Profit, loss, discount, simple interest — basic applications: Visual walkthrough

Step 1 — What "percentage of a base" means, drawn

WHAT. Before any formula, let us draw what " $20%$ of $100$ " looks like.

WHY. Every formula on this page divides one thing by a base and multiplies by $100$ . If you can see what that operation does, you never have to memorise the formula — you rebuild it. So we must first make the words base, percent, and compare into a picture.

PICTURE. Look at the figure. The tall bar is the base — the number we measure against. We slice it into $100$ equal thin strips. A percentage is simply how many of those 100 strips another quantity covers.

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

Here $\frac{Q}{B}$ answers "what fraction of the base?" and the $\times 100$ rescales that fraction so a full base reads as $100$ instead of $1$ . That is the only tool we use all page.

Step 2 — Profit: stacking a change on top of the base

WHAT. Draw the Cost Price as the base bar, and the Selling Price as a taller bar. The extra height is profit.

WHY. In buying and selling, the money you put in is the Cost Price — so it is the base (you ask "how well did my money work?"). We want a picture where profit is visibly extra height above CP, because that is exactly what " $S P - C P$ " is.

PICTURE. The left bar is $C P$ . The right bar is $S P$ , taller. The magenta slab sitting above the CP height is the profit — the part of $S P$ that rises past $C P$ .

Now apply the Step 1 tool with base $= C P$ :

$how many hundredths of CP Profit % = \frac{S P - C P the extra slab}{the base bar C P} \times rescale to per-hundred 100$

Each piece: the numerator is the magenta slab (the change), the denominator is the base bar (CP), and $\times 100$ turns the fraction into a percent — exactly Step 1.

Step 3 — Loss: the same picture, flipped

WHAT. Now let $S P$ be shorter than $C P$ . The missing height is loss.

WHY. We must cover both signs of the change. Profit and loss are not two formulas — they are the same slab, once pointing up, once pointing down. Drawing it stops you memorising a second formula.

PICTURE. $C P$ is still the base bar. $S P$ is now shorter, so there is a violet gap at the top of the CP bar that $S P$ fails to reach. That gap is the loss.

$Loss % = \frac{C P - S P the gap}{same base C P} \times 100$

Notice the base never changed — it is $C P$ in both directions. Only the order of subtraction flips so the change stays positive.

Step 4 — Going the other way: multiply to build SP

WHAT. Instead of measuring the slab, we now grow the CP bar by a chosen percent to find $S P$ .

WHY. Real problems say "sell at $25%$ profit — what is SP?" We know the base ( $C P$ ) and the slab's size in percent; we want the total height. So we add the slab-fraction back on top of the whole base.

PICTURE. Start with the full CP bar (that is " $1$ whole", or $100%$ ). Stack the profit slab, worth $\frac{p}{100}$ of that bar, on top. The combined height is $S P$ .

$S P = C P (keep whole CP 1 + stack profit slab \frac{p}{100})$

The $1$ = the entire CP bar you keep.
The $\frac{p}{100}$ = the extra slab, as a fraction of CP.
For a loss, use $1 - \frac{p}{100}$ (chop the slab off instead).

Step 5 — Reversing: why you divide, not multiply back

WHAT. Now the trap: given $S P = 1500$ at $25%$ profit, find $C P$ .

WHY. People "undo" $\times 1.25$ by doing $\times 0.75$ — but $0.75$ is a percent of the wrong bar. The $25%$ was measured on the unknown CP, not on SP. We must see that the two bars have different heights, so $25%$ of one is not $25%$ of the other.

PICTURE. Two bars side by side: the shorter unknown $C P$ with a $25%$ slab on top reaching $S P$ . The $25%$ slab is a slice of the short CP bar — so it is smaller than $25%$ of the tall SP bar. That size mismatch is why multiplying $S P \times 0.75$ overshoots.

Since $S P = C P \times 1.25$ , we isolate the base — this is a one-step linear equation:

$C P = \frac{S P}{1 + \frac{p}{100}} = \frac{1500}{1.25} = 1200$

Step 6 — Discount: the same slab, new base (MP)

WHAT. Draw a third, tallest bar — the Marked Price — and chop a slab off the top to reach $S P$ .

WHY. Discount is a shop decision measured from the tag the customer sees first. So its base is MP, not CP. The arithmetic is identical to Step 3 (chop a slab) — only the base bar changes. Seeing all three bars together is how you stop mixing MP and CP.

PICTURE. Three bars: $C P$ (base for profit), $M P$ (tallest, base for discount), and $S P$ landing between them. The orange slab chopped from the top of $M P$ is the discount; the magenta slab from the top of $C P$ up to $S P$ is the profit. Two different bases, one drawing.

$S P = M P (1 - \frac{d}{100}), Discount % = \frac{M P - S P}{M P} \times 100$

The customer pays the navy portion of the MP bar; the profit is still measured up from the $C P$ line. See Marked Price and Successive Discounts for chopping more than one slab.

Step 7 — Simple Interest: the same slab, once per year

WHAT. Draw the Principal as the base bar. Each year adds an identical thin slab of interest.

WHY. "Simple" means interest is always $R %$ of the original $P$ — never of last year's total. So every year's slab is the same height. That is why time just multiplies — no stacking-on-stacking (that would be Compound Interest).

PICTURE. The tall Principal bar, and then $T$ equal orange slabs stacked, each one being $R %$ of $P$ . Because they are all cut from the same $P$ , they are equal — a neat staircase, not a curve.

One year's slab $= P \times \frac{R}{100}$ (Step 1 with base $= P$ ). Repeat for $T$ years:

$S I = one year’s slab P \times \frac{R}{100} \times how many slabs T = \frac{P R T}{100}$

$A = P + S I = P (1 + \frac{R T}{100})$

$P$ = base bar. $\frac{R}{100}$ = one slab as a fraction of $P$ . $T$ = number of identical slabs.

Step 8 — Degenerate & edge cases (never get surprised)

WHAT. What happens at the boundaries — zero change, break-even, $100%$ discount, zero time?

WHY. The contract: the reader must never meet a case we did not draw. Edges are where slabs shrink to nothing or the whole bar vanishes.

PICTURE. Four mini-bars: (a) $S P = C P$ → no slab → $0%$ profit; (b) $S P = 0$ → whole CP bar is the loss → $100%$ loss; (c) $d = 100%$ → the entire MP bar chopped → free item, $S P = 0$ ; (d) $T = 0$ → zero interest slabs → $A = P$ .

Case	What the picture shows	Result
$S P = C P$	no slab	Profit $= 0%$ (break-even)
$S P = 0$	whole CP bar lost	Loss $= 100%$
$d = 100%$	MP fully chopped	$S P = 0$ (given away)
$T = 0$	no slabs added	$S I = 0, A = P$
$d > 100%$	impossible — can't chop more than the whole bar	shop would pay you; not allowed

The one-picture summary

One drawing, three problems. Every topic is: pick the base bar, then either read a slab off it (→ percent) or stack/chop a slab (→ new price). The base is the only thing that changes: $C P$ for profit, $M P$ for discount, $P$ for interest.

Recall Feynman: the whole walkthrough in plain words

Draw a tall bar — that's your starting number, the base. Slice it into a hundred thin strips; "percent" just counts strips. Profit: the base is the money you paid (CP). Sell higher, and the extra sticking out the top is profit; count what fraction of the CP bar it is, times 100. Sell lower and it's a gap instead — that's loss. Same bar, slab up or slab down. Building a price: keep the whole CP bar (that's the " $1$ ") and stack an extra slab of $\frac{p}{100}$ on top — that's SP. To go backwards you must divide, because the slab was cut from the short bar, not the tall one. Discount: just a taller bar (the tag, MP) with a slice chopped off the top. Different base, exact same chopping. Interest: the base is the seed money (P), and every year adds the same little slab — because simple interest never counts last year's slab. Multiply one slab by the number of years. One picture, one question every time: how big is the change compared to the starting bar?

Connections

Percentages — Step 1 is percentages; the whole page is one tool repeated.
Ratio and Proportion — every slab-over-base is a proportion (Profit : CP).
Linear Equations — Step 5's "back out the base" is a one-variable equation.
Marked Price and Successive Discounts — Step 6 with more than one slab chopped.
Compound Interest — Step 7 but each slab is cut from the grown bar, not the seed.