Foundations — Profit, loss, discount, simple interest — basic applications

Before you can rebuild the formulas in the parent note, you must own every symbol it throws at you. Below we build them one at a time, each with a plain meaning, a picture, and a reason the topic cannot live without it.

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0. The most important word: "base"

Nothing here makes sense until you know what a base is.

Picture a ruler that always starts at the base and stretches to the final amount. The gap between them is the "change"; the base is where the ruler begins.

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1. Percent — the symbol %

The picture: imagine the base split into 100 equal little coins. Saying "20%" means "grab 20 of those 100 coins". If your base is ₹800, each coin is worth ₹8, and 20 coins is ₹160.

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2. Turning a percent into an action: $\times (1+\frac{p}{100})$

A percent by itself is a size. To use it you attach it to a base with multiplication.

Where the "1" comes from: the $1$ is the whole base you keep, and $\frac{p}{100}$ is the extra you add (or the slice you remove). So a 25% increase is "keep all of it (1) plus a quarter more (0.25)" = $\times 1.25$.

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3. The three price symbols: CP, SP, MP

These are just three named amounts of money. The letters are labels, nothing scary.

The picture: think of a single item on three different signs.

The subtractions themselves:

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4. The rearrange move: ÷ to reverse a ×

Many parent problems give you the final number and ask for the base. That means undoing a multiplication.

The picture: multiplication stretches the base; division shrinks it back by the exact same factor — like walking forward 1.25 steps, then walking back over the same 1.25 steps.

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5. The interest symbols: P, R, T

Simple Interest just renames the same idea for money that grows over time.

The picture: a plant that gives the same fruit every year, because simple interest always uses the original seed $P$, never last year's fruit.

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6. How proportion ties it all together

Every percentage formula is one proportion (two equal ratios).

This single template is the Ratio and Proportion idea powering the whole chapter.

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Prerequisite map

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Equipment checklist

• What does % actually mean as a number? ::: "out of 100", i.e. $÷100$ (so $8\%=0.08$)
• What is the "base" in any percentage question? ::: the starting number the change is compared against
• Base for profit and loss? ::: Cost Price (CP)
• Base for discount? ::: Marked Price (MP)
• Base for simple interest? ::: Principal (P)
• How do you increase a base by $p\%$ in one multiplication? ::: multiply by $(1+\frac{p}{100})$
• How do you reverse a $\times 1.25$ to recover the base? ::: divide by $1.25$ (do NOT multiply by $0.75$)
• Do a $25\%$ increase and $25\%$ decrease cancel out? ::: no — $1.25\times 0.75=0.9375\ne 1$
• Convert 9 months into years for SI? ::: $9/12=0.75$ years
• The one master proportion behind every formula? ::: $\frac{\text{change}}{\text{base}}=\frac{\text{percent}}{100}$

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Connections

• Percentages — the engine; "out of 100" is defined and drilled there.
• Ratio and Proportion — the master proportion template used all through.
• Linear Equations — "solve for the unknown base" is a one-variable equation.
• Compound Interest — what happens when interest stacks on itself.
• Marked Price and Successive Discounts — MP with more than one discount.