PORCENTAGEM: Teoria e Exemplos | Matemática Básica - Aula 29

Understanding Percentages

Importance of Percentages

• Percentages are crucial for various exams, including the ENEM and other public entrance tests.

• The class aims to provide a comprehensive understanding of percentages, emphasizing their practical applications.

Forms of Representing Percentages

• 30% can be expressed as a fraction: 30/100 , which is also equivalent to 0.3 in unitary form.

• The three forms of representing percentages include percentage form (e.g., 30%), fractional form (e.g., 30/100 ), and unitary form (e.g., 0.3).

Converting Between Formats

• To convert from percentage to unitary rate, divide by 100; conversely, multiply by 100 to convert from unitary rate to percentage.

• Example: Converting 32% involves dividing by 100, resulting in 0.32; similarly, converting back from 0.15 gives you 15%.

Understanding Percentage of an Amount

• To find a specific percentage of a value (e.g., what is 30% of R$80?), recognize that the total amount corresponds to 100%. Cross-multiplication can be used here but there are simpler methods available.

Understanding Percentages and Their Applications in Problem Solving

Calculating Percentages

• To find the percentage of a value, convert the percentage into fractional form and multiply by the desired value. For example, to find 60% of a number, express it as 60/100 .

Example Problem: Finding a Value from Percentage

• In the problem where 60% of an unknown value equals 27, we set up the equation 0.6x = 27 . This leads to x = 27/0.6 = 45 .

Another Example: Relating Two Values with Percentages

• The next problem involves determining what percentage 24 is of 150. We can express this as x = 24/150 , simplifying to x = 16/100 = 16% .

Quick Methods for Calculating Simple Percentages

• A quick method for calculating percentages is moving the decimal point leftward:

• For example, to find 10% of 32.8, move the decimal one place left resulting in 3.28.

• Similarly, for finding 1% of a number like 123, divide by 100 which gives us 1.23.

Importance of Efficient Calculation Techniques

• Many students complicate calculations involving increases or discounts when simpler methods exist.

Understanding Increases and Discounts Using Percentages

• When increasing a value by a certain percentage (e.g., increasing $400 by 30%), calculate it as follows:

• Convert to final value: Original + Increase → 400 * (1 + 30/100) = 520.

Practical Application: Increasing Values with Percentages

• For an increase of $250 by 8%, calculate it as:

• Final Value = Original Value × (1 + Increase Rate)

• Thus, 250 * (1 + 8/100) = $270.

Understanding Percentage Increases and Decreases

Basic Calculations of Increases

• The initial value is 250; with an 8% increase, the new value becomes 270.

• To calculate a 50% increase, multiply by 1.5; for a 4% increase, multiply by 1.04.

• A 40% increase should be calculated as multiplying by 1.4 (not adding directly).

Understanding Discounts

• When applying a discount of x percent on an item worth 600, you subtract the percentage from the total: 100% - x%.

• A decrease of 40% means retaining only 60%, so for an original value of 600, the calculation is 600 times 0.6 = 360.

Further Discount Calculations

• For a further decrease on the new value (360), applying a discount of 15% results in retaining only 85%: 360 times 0.85 = 306.

• The calculations can be simplified through cancellation when working with percentages.

Final Value After Increases and Discounts

• To find the final value after a series of increases or discounts, use multiplication factors based on percentage changes.

• For example, to apply a discount of 14%, retain 86%: multiply by 0.86.

Successive Changes in Value

• When dealing with multiple increases and discounts over time, each change affects the subsequent calculations.

• An example involves starting with an initial amount that undergoes successive changes: first a 30% increase followed by two discounts (10%, then 20%).

Detailed Calculation Example

• Start with an initial value (x); after increasing it by 30%, multiply by 1.3.

• Apply discounts sequentially: first reduce by 10%, then another reduction of 20%.

Understanding Overall Impact

• The final result shows how much below the original value remains after all adjustments.

• If you end up below your starting point (less than x), it indicates there was an overall discount applied.

Importance of Order in Calculations

• It's crucial to understand that each step impacts subsequent values; thus, order matters significantly in these calculations.

Practice Problems and Application

Understanding Percentages in Agricultural Production and Population Statistics

Example 1: Agricultural Production Breakdown

• The example discusses the agricultural production of a region, where 68% of the total production was grains. Out of this, 75% was soybeans.

• To find the percentage of soybeans relative to total production, the calculation involves multiplying the total value (x) by 68% to get grain production.

• The soybean production is then calculated as 75% of the grain production, leading to a formula that avoids complex calculations like rule of three.

• The result shows that soybean production accounts for 51% of total agricultural output after simplifying the equation.

Example 2: Unemployment Statistics Over Time

• In another scenario from 2011, it states that there were 4.4 million unemployed individuals representing 22% of the population.

• Using cross multiplication, it determines that if 22% corresponds to 4.4 million, then the total population is calculated to be approximately 20 million.

• By comparing data from different years (2001 vs. 2010), it notes an increase in unemployment numbers despite a decrease in percentage terms due to population growth.

• In this case, while unemployment rose from 4.4 million to 5.4 million, the overall population increased significantly during these nine years.

Calculating Population Growth Percentage

• The discussion shifts towards calculating how much the country's population grew over time; with a jump from an estimated population of around 20 million to about 27 million.