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Margins and break-even

Margin against markup, a discount on a discount, profit and loss on cost, and how many units before a business breaks even.

  • Money
  • Untimed
  • Endless levels

How it plays

Each question is a price and a cost: the margin or the markup, a sale price after a discount, a cut on a cut, a mark-up then a discount, profit or loss as a share of cost, and the units a month that cover the fixed costs.

Every prompt says what a percentage is of — margin of the price, markup of the cost, profit and loss of the cost — because that is exactly where the mistakes live. New kinds unlock as you climb and the sums grow without end.

The wrong options are the real slips: the markup given for the margin, two discounts simply added, the cost found by taking the profit % off the price, selling at ±20% read as breaking even. The money is in the currency you pick.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes margin, discount and break-even arithmetic of these kinds quicker and surer. It is not business advice and claims nothing wider.

Learn

How to get better at margins and break-even: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Find what the percentage is “of”. The question always says: a margin is of the price, a markup is of the cost, profit and loss are of the cost.
  2. Work on 100 of that base when the sums are awkward: a cost of 100, or a price of 100. Percentages become plain numbers.
  3. For a chain — a cut on a cut, a markup then a discount — multiply the factors, never add the percentages.
  4. For break-even and target questions, find what each unit leaves after its own cost first; everything else is a division by that.

Techniques and tricks

  • Margin and markup from one profit

    Profit is price − cost. Divide it by the price for the margin, by the cost for the markup. The markup is always the bigger.

    e.g. Cost 80, price 100: profit 20, margin 20%, markup 25%.

  • Convert without a cost

    Markup u% is a margin of u ÷ (100 + u). Margin m% is a markup of m ÷ (100 − m).

    e.g. A 25% markup is a 20% margin; a 50% margin is a 100% markup.

  • Two cuts: a + b − ab ÷ 100

    A cut of a% then b% leaves (1 − a/100)(1 − b/100) of the price. The total cut is a + b less a small correction, ab ÷ 100.

    e.g. 20% then 10%: 20 + 10 − 2 = 28%.

  • Up then down, by factors

    A markup of u% then a discount of d% leaves (1 + u/100)(1 − d/100) of the cost. Above 1 is a profit, below 1 a loss.

    e.g. Up 25%, then 20% off: 1.25 × 0.8 = 1, no profit, no loss.

  • Same price, ±p%: lose p² ÷ 100

    Two items sold for the same price, one at p% profit and one at p% loss, always lose p² ÷ 100 per cent overall, because the loss is on the dearer item.

    e.g. ±20%: a loss of 400 ÷ 100 = 4%.

  • Divide back to the cost

    If a price includes p% profit on cost, the cost is the price ÷ (1 + p/100). Taking p% off the price is wrong, because the p% was of the smaller number.

    e.g. Sold for 1,380 at 15% profit: 1,380 ÷ 1.15 = 1,200, not 1,173.

  • Contribution does the counting

    Each unit leaves price − its own cost towards the fixed costs. Break-even is fixed costs ÷ that; for a target profit, add the target to the fixed costs first. For a margin, the price is cost ÷ (1 − margin).

    e.g. Fixed 12,000, contribution 50: 240 units; for 3,000 profit, 15,000 ÷ 50 = 300. Cost 600 at a 40% margin: 600 ÷ 0.6 = 1,000.

As the levels climb

  • Level 0 opens with three kinds: margin or markup, discounts and break-even. Level 1 adds markup to margin, mark up then discount, and profit and loss.
  • Level 3 adds two items at the same price and the cost from a price; level 5 adds units for a target profit and the price for a margin.
  • Between unlocks, and for ever after, the sums grow. From size 2 the discount questions include a cut on a cut, and from size 3 the conversions use any whole percentage, not only the tidy ones, so the formulas replace remembered pairs.
  • Bigger sums give a little more time, 8% more per notch for the first twelve. Work on 100 first, then scale, rather than dividing the big numbers directly.

Traps to avoid

  • The markup given for the margin, or the margin for the markup. Check which base the question names; the markup is always the larger of the two.
  • Two discounts simply added: 20% and 10% taken as 30% off. The second cut is on the smaller price.
  • A markup then a discount netted by subtraction: 25% up then 20% off taken as 5% profit. Multiply the factors instead.
  • The cost found by taking the profit percentage off the price, which takes it of the wrong amount.
  • Same price at ±p% read as breaking even. It is always a loss, of p² ÷ 100 per cent.

How to practise

  • Use “Kinds of question” to drill margin or markup together with markup to margin; mixing up the two bases is the root of most mistakes here.
  • Switch Answer by to “Typing the answer” once choosing is easy: the options include the wrong-base answers, which typing forces you to avoid yourself.
  • Pick your own Currency so the prices read like the ones you see.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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