Mathematics and Statistics · Ch 18 — Commercial Mathematics

Profit, Loss and their Percentages

Trade turns on the gap between what a trader pays for goods and what a trader receives for them.

Cost price, selling price, profit and loss

Cost Price (CP) — the price at which an article is bought (including buying expenses such as freight and cartage).

Selling Price (SP) — the price at which it is sold.

If $\text{SP}>\text{CP}$ , there is a profit (gain) $=\text{SP}-\text{CP}$ .

If $\text{SP}<\text{CP}$ , there is a loss $=\text{CP}-\text{SP}$ .

Profit and loss as percentages

Profit and loss are almost always quoted as a percentage of the cost price (the base is the CP unless a problem states otherwise).

Note

The core formulae

$\text{Profit}\%=\frac{\text{Profit}}{\text{CP}}\times100,\qquad \text{Loss}\%=\frac{\text{Loss}}{\text{CP}}\times100.$

Using multiplying factors:

$\text{SP}=\text{CP}\left(1+\frac{\text{Profit}\%}{100}\right),\qquad \text{SP}=\text{CP}\left(1-\frac{\text{Loss}\%}{100}\right).$

Rearranging recovers the cost price from a known SP:

$\text{CP}=\frac{\text{SP}}{1+\dfrac{\text{Profit}\%}{100}}\quad\text{or}\quad \text{CP}=\frac{\text{SP}}{1-\dfrac{\text{Loss}\%}{100}}.$ …

Definition 1Cost Price (CP) and Selling Price (SP)

CP is the buying price (with buying expenses); SP is the selling price. Profit $=\text{SP}-\text{CP}$ when SP exceeds CP; Loss $=\text{C …

Definition 2Profit\% / Loss\%

$\text{Profit}\%=\dfrac{\text{Profit}}{\text{CP}}\times100$ and $\text{Loss}\%=\dfrac{\text{Loss}}{\text{CP}}\times100$ — both mea …

Definition 3SP from CP (and back)

$\text{SP}=\text{CP}\left(1\pm\tfrac{\%}{100}\right)$ ( $+$ for profit, $-$ for loss); invert to get $\text{CP}=\dfrac{\t …

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