## Chart-based math drills

This is a sample of a practice chart-based math drills. You might want to go back to the Consulting Math overview article for more in-depth information on improving your quantitative capabilities 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 below.

Exhibit 1

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

*Note: The information provided for each city type is 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.

#1: 10.64%

Assuming that the total urban population in 2010 is Y, we have the following calculation:

Total Urban population in 2030: 1.05 * Y

Advanced cities population in 2030: 0.21 * 1.05Y

Advanced cities population in 2010: 0.15 * Y

The increase of population in advanced cities from 2010 to 2030: 0.21 * 1.05Y – 0.15Y = 0.0705Y

The increase in urban population in advance cities from 2010 to 2030 is 5% (overall urban population increase) of 15%Y (the portion of urban population in advance cities): 0.05 * 0.15Y

Portion (in %) of the increase in urban population in advanced cities from 2010 to 2030 caused by national urban population growth: 0.15 * 0.05Y / 0.0705Y = 10.64%

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

Assume X is the total urban population of Canada.

The urban population in 13 advanced cities 0.15X.

The average urban population in an advanced city: 0.15X/13 = 0.012X

The urban population in 80 developing cities 0.26X

The the average urban population in a developing city: 0.26/80 = 0.003X

The urban population in 369 emerging cities is 0.45X

The average urban population in an emerging city: 0.45X/369= 0.0012X,

The urban population in 188 lagging cities is 0.14X

The average urban population in a lagging city: 0.14X/188 = 0.0007X

#3: \$121.25 billion

The total urban GDP in 2010: 1500B x 85% = \$1,275 billion

The 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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