c% on the SP: the owner keeps (100 − c)%, so SP = amount needed ÷ (1 − c/100).
Wastage: 5% of 400 kg spoils, so the whole cost and profit must come from the other 380 kg.