Index numbers, the CPI and rates

The price of a bus ticket was $2,50\ \text{ €}$ in 2023 and $2,65\ \text{ €}$ in 2024.

1. a) Calculate the price index number in 2024 with base year 2023.

1. b) What was the percentage change in price?

The table shows the price of a video game across three years: 2022: $60\ \text{ €}$; 2023: $66\ \text{ €}$; 2024: $69\ \text{ €}$.

1. a) Construct the series of index numbers with base year 2022.

1. b) In which year did the price increase the most, as a percentage, compared with the previous year?

The series of index numbers for the price of urban transport, with base year 2018, is: 2018: $100$; 2019: $103,2$; 2020: $107,5$; 2021: $112,1$.

1. a) Construct the same series using 2021 as the base year (index $100$ in 2021), rounding to one decimal place.

1. b) What information is preserved and what changes when changing the base year?

A supermarket chain has a series of index numbers for its sales with base year 2020 and another with base year 2023, but is missing the link between them. The known data are:

• Base 2020: 2021 index $= 104$, 2022 index $= 109$, 2023 index $= 115$.

• Base 2023: 2024 index $= 106$, 2025 index $= 110$.

1. a) Calculate the 2024 and 2025 indices with base 2020 (that is, link the two series).

1. b) What was the total percentage change in sales between 2020 and 2025?

1. c) And what was the average annual change in that period? Express it as a percentage rounded to one decimal place (use the root of the overall index).

The annual CPI for three consecutive years was $3,5 \%$, $2,8 \%$ and $4,1 \%$.

1. a) Calculate the chained index of variation for the entire three-year period and the total cumulative rise, rounded to two decimal places.

1. b) If a litre of milk cost $0,95\ \text{ €}$ at the start of the first year and its price has increased by exactly the CPI, how much does it cost at the end of the third year? Round to the nearest cent.

The CPI of a country is calculated as a weighted average of the indices of several expenditure groups. In a given year, the data are:

• Food and drink: weighting $25 \%$, rise of $9 \%$.

• Housing and utilities: weighting $30 \%$, rise of $4 \%$.

• Transport: weighting $15 \%$, rise of $6 \%$.

• Other groups: weighting $30 \%$, rise of $2 \%$.

1. a) Calculate the overall CPI for the year, rounded to two decimal places.

1. b) A worker whose salary rose by $3,5 \%$ that year claims that they have «lost purchasing power». Are they right? Justify your answer by calculating the real index of variation of their salary.

1. c) Another worker spends half of their salary on housing and the other half equally among the remaining items. Explain why their «personal inflation» may be different from the official CPI.

In a country of $2,1$ million inhabitants, the birth rate is $7,1$ births per $1.000$ inhabitants.

1. a) Approximately how many births were recorded that year?

1. b) If the death rate was $9,3$ per $1.000$ inhabitants, is the population growing or shrinking due to natural causes? Quantify it.

The table shows approximate data from the Labour Force Survey for two regions in a quarter:

• Region A: active population $2.400.000$ people, unemployed $312.000$, employed $2.088.000$.

• Region B: active population $950.000$ people, unemployed $123.500$, employed $826.500$.

1. a) Calculate the unemployment rate for each region, rounded to two decimal places.

1. b) In region A, the population aged $16$ or over is $4.300.000$ people. Calculate its participation rate, rounded to two decimal places.

1. c) A headline reads: «Region B has less unemployment because it has fewer unemployed people». Critique the headline using the calculated rates and explain why rates allow us to compare populations of different sizes.