Equal selling prices and false weights

Here are two results about profit and loss that sur­prise most peo­ple. The first: if you sell two things for the same price, gain­ing 10% on one and los­ing 10% on the other, you do not break even. You lose. The sec­ond: a trader who says he sells at cost price can still make a profit, if his weights are false. This les­son proves both results and shows you how to use them.

Result 1: same sell­ing price, equal profit and loss#

Result. Two arti­cles are sold at the same sell­ing price. One is sold at a profit of $a\%$ and the other at a loss of $a\%$. Then there is always a loss on the whole deal, and

$$\displaystyle \text{loss}\% = \left(\frac{a}{10}\right)^2\% = \frac{a^2}{100}\%$$

First see it with num­bers#

Two phones are each sold for ₹9900. One is sold at a 10% profit and the other at a 10% loss.

Cost price of the first phone:

$$\displaystyle \text{C.P.} = \frac{100}{110} \times 9900 = 9000$$

Cost price of the sec­ond phone:

$$\displaystyle \text{C.P.} = \frac{100}{90} \times 9900 = 11000$$

Total cost price $= 9000 + 11000 = 20000$ rupees.

Total sell­ing price $= 9900 + 9900 = 19800$ rupees.

The sell­ing price is less, so there is a loss of $20000 - 19800 = 200$ rupees.

$$\displaystyle \text{loss}\% = \frac{200}{20000} \times 100 = 1\%$$

And the result says $\displaystyle \left(\frac{10}{10}\right)^2 = 1$, so 1%. It agrees.

Why a loss? The 10% loss is taken on the dearer phone (₹11,000), but the 10% profit is taken on the cheaper one (₹9000). Ten per cent of the big­ger cost is more money.

Proof#

Let the sell­ing price of each arti­cle be $x$ rupees.

For the arti­cle sold at $a\%$ profit:

$$\displaystyle \text{C.P.} = \frac{100}{100 + a}x$$

For the arti­cle sold at $a\%$ loss:

$$\displaystyle \text{C.P.} = \frac{100}{100 - a}x$$

The total sell­ing price is $2x$. The total cost price is

$$\displaystyle \begin{aligned} \frac{100x}{100 + a} + \frac{100x}{100 - a} &= 100x\left(\frac{1}{100 + a} + \frac{1}{100 - a}\right) \\ &= 100x\left(\frac{(100 - a) + (100 + a)}{100^2 - a^2}\right) \\ &= 100x \times \frac{200}{100^2 - a^2} \\ &= \frac{100^2 \times 2x}{100^2 - a^2} \end{aligned}$$

The bot­tom, $100^2 - a^2$, is smaller than $100^2$. So the total C.P. is more than $2x$, the total S.P. There is a loss.

$$\displaystyle \text{loss}\% = \frac{\text{C.P.} - \text{S.P.}}{\text{C.P.}} \times 100 = \left(1 - \frac{\text{S.P.}}{\text{C.P.}}\right) \times 100$$

Now $\displaystyle \frac{\text{S.P.}}{\text{C.P.}} = 2x \div \frac{100^2 \times 2x}{100^2 - a^2} = \frac{100^2 - a^2}{100^2}$. So

$$\displaystyle \text{loss}\% = \left(1 - \frac{100^2 - a^2}{100^2}\right) \times 100 = \frac{a^2}{100^2} \times 100 = \frac{a^2}{100}$$

That is $\displaystyle \left(\frac{a}{10}\right)^2$ per cent.

Result 2: false weights#

Result. A trader says he sells his goods at cost price, but he uses a false weight. Then he gains, and

$$\displaystyle \text{gain}\% = \left(\frac{\text{error}}{\text{true value} - \text{error}} \times 100\right)\%$$

The error is how much the false weight is short of the true weight.

First see it with num­bers#

Rice costs the trader ₹20 per kg. He says he sells it at cost, ₹20 per kg. But his "1 kg" weight is really only 900 g. The error is $1000 - 900 = 100$ g.

For every ₹20 he takes, he gives only 900 g. What did those 900 g cost him?

$$\displaystyle \text{C.P. of 900 g} = \frac{900}{1000} \times 20 = 18$$

He sells them for ₹20. So he gains ₹2 on a cost of ₹18.

$$\displaystyle \text{gain}\% = \frac{2}{18} \times 100 = \frac{100}{9} = 11\frac{1}{9}$$

The result gives the same: $\displaystyle \frac{100}{1000 - 100} \times 100 = \frac{100}{900} \times 100 = 11\frac{1}{9}$, so $\displaystyle 11\frac{1}{9}\%$.

Proof#

Let the cost price of $p$ kg be $x$ rupees, and let the false weight be short by $f$ kg. So when he claims to give $p$ kg, he really gives $(p - f)$ kg.

When the trader buys: $p$ kg cost $x$ rupees, so $(p - f)$ kg cost him $\displaystyle \frac{x}{p}(p - f)$ rupees.

When the trader sells: he gives $(p - f)$ kg and takes $x$ rupees.

So for $(p - f)$ kg, S.P. $= x$ and C.P. $\displaystyle = \frac{x}{p}(p - f)$. S.P. is big­ger, so he gains.

$$\displaystyle \begin{aligned} \text{gain}\% &= \frac{\text{S.P.} - \text{C.P.}}{\text{C.P.}} \times 100 \\ &= \frac{x - \frac{x}{p}(p - f)}{\frac{x}{p}(p - f)} \times 100 \\ &= \frac{1 - \frac{p - f}{p}}{\frac{p - f}{p}} \times 100 \\ &= \frac{\frac{f}{p}}{\frac{p - f}{p}} \times 100 \\ &= \frac{f}{p - f} \times 100 \end{aligned}$$

Here $f$ is the error and $p$ is the true value. That is the result.

Exam­ple 1#

A shop­keeper says he sells at cost price, but he uses a weight of 960 g for 1 kg. Find his gain per cent.

The error is $1000 - 960 = 40$ g.

$$\displaystyle \text{gain}\% = \frac{40}{1000 - 40} \times 100 = \frac{40}{960} \times 100 = \frac{25}{6} = 4\frac{1}{6}$$

His gain is $\displaystyle 4\frac{1}{6}\%$.

Exam­ple 2#

Two scoot­ers are each sold for ₹48,000. One is sold at a 20% profit and the other at a 20% loss. Find the over­all loss per cent.

By Result 1, $\displaystyle \text{loss}\% = \left(\frac{20}{10}\right)^2 = 4$, so the loss is 4%.

Check: the costs are $\displaystyle \frac{100}{120} \times 48000 = 40000$ and $\displaystyle \frac{100}{80} \times 48000 = 60000$. Total C.P. is ₹1,00,000, total S.P. is ₹96,000, and the loss of ₹4000 is 4% of ₹1,00,000.

Prac­tice#

1. Two cycles are each sold for ₹4800. One is sold at a 20% profit and the other at a 20% loss. Find the over­all gain or loss per cent.
2. Two arti­cles are each sold for ₹1920, one at a 25% profit and the other at a 25% loss. Find the loss per cent.
3. Two watches are each sold for ₹990, one at a 10% gain and the other at a 10% loss. Find the loss in rupees.
4. A trader says he sells at cost price, but uses an 800 g weight for 1 kg. Find his gain per cent.
5. A gro­cer sells at cost price but uses a weight of 875 g for a kilo­gram. Find his gain per cent.
6. A milk­man sells at cost price but gives only 750 ml for every litre. Find his gain per cent.
7. A trader sells at cost price but gains 25% by using a false weight. What weight does he use for 1 kg?
8. Two arti­cles with the same sell­ing price are sold at $a\%$ profit and $a\%$ loss. The over­all loss is 2.25%. Find $a$.

Answers#