This is the third theory session of the Problem-Solving Test Course. I introduce fact questions, explain an algorithm to solve them, showcase a few quick solutions and complete solution techniques, and put them into practice.
This is the third theory session of the problem-solving test course. And it is devoted to fact questions. I will explain what actions fall into this category.
After that we’ll discuss an algorithm to solve Fact questions as quickly as possible but not too quickly to avoid some unnecessary and stupid mistakes. And after that we’ll look at several examples.
Recall that there are four major questions types: premise, fact, conclusion and the last 13 percent.
Today we’re talking about the second type: fact. Fact questions fall into four subtypes.
One: estimation. Simple calculations which do not require a precision.
Two: precise computation. Simple, yet time consuming calculations or calculations requiring high precision.
Three: word problem. Building equation or formula based on the data in the text and exhibits than solve or computed.
Four: ranking. Rank several options by their values.
Recall the differences among question types. A premise doesn’t have to be true or based on the data in the text or exhibits, it might be a completely new piece of information.
However, if assume it’s truth it allows us to make certain conclusions about a fact stated in the text. In comparison a fact is true based on the data. You just need to interpret it.
And the conclusion has to follow from the data. Let’s look at an example of fact question. What is the average productivity of all workers in Table 1.
First of all, what is productivity? Productivity is output divided by input, right? Output is items made, while input is hours spent? Now we need average productivity. In PST that means we need to sum up all outputs and divide by all inputs. Worker by worker division is not necessary. It would take just too much time. Then the answer is just a hundred over a hundred thirty.
Time to do some calculations, right? Oh, wait a sec. Can we approximate here? The answer options are far apart. 100 over 130 is definitely less than 1. That means Option C and D are out. But similarly, 100 over 130 is definitely more than half. So, A’s out. So, our answer is B and we do not need to be any more precise than that.
The algorithm we will use for solving questions of fact type has four steps.
Understand the question.
Gets the data.
Try to find a quick solution and go for a complete solution if absolutely necessary.
At the first step we need to understand what question we have to answer.
The tricks of most questions lie in details rather than computational complexity.
So, pay attention to the details, subjects and their actions, numbers, references and directions.
In the meanwhile, notice the range of answer options. Is it wide or narrow? If it is wide, most likely you can solve the question easily by quick approximations.
If it is narrow, however, you will need to invest more time in the question. This knowledge matters for time management, especially at the end of the test.
At the second step of the algorithm, we need to collect all the data necessary for solving the question. The most important part of exhibits are: title, exhibit type, legend, units of measurement and references to other exhibits.
The text usually provides useful information on the variables, including their definitions and sometimes values.
As an example, let us read an exhibit from an official McKinsey PST. Here it it is.
First thing we notice is the title of the exhibit. Housing starts by quarter.
Then the legend explains that three different lines are three possible scenarios for the dynamics of housing starts. Then we check the axes. Horizontal axis depicts the timeline in years while the vertical axis shows the numbers of housing starts. At this point you might wonder what the housing start is, so we go back to the text and search for a definition of the term.
Additionally, we discover that vertical lines are first quarters.
This is clear from the exhibit itself, but our understanding is now confirmed. Once we’ve analyzed the data we’re ready for the third step, search for a quick solution.
The two major pieces of advice here are estimate and eliminate. Estimate means be lazy about computations.
Figure out what needs to be calculated and searched for approximations and roundabout. Round numbers as much as possible but not too much to avoid mixing of options.
Eliminate means test options and get rid of obviously the wrong ones, those which are too large or too small. Drawing values, for example, averages on exhibits, might also help eliminate some options.
Let’s have a look at an example.
In this question we need to increase the share of left handed employees at G&P, the first column, so that the corresponding absolute value reaches that for Rams, the second column. Notice that the options are quite far apart which is great for us. We can approximate. To find the percentage increase we need to calculate the number of left handed employees at each company,
take the difference and translate it into percentage points. The number of left handed employees at G&P is 9k times 30 percent equals 2.7k. The number of left handed employees at Rams’ is 12k times 60 percent, equals 7.2k.
At this point, we could dive into calculation of 7.2k minus 2.7k over 9k.
This solution is correct but can we find a quick alternative?
Yes, if we use elimination.
Notice that we need to more than double the number of left handed employees at G&P to reach the level of Rams. That means the difference will be more than 30 percentage points, because the current share of left handed employees at G&P is exactly 30 percent.
So, options A and B are wrong. There are just too small, but the option D is too large: 30 percent plus 70 percent equals hundred percent, so in this option all of G&P 9k employees have to be left handed which is an overkill, so option D is wrong too.
So, the only possible option here is C. In this way we found a quick solution and avoided straightforward calculations due to illuminating answer options.
Another opportunity to practice quick solutions is this ranking exercise.
We need to order five companies according to their profits revenue times profit margin from largest to smallest.
Again, you might want to pause the video here to try and solve the question on your own.
This straightforward approach is to calculate all profits, then order them from largest to smallest, then look at the answer options and see which fits best.
A quick approach is to look at the options, understand which pairwise comparisons we need, do them and eliminate all but one option.
Notice that options A and D rank company 2 above company 1 while options B and C do the opposite. Both cannot be correct if we compare one to two and learn which is bigger we can get rid of two options. 15 percent of almost 400000 is more than 10 percent of less than 600000.
So, profit of company 2 is larger than profit of company one and options B and C are out. What we need to compare next is options A and D.
Similarly compare profit of companies 3 to profit of company 4 and find that the former is greater than the later.
So, D is out, and the answer is A. In some cases, unfortunately a quick solution won’t do or just won’t be enough.
Then we’ll will have to take the fourth step. Complete solution.
Instead of jumping in the numbers randomly we need to figure out our strategy first and only then execute it.
Start with a generic formula or a question like: revenues equals costs plus profit.
Then adds details replacing, for example, revenues with price times quantity and cost with the same variable fixed cost.
At this point who might want to work with letters P, Q, R and so on, rather than numbers. Observe which point is unknown from the data and if needed,
select the most convenient variable for the unknown. You get a pretty detailed formula or equation.
Now it’s high time for execution. Insert numbers, simplify and solve for the unknown.
Finally, the transition from the unknown value to the value asking the question is critical.
We might have chosen a monthly revenue as x, but the question asks for annual revenues, then the answer is 12 x rather than just x.
Most questions try to catch us off guard this trick, so be careful. To test the approach, let us take an example similar to what we’ve sold before: the Rams and G&P story.
However, this time the numbers are a bit more complicated,
while the answer options are too similar for quick approximations. We have to be more careful now.
You might want to pause the video here to try and solve the question on your own.
Let us figure out the strategy first. Just like before, we need to compute the number of left handed employees a G&P, the number of left handed employees at Rams, take the difference and divided by the number of employees at G&P. In the question before we did not pay much attention to it. The answer is this time the calculations will be more time consuming and we cannot afford to go in the wrong direction.
Now execute computations.
Notice that even in situation of seemingly precise computations, there might be some room for approximations, but these approximations have to be more precise than in the estimation case. This is how I would do the calculations. If this technique is not familiar to you, I suggest you read the book on mental math. Our consulting maths course book covers this topic extensively too.
So, the answer is B because A is too small while C and D are too large.
And here’s a word problem example.
Find the size of the combined air fleet of 2 hypothetic states.
The problem might sound daunting at first but it becomes much easier to solve, once we tackle it step by step.
Let’s find a generic equation first. What’s remains unchanged in the setting of the question?
The number of helicopters expressed separately and simultaneously for the two states.
Insert the data from the problem to make the question more specific. Helicopters make up 95 percent of the fleet of North Ikota.
86 percent of the fleet of South Ikota and 100 in minus 10 equals 90 percent of the combined fleet.
The combined fleet can be expressed by the fleet of just one state. Let’s write it down.
The most convenient unkonown here would be a 9: the size of the air fields of North Dakota.
Notice that this variable is not what is being asked in the question, but it will help us find the answer. Okay, now it’s time to solve this equation.
Simplify, solve and find the inknown, and then, find the desired value: the size of the combined fleet. The answer is D. Well that was too quick of course but not rocket science, either right?
Let us synthesize what we have learned today.
A fact is just what we get from the data. It is true by definition, we just need to get the right data and we can mix it in the right way to establish this fact.
The way we do this tricking mixing is via four steps.
One: understand the question and find key words.
Two: get the data from exhibits and text.
Three: try to solve quickly through estimation and elimination.
And four: if necessary, build a complete solution defining your strategy and carefully executing it.
That’s all for today.
If you have any questions as always feel free to leave comments below.
Thanks for watching and see you next time.