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Chart-based Math Drills

Chart-based Math Drills

This is a sample of a practice chart-based math drill. You may go back to the main Consulting Math article for more in-depth information on improving your quantitative capabilites for the McKinsey PST and Case Interviews!
Population and Income in Canada

Based on a McKinsey & Company research

A McKinsey & Company research on the trend of urban population and income by different city types in Canada was conducted in 2010. Both the extracted and projected data are shown in Exhibit 1:

Exhibit 1
Population and Income data by different city types, 2000 – 2010, 2030 projected

Exhibit on Population and Income data by city type

*Note: Information provided for each city type are from 2010

#1: It is projected that between 2010 and 2030, the total national urban population will increase by 5% and advanced cities will contribute 21% of the total urban population in 2030.

What portion (in %) of the increase in urban population in advanced cities from 2010 to 2030 is caused by the overall national urban population growth (as opposed to the gaining population of other city types) ?

#2: What is the appropriate ranking (from high to low) of the urban population density of each type of city? (Population density of each type of city is represented by the average urban population of each city).

#3: Given that GDP of Canada in 2010 was approximately $1,500 billion, 85% of which was urban GDP. Calculate the total annual income from the “poor” category of this country in 2010.

Click to see answer keys

#1: 10.64%

The total urban population in 2010 is Y; then we have the following information: Total Urban population in 2030: 1.05 x Y. Advanced cities population in 2030: 0.21 x 1.05Y Advanced cities population in 2010: 0.15 x Y The increase of population in advanced cities from 2010 to 2030: 0.21 x 1.05Y – 0.15Y = 0.0705Y Portion (in %) of the increase in urban population in advanced cities from 2010 to 2030 caused by national urban population growth: 0.15 x 0.05X / 0.0705X = 10.64%

#2: Advanced cities – Developing cities – Emerging cities – Lagging cities

If X is the total urban population of Canada, thus 0.15X is the urban population in 13 advanced cities as the average urban population in an advanced city is 0.012X 0.26X is the urban population in 80 developing cities as the average urban population in a developing city is 0.003X 0.45X is the urban population in 369 emerging cities as the average urban population in an emerging city is 0.0012X, and 0.14X is the urban population in 188 lagging cities as the average urban population in a lagging city is 0.0007X

#3: $121.25 billion

Total urban GDP in 2010: 1500B x 85% = $1,275 billion Total annual income from the “poor” category in 2010: (1,275 * 0.27 * 0.03) + (1,275 * 0.35 * 0.06) + (1,275 * 0.33 * 0.15) + (1,275 * 0.05 * 0.33) = $121.25 billion

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  • Anh Thai

    Hello,

    I find the answer for question 1 is not accurate. I think the answer should be 7.05% which is the difference in percentage of population between 2010 and 2030 in Advanced Cities that caused by overall growth population. What is X in the answer? And what does 0.15*0.05 represent for? I can’t find any relationship between 15% and 5% here

    • Cameron Duncan

      The TOTAL national urban population grew by 5% from 2010-2030. However, as we can see from the chart, only 15% of the national urban population lived in advanced cities in 2010 – this is important because raw population growth in this case would be in terms of advanced cities only. Thus, the change in advanced population from 2010-2030 based solely on the overall population growth is the % of national urban population in 2010 (because we assuming no cannibalization) multiplied by the growth. Thus, 15%*5%. I believe the confusing part is the author forgot to include a parenthesis in the equation. It should be this:

      = (0.15Y*0.05)/(0.21*1.05Y-0.15Y)
      = 0.0075Y/0.0705Y
      = 0.1064

  • Phil 1986

    Hi, I have to agree with Anh Thai. I and all student fellows I have been discussing this issues with are not capable of reproducing this calculation. Anyhow, we think that the X in your final step of question 1 has to be replaced with Y. Is that correct?

  • Cameron Duncan

    Would it not be much easier in number 3 to remove the common factor of 1,275 from the equation and multiply at the end like so?

    1,275[0.27(0.03)+0.35(0.06)+0.33(0.15)+0.05(0.33)]

    Effectively removing 4 calculations from the problem ;).

    Thanks for all of the great, challenging content Kim!