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How many aeroplanes are in the sky at this very moment? How much does a Boeing 747 weigh? And how many ping pong balls could you fit inside the same plane?

They may sound like cruel questions designed to catch you off-guard. But brainteasers or guesstimates, the seemingly random and ridiculous questions you’re asked during the interview process, boil down to (relatively) simple logic-based problem solving tasks – as long as you keep your cool.

Consulting is all about solving problems and developing strategies to improve efficiency and performance in businesses or functions. Many consulting firms also use business-related case studies to test candidates’ ability to think commercially, but brainteasers isolate your core critical thinking abilities from your knowledge of the industry you’re hoping to join. This means interviewers can assess how you respond to unfamiliar situations and your raw ability to succeed in the job. Nevertheless, when you begin to work them out, you’ll probably find that the problems have more in common with the line of work you want to go into than you first thought.

Solving brainteasers isn’t about knowing the answer, or working out the correct answer. PA Consulting Group graduate recruiter Gemma Guy says: "If you come out with the answer and it’s right, it’s not going to get you any points at all. For us it’s about seeing how you logically work something out.

"The estimation you make can be wrong, as long as it’s not hugely off key – if it is, we’d question your common sense. As long as you’ve gone though logical steps and explain your reason, you should get something that’s plausible," she adds.

So how can you prepare? "There aren’t any specific facts you need to know to answer a brainteaser," says Gemma. But lots of common questions are based on market-sizing problems, so it’s a good idea to know the UK’s population (62.6 million, which can be rounded off to 60 million), life expectancy (78 for men and 82 for women, 80 years average) and household size (three people, so there are around 20 million in the UK).

- The UK population is approximately
**60 million**. - Average life expectancy is
**80 years**, so assuming a flat demography, there will be 750,000 people in each year of age between 0 and 80 years (60 million divided by 80).

This means that approximately **750,000 babies are born each year**.

- The responsibility of midwives goes far beyond just delivering babies. Midwives’ time is spent booking appointments, conducting scans, providing antenatal care, care during labour and delivery, and postnatal care of both mother and baby.

- Let’s assume that for an average birth midwives’ work can be broken down as follows: Initial appointment -
**1 hour;**Scans -**2 hours;**Antenatal care -**9 hours;**Care during labour -**10 hours;**Delivery -**5 hours;**Postnatal care -**9 hours; Total 36 hours.** - If a full-time midwife works for 8 hours per day, 5 days per week, for 45 weeks per year (this accounts for holiday and time off) – that’s
**1,800 working hours per year**. - We can therefore assume that up to
**50 babies per year**(1,800 divided by 36) can be delivered per full-time midwife.

- 750,000 babies per year divided by 50 births per midwife gives us
**15,000 full-time midwives**. - But of course not all midwives work full-time. It’s safe to assume that some work part-time or combine midwifery with other healthcare services. To keep the maths simple, let’s say that two-fifths of midwives work at half of full-time capacity. This revises our births-per-midwife down to
**40 per year.** - Our final estimate is therefore 750,000 divided by 40, giving
**18,750 midwives in the UK**.

Here are three more for you to practise at yourself. We recommend attempting them using just a pen and paper, and then turning this page around to find out how your approach differed from our solutions.

Assume there are 30 pods, holding 25 people each. That means 750 people can ride the London Eye at one time. One rotation lasts 30 minutes, so it can carry 1,500 people per hour. If the attraction is open from 10am to 7pm, the capacity is 13,500 people per day. Let’s say it operates at two-thirds of its capacity on an average day, so **9,000 people** go on the London Eye every day.

Assume one toaster per household. The average British household consists of 3 people. Given that the UK population is 60 million, there are 20 million households. Assuming a toaster needs to be replaced every four years, one quarter of all households are going to buy a toaster every year. This puts UK toaster sales at **5 million** per year.

Assume that the diameter of a ping pong ball is 4cm. Because the balls do not tessellate, each ball occupies a volume of 4cm cubed. One cubic meter will therefore hold 25 by 25 by 25 balls, which is approximately 16,000. To calculate the volume of the aeroplane’s fuselage we’re going to approximate it as a cylinder. There are ten seats, each 50cm wide, and two aisles, also 50cm wide, in a row. This gives our cylinder a diameter of 6m (0.5m multiplied by 12), and a cross-sectional area of 27m^{2} (3 squared is 9, multiplied by pi, which we’ll round to 3). Let’s suppose there are 50 rows, covering an average distance of 1.4m (we’ve over-estimated to account for toilets and kitchens). This gives a length of 70m. We can now calculate the volume of the fuselage to be approximately 1,900m^{3} (27m^{2} multiplied by 70m). This can be filled with **30 million** ping pong balls (1,900m^{3} multiplied by 16,000 balls).

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