Solving Percentage Problems 33 Is 55% Of What And What Percent Of 40 Is 26

Navigating the world of percentages can often feel like traversing a complex maze. Percentages are fundamental in various aspects of daily life, from calculating discounts while shopping to understanding financial statements and interpreting statistical data. Mastering the art of solving percentage problems is a crucial skill that empowers individuals to make informed decisions and confidently tackle mathematical challenges. In this comprehensive guide, we will dissect two intriguing percentage questions: (a) 33 is 55% of what? and (b) What percent of 40 is 26? By meticulously examining each question and employing a step-by-step approach, we will demystify the underlying concepts and equip you with the knowledge and techniques necessary to conquer any percentage problem that comes your way.

(a) 33 is 55% of What?

Deconstructing the Problem

The question "33 is 55% of what?" presents a classic percentage problem that requires us to determine the whole quantity when a specific percentage of it is known. To effectively tackle this problem, it is essential to break it down into smaller, manageable parts and translate the verbal statement into a mathematical equation. The key to solving this type of problem lies in recognizing the relationship between the percentage, the part, and the whole. In this case, 33 represents the part, 55% represents the percentage, and we need to find the whole.

Transforming Words into Equations

To solve this problem mathematically, we can translate the words into an equation. Let's represent the unknown whole quantity with the variable "x". The phrase "55% of what?" can be expressed as 0.55 multiplied by x (since 55% is equivalent to 0.55 in decimal form). The word "is" in the question signifies equality, so we can set up the equation as follows:

33 = 0. 55 * x

This equation now forms the foundation for solving the problem. It succinctly captures the relationship between the given information and the unknown quantity we seek.

Solving for the Unknown

Now that we have the equation 33 = 0.55 * x, our objective is to isolate the variable "x" to determine its value. To achieve this, we can employ the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain balance. In this case, we need to undo the multiplication by 0.55. The inverse operation of multiplication is division, so we will divide both sides of the equation by 0.55:

33 / 0. 55 = (0.55 * x) / 0.55

This step effectively isolates "x" on the right side of the equation. Performing the division on the left side, we get:

60 = x

Therefore, the solution to the problem is x = 60. This means that 33 is 55% of 60.

Validating the Solution

To ensure the accuracy of our solution, it's always prudent to verify the answer. We can do this by calculating 55% of 60 and checking if it equals 33. To find 55% of 60, we multiply 60 by 0.55:

0. 55 * 60 = 33

The result confirms that our solution is correct. 33 is indeed 55% of 60.

(b) What Percent of 40 is 26?

Understanding the Question

The question "What percent of 40 is 26?" challenges us to find the percentage that 26 represents out of the total quantity of 40. This type of percentage problem requires us to determine the ratio between the part (26) and the whole (40) and express it as a percentage. The key here is to identify the part, the whole, and the unknown percentage.

Setting up the Proportion

To solve this problem, we can use the concept of proportions. A proportion is an equation that states that two ratios are equal. In this case, we can set up a proportion to relate the part (26) to the whole (40) and the unknown percentage to 100 (since percentages are expressed as parts out of 100). Let's represent the unknown percentage with the variable "p". The proportion can be written as:

26 / 40 = p / 100

This proportion elegantly captures the relationship between the given numbers and the unknown percentage.

Solving the Proportion

To solve for "p", we can use the technique of cross-multiplication. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal. Applying cross-multiplication to our proportion, we get:

26 * 100 = 40 * p

This simplifies to:

2600 = 40p

Now, to isolate "p", we divide both sides of the equation by 40:

2600 / 40 = (40p) / 40

This gives us:

65 = p

Therefore, the solution to the problem is p = 65. This means that 26 is 65% of 40.

Verifying the Answer

To ensure the accuracy of our solution, we should verify that 65% of 40 is indeed 26. To calculate 65% of 40, we multiply 40 by 0.65 (since 65% is equivalent to 0.65 in decimal form):

0. 65 * 40 = 26

The result confirms that our solution is correct. 26 is indeed 65% of 40.

Conclusion

In this comprehensive guide, we have meticulously dissected two fundamental percentage problems: (a) 33 is 55% of what? and (b) What percent of 40 is 26? By translating verbal statements into mathematical equations, employing algebraic techniques, and verifying our solutions, we have successfully navigated the intricacies of percentage calculations. These skills are not only valuable in academic settings but also in everyday scenarios where understanding percentages is crucial for informed decision-making. Mastering the art of solving percentage problems empowers you to confidently tackle mathematical challenges and apply these concepts to real-world situations.