Percentage Calculations

Starting from the fifth grade, you have solved many applied problems involving percentages. You are familiar with the following types of percentage problems:

finding a percentage of a number;
finding a number given its percentage;
finding the percentage ratio of two numbers.

Three Basic Types of Percentage Problems

You know how to construct mathematical models of these problems using the following expressions:

📐Definition — Basic Percentage Formulas

$\dfrac{a \cdot p}{100}$ — finding $p\,\%$ of a number $a$ ;
$\dfrac{a \cdot 100}{p}$ — finding a number whose $p\,\%$ equals $a$ ;
$\dfrac{a}{b} \cdot 100\,\%$ — finding the percentage ratio of number $a$ to number $b$ .

Compound Interest Formula

Let us consider an applied problem that bank employees, as well as anyone who keeps money in a bank at interest, frequently encounter.

✎Example — Bank Deposit

Problem statement. A depositor placed 100,000 UAH in a bank at 10% per annum. What amount will be in their account after 7 years, assuming the depositor does not withdraw any money during this period?

Solution.

Let $a_0$ be the depositor’s initial capital, i.e., $a_0 = 100{,}000$ UAH.

Let $a_1, a_2, \ldots, a_7$ denote the amount of money in the account at the end of the first, second, …, seventh year respectively.

At the end of the first year, the initial capital $a_0$ increased by 10%. Thus $a_1$ is 110% of the initial capital $a_0$ :

$a_1 = a_0 \cdot 1.1 = 100{,}000 \cdot 1.1 = 110{,}000 \text{ (UAH)}$

At the end of the second year, $a_1$ in turn increased by 10%. Thus $a_2$ is 110% of $a_1$ :

$a_2 = a_1 \cdot 1.1 = a_0 \cdot 1.1^2 = 110{,}000 \cdot 1.1 = 121{,}000 \text{ (UAH)}$

At the end of the third year, $a_2$ increased by 10%. Thus $a_3$ is 110% of $a_2$ :

$a_3 = a_2 \cdot 1.1 = a_0 \cdot 1.1^3 = 100{,}000 \cdot 1.1^3 = 133{,}100 \text{ (UAH)}$

It now becomes clear that:

$a_7 = a_0 \cdot 1.1^7 = 100{,}000 \cdot 1.1^7 = 194{,}871.71 \text{ (UAH)}$

Answer: 194,871.71 UAH.

The problem is solved analogously in general form, when an initial capital of $a_0$ is placed in a bank at $p\,\%$ per annum.

Indeed, at the end of the first year the initial capital increases by $\dfrac{a_0 \cdot p}{100}$ and equals:

$a_1 = a_0 + \frac{a_0 \cdot p}{100} = a_0\left(1 + \frac{p}{100}\right)$

that is, it increases by a factor of $\left(1 + \dfrac{p}{100}\right)$ .

It is clear that at the end of the second year the sum again grows by the same factor and equals:

$a_2 = a_1\left(1 + \frac{p}{100}\right) = a_0\left(1 + \frac{p}{100}\right)^2$

Therefore, at the end of the $n$ -th year:

⚡Formula — Compound Interest Formula

$a_n = a_0\left(1 + \frac{p}{100}\right)^n$

where $a_0$ is the initial capital, $p$ is the interest rate, $n$ is the number of years, and $a_n$ is the amount after $n$ years.

This formula is called the compound interest formula.

Avoiding Ambiguity in Percentage Problems

Problems involving changes in percentage rates can cause certain complications. A percentage rate is a quantity just like any other variable: speed, distance, price, etc. The only difference is that this quantity is itself expressed as a percentage. Therefore, when discussing changes in this quantity, ambiguous interpretations may arise.

Compare:

Change in price $x$	Change in percentage rate $x$	Mathematical model
Price increased by 10 UAH	Percentage rate increased by 10%	$x + 10$
Price increased by 10%	Percentage rate increased by 10%	$1.1x$

We see that for a percentage rate, the verbal description for different mathematical models turns out to be identical.

📐Definition — Percentage Points

To avoid this ambiguity, in economics and other fields where percentage calculations are widely used, the concept of percentage points is employed.

For example: in the ninth-grade classes there are 100 students, of whom 20% at the beginning of the school year were honour students.

If we say that by the end of the year the number of honour students increased by 5%, this means the number of honour students (expressed as a count of people) increased by 5% of that quantity. The number of honour students in this example was 20 people; when this number grew by 5%, it became 21 people.
If we want to say that the indicator “20%” increased and now equals “25%”, we need to use the term “percentage points”: “by the end of the year, the number of honour students increased by 5 percentage points.” With this formulation, the number of honour students at the end of the year would be 25 people.

Percentage points are often abbreviated as “pp”.

For comparing percentage rates where the difference is very small, another conventional unit has been introduced: the basis point, which equals 0.01 of a percentage point (in other words, one percentage point contains 100 basis points). It is denoted “bp” or “bps”.

Exercises

✏Exercise 21.1

The price of a product was increased by 25%. By what percentage must it now be reduced to return to the original price?

✏Exercise 21.2

A depositor placed 5000 UAH in a bank at 8% per annum. How much money will be in the account after three years?

✏Exercise 21.3

Four years ago, a factory produced 10,000 units of a certain product per year. Thanks to modernisation and increased labour productivity, annual production growth of 20% was achieved. How many units of the product will be manufactured this year?

✏Exercise 21.8

There were 300 g of a 6% salt solution. After some time, 50 g of water evaporated. What is the new percentage concentration of salt in the solution?