Solving for price incease

Question

The following table lists the basket of goods in the Consumer Price Index for the nation of Astro​ (Assume 2014 is the base​ year.)

  Table 6.2  
             2014     2014  2015     2015  2016     2016
             quantity price quantity price quantity price
  Sunglasses  8        $9      10      $11   7       $10
  Toothpaste  15        3      18       4    15       4
  Ferrets     6         12     4        15   9        18

Using the information in Table​ 6.2, the percent increase in prices over the two year period from 2014 to 2016 is approximately

The correct answer is 31% but I kept on getting 35%. My solution was (CPI of 2016 - 100)/100

Answer 1

The Consumer price index of a country is defined as:

$$CPI_{t}^{t_0}={{P_{1t} Q_{1t_0}+P_{2t}Q_{2t_0}+P_{3t}Q_{3t_0}}\over{P_{1t_0}}Q_{1t_0}+P_{2t_0}Q_{2t_0}+P_{3t_0}Q_{3t_0}} \times 100$$

where $P$ is price, $Q$ is quantity, $t$ is reference to current year and $t_0$ is reference to the base year.

So using this formula we get:

$${{10\times8 +4 \times 15+18 \times 6}\over{9 \times 8+3 \times 15 +12 \times 6}} \times 100 \approx 131.2 $$

Now finding the inflation rate we can see (CPI of the base year is always going to be $100$)

$${{131.2-100}\over{100}} \times 100 \% \approx 31\%$$

Hope this is helpful!

Answer 2

Presumably if you are to assume 2014 is the base year, you should weight prices in 2016 and 2014 by 2014 quantities, giving

$$\dfrac{10\times 8+4\times 15+18\times 6}{9\times 8+3\times 15+12\times 6} \approx 1.312$$

i.e. a percentage increase of about $31\%$