KCSE 2025 Maths P1 Q11 — Profit and Loss
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The Question
“A shopkeeper bought an item from a wholesaler. If the shopkeeper sells the item at Ksh 2,740, he would make a profit of Ksh 3x. If the shopkeeper sells it at Ksh 2,340, he would make a loss of Ksh 2x. Determine the amount that the shopkeeper paid for the item.”
Set up the profit equation
Profit is what you have left after covering your cost, so it equals the selling price minus the cost price. Let the cost price be C. Selling at 2,740 gives a profit of 3x, which means the selling price is bigger than the cost by that amount. Writing this as an equation lets us pin down the relationship between C and x.
Set up the loss equation
A loss happens when the selling price falls short of the cost, so the loss equals the cost price minus the selling price. Selling at 2,340 gives a loss of 2x. This gives a second equation linking the same two unknowns, C and x, which is exactly what we need to solve for them.
Add the two equations to eliminate C
Notice that equation 1 has a minus C and equation 2 has a plus C. If we add the equations together, those two terms cancel out completely, and the unknown cost price disappears for a moment. This leaves a single equation in x only, which is easy to solve. This is the clever move that makes the whole problem quick.
Solve for x
Now divide both sides by 5 to get x on its own. This tells us the value that the profit and loss amounts are built from.
Substitute back to find the cost price
Put x equals 80 back into equation 2 to recover the cost price. The loss was 2x, so that is 2 times 80, which is 160. Adding this to the 2,340 selling price gives the amount the shopkeeper actually paid.
Final Result
The shopkeeper paid Ksh 2,500 for the item.
Why this method works
The trick works because both statements describe the same unknown cost price from two directions: one selling price sits above the cost (a profit) and the other sits below it (a loss). Writing profit as selling price minus cost, and loss as cost minus selling price, gives two equations where the cost appears with opposite signs. Adding them cancels the cost and isolates x, so we can solve one unknown at a time rather than juggling both at once. This is the standard elimination method for simultaneous equations, applied to a real money situation.
Check: at 2,740 the profit is 2,740 - 2,500 = 240 = 3x = 3 times 80; at 2,340 the loss is 2,500 - 2,340 = 160 = 2x = 2 times 80. Both match, so Ksh 2,500 is correct.