1.1.21 · D5Arithmetic & Number Systems
Question bank — Profit, loss, discount, simple interest — basic applications
True or false — justify
A 10% profit followed by a 10% loss on the same item returns you to the original CP
False — the loss acts on the bigger number (SP), not on CP, so ; you end up 1% below CP.
Raising the marked price and then giving a bigger discount always leaves the customer worse off
False — it depends on the numbers; you can inflate MP then "discount" back to the same SP, so the customer may pay exactly the same. Only the final SP matters to the buyer.
If Profit% is measured against CP and Discount% against MP, then Profit% and Discount% of the same sale can never be equal
False — they are just percentages of different bases; nothing stops the values from coinciding for particular numbers. The bases differ, not the possible values.
Doubling the time in simple interest doubles the total amount
False — doubling time doubles the interest, but ; the principal does not double, so grows by less than a factor of 2.
A 0% discount and a 0% profit describe the same situation
False — 0% discount means (no reduction from tag); 0% profit means (break-even on cost). These are different equalities unless .
Selling below the marked price always means the shopkeeper made a loss
False — MP is a tag chosen above CP; even after a discount the SP can still exceed CP, giving profit. Loss is about CP, not MP.
If SI for one year is ₹200, then SI for three years is exactly ₹600
True — simple interest charges on the same original principal every year, so the yearly interest is constant and just multiplies by the number of years.
A larger Marked Price guarantees a larger profit
False — profit depends on the final SP versus CP; a high MP with a deep discount can leave SP low. MP is only the starting point for the discount, not the profit.
Spot the error
"SP = ₹1500 at 25% profit, so CP = ."
The 25% sits on the unknown CP, not on SP: , so . Percentage-up is undone by dividing, not by subtracting the same percent.
"Profit was ₹120 on an ₹920 sale, so profit% = ."
The base for profit is CP, not SP. If CP was ₹800 the correct figure is .
"Discount is 10%, so I take 10% off the cost price."
Discount lives on the marked price, never on CP. Take 10% off MP to get SP; profit is a separate comparison against CP.
"9 months at 6% gives ."
The rate is per year, so time must be in years: months years. Using inflates the interest twelvefold.
"Two successive 20% discounts equal one 40% discount."
The second 20% is taken off the already-reduced price, so the true factor is , i.e. a 36% discount, not 40%. See Marked Price and Successive Discounts.
"To reverse a 25% profit I apply a 25% loss."
A 25% increase then a 25% decrease gives , not 1. Up and down by the same percent are not inverse operations because they act on different bases.
"Amount for R = 8%, T = 3 gives ."
The R inside the bracket must be the decimal or divided by 100: . Forgetting the /100 treats 8% as 8.
Why questions
Why is CP the base for profit and loss rather than SP?
You invested the CP, so profit measures how hard your money worked — gain per rupee put in. That reference point is what you paid, i.e. CP. See Percentages.
Why is MP (not CP) the base for a discount?
A discount is a reduction the shopkeeper makes from the price the customer sees first — the tag. So it is naturally measured against the marked price the customer starts from.
Why do we divide (not multiply) to back out CP from an SP-and-profit%?
The percent is written on the unknown CP: . To isolate CP you undo a multiplication, which means dividing by .
Why does simple interest never "stack"?
By definition it charges only on the original principal each year — like a plant giving the same fruit yearly from the first seed. Interest earned is set aside, not re-invested, unlike Compound Interest.
Why is profit% just a proportion in disguise?
"For every 100 rupees of CP, how many did I gain?" is the ratio Profit : CP scaled to 100, exactly a proportion. That is why Ratio and Proportion underlies the formula.
Why must time be converted to years in the SI formula?
The rate R is defined per year, so R and T must share the same time unit; expressing months as a fraction of a year keeps them consistent.
Why can "back out the CP" problems be written as linear equations?
SP, MP, or A relate to the unknown base by a single multiplication like — one unknown, one linear relation. See Linear Equations.
Edge cases
What is the profit% when SP exactly equals CP?
Profit , so profit% — the break-even point, neither gain nor loss.
What does a 100% discount mean, and is it possible?
, i.e. the item is given away free. It is mathematically valid but only sensible in giveaways or clearances.
Can a discount percentage exceed 100%?
No — that would force , meaning the shop pays the customer to take it. Discount% is bounded between 0 and 100.
What happens to SI when the rate R is 0%?
for any time, and the amount stays — money kept but never grows.
What happens to SI when time T is 0 years?
No time has passed, so and ; interest needs elapsed time to accrue.
If SP is 0 (item given away free), what is the loss%?
Loss , so loss% — you lost your entire investment.
Can loss% ever exceed 100%?
Not from a normal sale — the most you can lose is the whole CP (SP = 0), which is 100%. A greater loss would need extra costs beyond CP, outside this basic model.
What is the SP when profit% equals loss% is impossible — why?
Profit and loss are opposite signs of the same comparison ; a single sale is one or the other (or zero), never both, so the two percentages can't co-exist for one transaction.
Connections
- Percentages — every base-and-comparison idea traps here trace back to this.
- Ratio and Proportion — profit% as the proportion Profit : CP.
- Marked Price and Successive Discounts — the "two 20%s ≠ 40%" trap.
- Compound Interest — the sibling where interest does stack.
- Linear Equations — the "back out the base" one-variable traps.
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