Applied Math - Percentages - Inflation

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Introduction

Inflation measures how much your dollar could buy today compared to how much your dollar could buy in the past. Maybe your dollar today can buy you two bananas. A dollar in the past could by three bananas. Further in the past, a dollar could buy four bananas. Inflation has been positive over time, meaning our one dollar can buy less and less things.

Wages have also increased over that time. In theory, wages should always increase with inflation so that employees can have the same purchasing power, but that is not always the case. An employee should receive a wage equal to the inflation rate each year (or a raise every two years equal to the combined inflation for two years, etc.), or they are effectively receiving a pay decrease.

The federal government monitors inflation by calculating the computer price index. Federal government workers call grocery stores, restaurants, clothing stores, etc. and ask for their current prices of items, and compare those prices to previous year prices.

Example:

1.

The cost of a particular box of cereal was $2.00 in the year 2000. The cost of the same box of cereal in 2023 was $3.50. What was the percentage increase of this box of cereal?

To calculate a percentage, divide the amount you are interested in by the starting amount (or total), then multiply by 100.

To calculate the percent increase, we divide the amount of the increase by the starting price, then multiply by 100.

\[\frac{\$1.50}{\$2.00} \cdot 100 = 75\%\]

The price of this cereal increased by 75% over the 23 year period.

2.

Assume an apple costs $1 in 2022. An employee makes $40,000 a year. Assume the price of the apple rose to $1.10 in 2023 due to inflation. If the employee used every dollar they had to buy apples, how many less apples can they buy in 2023 than in 2022?

In 2022, they could buy

\[\$40,000 \cdot \frac{1 \text{ apple}}{\$1} = \cancel{\$}40,000 \cdot \frac{1 \text{ apple}}{\cancel{\$}1} = 40,000 \text{ apples}\]

In 2023, they could buy

\[\$40,000 \cdot \frac{1 \text{ apple}}{\$1.10} = \cancel{\$}40,000 \cdot \frac{1 \text{ apple}}{\cancel{\$}1.10} \approx 36,364 \text{ apples}\]

This person can buy \(40,000 - 36,364 =3,636\) less apples with their salary due to inflation.

3.

A person makes $40,000 a year currently. If the yearly inflation rate for the year is 5%, what salary should they have next year so that they have the same purchasing power?

We need to increase the person’s salary by 5%. We could calculate their raise and then add it to their salary,

\[\$40,000 \cdot 0.05 = \$2,000\]

\[\$40,000 + \$2,000 = \$42,000\]

but a quicker way would be to multiply their previous salary by \(1.05\) (1 added to the decimal form of the percentage).

\[\$ 40,000 \cdot 1.05 = \$42,000\]

Practice Problems

1. The cost of a particular candy bar was $1.00 in the year 2000. The cost of the same box of cereal in 2023 was $2.25. What was the percentage increase of this candy bar?

2. Assume a box of pasta costs $1 in 2019. An employee makes $50,000 a year. Assume the price of the apple rose to $1.25 in 2023 due to inflation. If the employee used every dollar they had to buy boxes of pasta, how many less boxes of pasta can they buy in 2023 than in 2019?

3. A person makes $50,000 a year currently. If the yearly inflation rate for the year is 8%, what salary should they have next year so that they have the same purchasing power?

Theory Questions

1. If a person holds on to a large amount of cash, the cash will have less spending power every year (assuming inflation is positive). What can a person do to keep their purchasing power with this cash?

2. What is the relationship between the spending power of the dollar and time - is it linear, exponential, logarithmic, etc.?