McKinsey PST 07: Rule of 72

This is the seventh theory session of the Problem Solving Test Course. I show a way of calculating how much time it would take for a quantity to double, triple, quadruple and so on. Then I explain how you can quickly derive similar rules on your own and apply this theory to a few PST questions

[TRANSCRIPT]

This is the seventh theory session of the problem-solving test course. We’ll continue to talk about compound growth estimation techniques.

This time we will focus on the Rule of 72 and also, we’ll focus on the rule of 42, 96, 114, 132, and other rules. We will learn how to derive those rules as well as how to apply them to McKinsey problem-solving test questions.

Let us start with the following example.

As always you might want to pause the video now and try to solve the question on your own and then check. Which of the following would be the minimum required annual growth in the revenue of Shadow Inc., that would see represent more than half of ABZ Industries revenue in ten years.

And the text tells us that the revenue ABZ Industries is eleven points sixty-four billion dollars and the revenue of the Shadow Inc., its subsidiary is two points eighty-three billion dollars and the growth of ABZ industries is 5 percent per annum.

So, here’s our problem, and we have two ways of solving it: the straightforward way and a roundabout.

First of all, we would find the revenue of ABZ, perhaps using the contribution margin. Then we would calculate the revenue or Shadow as the half of the revenue ABZ.

Then we would need to calculate the growth rate of Shadow, that will help it reach the desired revenue.

But the problem is that in this case beside the long computations we are also need to compute these guys here, which might be a nightmare in a PST, so I wouldn’t really advise you to take this approach.

But here’s another way of tackling the question.

It’s called Rule of 72.

We know that the revenue of Shadow now is about 25 percent of the revenue of an ABZ, right? And we know that this share has to double to 50 percent.

So, it has to double in 10 years.

That means that we can use the rule of 72, which tells us that X minus five times 10 equals 72. Or that X equals twelve points two.

And now our answer is B. Well I guess at this point it sounds a bit like magic but I will explain everything in a moment.

Here’s what the rule of 72 says: if some quantity grows by X percent for n periods and doubles then X approximately equals 72 provided that X is not greater than 20.

Here are a few examples how we can use it. In all the cases we noticed that the number almost doubles or doubles, when grows at a certain percent for n years.

That means that the percent’s times n equals approximately seventy-two and we can easily find n – the number of years. For some math nerds and most consultants are math nerds anyways, here’s a proof of this result.

This proof suggests that the rule of 72 is not the only one of its kind and indeed there are similar rules as well.

For example, the rule of 42 the rule of 72,

the rule of 96, 114, 132, 144, 156, 168.

And so on.

And here’s how we can build similar rules on your own. For example, start with a rule of 42 multiplication by one point five.

Combine it with the rule of 72, multiplication by two. In combining we sum up the anchor numbers which is forty-two and seventy-two and take a product of the multipliers – two and one point five, and we arrive at the rule of hundred fourteen – multiplication by three.

We’ve got three from multiplying two by one point five and 114 from summing up forty-two and seventy-two.

Here’s another example.

Take the rule of hundred thirty-two,

Add the rule of 72, combine and get the rule of 204 multiplying by 7.

Well now it’s high time to test this knowledge connection right.

Again, pause the video and try to tackle the question on your own and then, check yourself.

The cost to attending Harvard College is 75 k per year and is growing at six-point six percent per annum. At typical undergraduate degree at Harvard comprises four years of education.

If this trend continues, what is the minimum number of years which need to pass, before the cost of a Harvard undergraduate degree climbs over 400k?

So, if one year of college education right now costs 75 k, then four years cost 300k and the maths question that we face is the following: 300k times one plus six-point six percent to the power n

When N is unknown equals 400k?

Okay. Let us simplify it a bit, and we see that this is a great opportunity for the Rule of Four Thirds.

Let’s make it up. The rule of multiplication by 4 the anchor number is 144.

Rule of multiplication by 3.

The anchor numbers 114.

And then our, anchor number here is the difference of these numbers and is number 30.

So, our newly arrived rule of 30 tells us that six-point six n should roughly equal 30 and that n is about 4.5.

And then our answer is C. There is one more similar rule we need to learn: the rule of 72 for shares.

Start with the rule for absolute values and arrive at a similar one for shares. In simple words,

if we want to grow the share of a certain object in its base, we must subtract the growth of the base from its growth and apply the rule of 72, as always. The intuition is simple: if we are running at six miles per hour and the dog is running behind us at 10 miles per hour the dog is catching up with us at ten minus six equals four miles per hour.

And again, for those who is interested, here is a proof of this result. Here’s another example this time from an official practice test. Paused the video and try to solve it. If the total Innovation Capital in Exhibit 1 were to grow by 5 percent per year in the future, which of the following would be the minimum required annual growth in human capital, that would see it represent more than half of the total innovation capital in 10 years?

So, from reading the text we learned about human capital is and learn we will need to sum up three numbers.

1.01, 0.7 and 1.5, so total human capital right now is three points three.

So, we know that right now human capital is about 25 percent of total innovation capital and this number needs to double in 10 years.

Therefore, by rule of 72 for shares we know that X minus five times ten roughly equals 72. X minus 5 roughly equals 7.2 and X is about 12.2.

And here are answers B.

Let us think to size what we learned today.

The rule of 72 says: if some quantity grows by X percent for end periods and doubles, then Xn then approximately equals 72 provided that X is not greater than 20.

There are all kinds of similar rules.

Rule 42, 72, 96, 114, 132, 144, 156, 168, and so on. The way you derive them: is by summing the anchor numbers and by taking a product of the multipliers. A similar rule for share says: if you want to grow the share of a certain object in its base we must subtract the growth of the base from its growth and apply the rule of 72 as always.

That’s it for now.

Any questions or comments please leave them below the video and I’ll be happy to reply.

Thanks for watching the video.

And then to next time.