Percentages

Percentage refers to expressing a quantity as a fraction with a denominator of 100.  $$ p \% = \frac{p}{100} $$

In mathematical terms, the % symbol represents dividing a number by 100.

For instance, writing 50% means representing the fraction \( \frac{50}{100} \), which equals one-half.

$$ 50 \% = \frac{50}{100} = \frac{1}{2} $$

Percentages are essentially fractions that allow for quick and easy comparisons of proportions or parts of different quantities.

Here are a few more examples:

• 75% equals \( \frac{75}{100} = \frac{3}{4} \), or three-quarters of the total.
• 25% corresponds to \( \frac{25}{100} = \frac{1}{4} \), or one-quarter.
• 10% is \( \frac{10}{100} = \frac{1}{10} \), or one-tenth.

Since percentages are fractions, they provide a simple and intuitive way to represent proportions consistently, regardless of the size of the numbers involved.

Percentages play a significant role in everyday life: from sales discounts and interest rates to school performance statistics. Understanding how they work is essential, not just for solving math problems, but also for interpreting the world around us.

Practical Example of Percentages

In a class of 20 students, 60% are boys.

$$ 20 \cdot 60 \% $$

By performing the calculation, after converting the percentage into a fraction, we find that 12 students are boys.

$$ 20 \cdot 60 \% = 20 \cdot \frac{60}{100} = 2 \cdot 6 = 12 $$

This problem can also be solved using a proportion: "60% is to 100% as x is to 20" $$ 60 \% : 100 \% = x : 20 $$ From this, we can determine the value of x: $$ x = \frac{60 \cdot 20}{100} = 12 $$ The final result is the same.

The remaining 40% of the class are girls.

$$ 20 \cdot 40 \% = 20 \cdot \frac{40}{100} = 2 \cdot 4 = 8 $$

When percentages refer to the same total, they can also be added or multiplied as needed.

For example, in a class of 20 students, 60% are boys, and 50% of them passed.

$$ 20 \cdot 60 \% \cdot 50 \% = 20 \cdot \frac{60}{100} \cdot \frac{50}{100} = 20 \cdot \frac{6}{10} \cdot \frac{5}{10} = 20 \cdot \frac{30}{100} = 6 $$

This means 6 boys passed.

Common Types of Percentage Problems

Percentages can be applied to solve various types of problems. Here are a few examples:

1] Finding a Part Given the Total and a Percentage

If the total and percentage are known, you can calculate the corresponding part by multiplying the total by the percentage in decimal form.

For example, to calculate 40% of 60:

$$ 40\% = 0.4, \quad 0.4 \cdot 60 = 24 $$

The result is 24.

2] Finding the Total Given a Part and a Percentage

When a part and its percentage are known, the total can be determined by dividing the part by the percentage in decimal form. For example:

If 60% of a total is 12, what is the total?

Knowing that \( x \cdot 60 \% = 12 \), we solve for \( x \):

$$ x = \frac{12}{60 \%} = \frac{12}{ \frac{60}{100} } = \frac{12}{0.6} = 20 $$

Thus, the total is 20.

3] Determining the Percentage Given the Total and a Part

If the total and a part are known, the percentage can be calculated by dividing the part by the total and multiplying by 100.

For instance, in a school with 200 students, 160 passed. What percentage of students passed?

$$ \frac{160}{200} \cdot 100 = 80 \% $$

So, 80% of the students passed.

4] Increasing or Decreasing a Quantity by a Percentage

To increase or decrease a quantity \( x \) by a percentage \( p\% \), use the following formulas:

• For an increase of \( p\% \): \[ x \cdot \left( 1 + \frac{p}{100} \right) \] Here, \( 1 + \frac{p}{100} \) accounts for the original value (\( 1 \)) plus the proportional increase (\( \frac{p}{100} \)).

  Example: Increasing \( x = 200 \) by \( 15\% \): $$ 200 \cdot \left( 1 + \frac{15}{100} \right) = 200 \cdot 1.15 = 230 $$ Alternatively: $$ 200 \cdot \frac{115}{100} = 200 \cdot 1.15 = 230 $$

• For a decrease of \( p\% \): \[ x \cdot \left( 1 - \frac{p}{100} \right) \] Here, \( 1 - \frac{p}{100} \) accounts for the original value (\( 1 \)) minus the proportional decrease (\( \frac{p}{100} \)).

  Example: Decreasing \( x = 200 \) by \( 20\% \): $$ 200 \cdot \left( 1 - \frac{20}{100} \right) = 200 \cdot 0.80 = 160 $$ Alternatively: $$ 200 \cdot \frac{80}{100} = 200 \cdot 0.80 = 160 $$

These formulas are particularly helpful in avoiding errors, especially in financial calculations or when dealing with successive percentage changes.

Common Mistakes with Percentages

Working with percentages requires careful attention to avoid common pitfalls. Here are some frequent mistakes:

1] Adding Percentages from Different Bases

A common error is adding percentages that refer to different totals.

For instance, in one class of 20 students, 30% are boys, while in another class of 20 students, 20% are boys.

What is the overall percentage of boys across the two classes? You cannot simply add \( 30\% + 20\% \) because the percentages are based on different totals.

The correct approach is to calculate the absolute numbers:

• In the first class: \( 30\% \cdot 20 = 6 \)
• In the second class: \( 20\% \cdot 20 = 4 \)

In total, there are \( 6 + 4 \) boys out of \( 20 + 20 = 40 \) students.

The percentage of boys is therefore:

$$ \frac{6+4}{40} = \frac{10}{40} = 25 \% $$

2] Miscalculating Successive Percentage Changes

Another common error involves successive percentage changes.

For example, if a price increases by 10% and then decreases by 10%, it’s incorrect to assume the price remains unchanged.

$$ +10\% -10\% = 0\% \quad (\text{unchanged price}) $$

This is wrong because the 10% decrease applies to the increased price, not the original one.

The mistake happens when the decrease is calculated based on the original value rather than the increased value, leading to an inaccurate result.

Suppose the initial price is €100. After a 10% increase, the price becomes:

$$ 100 \cdot 1.10 = 110 \, \text{€} $$

After the 10% decrease, the calculation applies to the new price (€110):

$$ 110 \cdot 0.90 = 99 \, \text{€} $$

Thus, the correct final price is €99, not €100.

In summary, percentages are versatile tools for representing proportions and changes.

However, as we’ve seen, it’s crucial to use them correctly to avoid common errors.

Understanding the context and the reference total is key to applying percentages accurately and meaningfully.