Consulting Interview Questions: Mental Math Riddles Solved
These riddles train you to set up a numerical case, work out a break-even point and estimate an order of magnitude in your head. Here are 4 riddles, each with a detailed solution.
Break-Even Point: How Many Units to Sell at €20 Each?
Consulting case study on break-even point and contribution margin
A product sells for €20 per unit. Its variable cost is €12 per unit sold, and its fixed costs amount to €4,000 for the period. How many units must be sold to reach the break-even point, that is, to exactly cover all its costs?
Need a hint?
First calculate what each unit sold brings in after paying its variable cost, then see how many units it takes to repay the fixed costs.
Show the answer and the solution
Answer: 500 units, for €10,000 in revenue: the €8 unit margin covers the €4,000 of fixed costs.
The reasoning
The contribution margin per unit is the selling price minus the variable cost: €20 minus €12 = €8. Each unit sold therefore contributes €8 toward covering the fixed costs. The break-even point is reached when the sum of these contributions equals the fixed costs: €4,000 / €8 = 500 units. The corresponding revenue is 500 × €20 = €10,000, and profit is zero at this level. Beyond 500 units, each additional sale earns a profit of €8.
Revenue Up 20%, Costs Up 30%: What Happens to Profit?
Consulting case study on combined changes in revenue and costs
A company has revenue of €1,000 and costs of €800, so a profit of €200. The following year, its revenue rises by 20%, but its costs rise by 30%. What does its profit become, in euros and as a percentage change?
Need a hint?
Recalculate each amount separately, applying its own increase to each, before taking the difference.
Show the answer and the solution
Answer: Profit falls to €160 from €200, a 20% decline, even though revenue increased.
The reasoning
The new revenue is 1,000 × 1.20 = €1,200. The new costs are 800 × 1.30 = €1,040. Profit therefore becomes €1,200 minus €1,040 = €160, instead of €200. The change is (160 - 200) / 200 = -20%. Revenue grows by 20%, but profit falls by 20% because costs rose faster than sales. A rise in revenue does not guarantee a rise in profit, so you must compare the growth rates of revenues and costs.
Bird Between Two Trains Riddle: How Far Does It Fly?
Mental math interview question on relative speed and simplifying a problem
Two trains are 60 km apart and are heading toward each other, each at 30 km/h. A bird leaves the first train at the moment the start is given, flies at 60 km/h toward the second, immediately turns around when it reaches it, and keeps going back and forth between the two trains until they collide. What total distance does the bird fly?
Need a hint?
Do not calculate each back-and-forth trip: ask how long the journey between the two trains takes, then how far the bird covers in that time.
Show the answer and the solution
Answer: 60 km: the trains meet after one hour, and the bird flies for that hour at 60 km/h.
The reasoning
The two trains approach each other at 30 + 30 = 60 km/h, since their speeds add up. Starting 60 km apart, they meet after 60 / 60 = 1 hour. The bird flies continuously for this whole duration at 60 km/h, regardless of how many times it turns around. It therefore covers 60 × 1 = 60 km. There is no need to sum the infinite series of back-and-forth trips, whose successive lengths shrink: knowing the total time is enough.
Rule of 72: How Many Years for Money to Double at 6%?
Mental math question in finance
A sum of money is invested at 6% per year, with compound interest, meaning that interest is added to the capital and itself earns interest. Roughly how many years will it take for this sum to double? You may give a quick estimate, in your head.
Need a hint?
Think of a mental math rule that uses an easy number divided by the interest rate expressed as a percentage.
Show the answer and the solution
Answer: About 12 years, using the rule of 72: 72 divided by 6 gives 12, while the exact value is about 11.9 years.
The reasoning
The rule of 72 gives a quick estimate: the number of years needed to double a sum with compound interest is roughly 72 divided by the annual rate in percent. Here, 72 / 6 = 12, so the sum doubles in about 12 years. The exact calculation uses logarithms: we need a number n such that 1.06 to the power n equals 2, so n = ln 2 / ln 1.06, or about 11.9 years. The estimate is therefore very close to the exact value. The rule remains reliable for rates roughly between 6% and 10%, and loses precision outside this range.
Frequently asked questions
What kinds of riddles does this page offer for consulting?
This page offers four consulting and mental math riddles: a break-even point, a profit that falls while revenue rises, a train and bird problem, and the rule of 72. Each is solved in short steps and rough orders of magnitude.
Do you need a device to solve these riddles?
No, they are solved in your head: round the numbers, split the operations and check the result against an order of magnitude. The rule of 72, for example, only needs a single division.
How can you get better at mental math for these riddles?
Redo each riddle without paper, rounding the numbers, then check the result against an order of magnitude. Writing down the steps of a unit margin or a break-even point helps you recall the method when you need to answer.
What next?
Reasoning improves like memory: by practising often. Next, test yourself on our general knowledge quizzes.