Question

Explain the mistake, then solve the problem correctly. \(5 \%\) of what number is \(24 ?\) $$ \begin{array}{l} \frac{5}{100}=\frac{n}{24} \\ \frac{120}{100}=\frac{100 n}{100} \\ 1.2=n \end{array} $$

Short answer

Answer: 480

Step-by-step solution

1. Identify the mistake

The given solution starts with an incorrect equation based on the problem statement. The equation should be \(\frac{5}{100}n=24\), which means that 5% of some number 'n' is equal to 24.

2. Write the correct equation

We are asked to find the number 'n' such that 5% of it is equal to 24. So, we will write down the correct equation: $$\frac{5}{100}n=24$$

3. Solve for 'n'

Now we will solve for 'n' in the equation \(\frac{5}{100}n=24\). To do this, we will first multiply both sides of the equation by 100 to eliminate the fraction: $$5n=2400$$ Now, we will divide both sides of the equation by 5 to find 'n': $$n=\frac{2400}{5}$$

4. Calculate the value of 'n'

Now we can calculate the number 'n' by dividing 2400 by 5: $$n=480$$ Therefore, 5% of 480 is equal to 24.

Key concepts

Solving Percentage Equations

When dealing with percentage problems in maths, the key to solving such equations is understanding the relation between the percentage, the whole, and the part. The general form of a percentage equation is \( P \% \times W = P \), where \( P \% \) is the percentage, \( W \) is the whole, and \( P \) is the part. To find an unknown part of the equation, one must isolate the variable.
For example, if the problem is to find what number (the whole) 5% represents the part equal to 24, we can set up the equation as \( 0.05 \times W = 24 \)—noting that 5% is equivalent to \( 0.05 \) when expressed as a decimal. This part of algebraic problem-solving is crucial to understand as we use decimals to simplify percentage equations. We then solve for \( W \) by dividing both sides by 0.05, yielding \( W = \frac{24}{0.05} \) which calculates to 480. So, 5% of 480 is equal to 24. Understanding how to manipulate these equations is a fundamental mathematical skill, needed for various real-world applications, from calculating discounts to understanding interest rates.

Mathematical Mistake Identification

Spotting errors during algebraic problem-solving is a skill worth developing, as it not only helps in achieving the correct answer but also enhances one's understanding of mathematical concepts. In the provided exercise, an initial mistake was made in setting up the equation. Such errors can be conceptual—like misunderstanding the relationship between values in a percentage problem—or procedural, such as miscalculations during the solving process.
To identify mistakes, it's important to:

• Review your understanding of the mathematical concept to ensure the equation is set up correctly.
• Check each step methodically, looking for any mishandling of mathematical operations.
• Use inverse operations to verify if the obtained solution, when plugged back into the original equation, produces the expected result.

Correcting the mistake requires rewriting the equation properly as \( \frac{5}{100}n=24 \) and following the correct algebraic steps, resulting in the correct solution, where \( n \) is 480. Being able to identify and correct such mistakes is what turns a good maths student into a great one, and it’s particularly useful in exams or real-life problem scenarios.

Algebraic Problem-Solving

Algebraic problem-solving involves working through equations systematically to find an unknown variable. The process typically includes establishing variables, forming an equation, and manipulating the equation to isolate the unknown. Using operations such as addition, subtraction, multiplication, and division, you modify both sides of the equation in a balanced manner, eventually allowing the variable to be evaluated alone.
In the context of the 5% problem, this translates to:

• Defining the unknown (in this case, the whole number represented by 'n').
• Creating an accurate equation that represents the problem (\( \frac{5}{100}n=24 \) after correction).
• Performing algebraic operations to solve for 'n' (multiplying by 100 to get rid of the denominator and then dividing by 5).

Upon solving correctly, we find that 'n' is 480. Each algebraic step is an application of the fundamental properties of equality—what you do to one side of the equation, you must do to the other. This maintains the 'balance' of the equation. Students must hone their skills in these operations, particularly with percentages, as they are commonplace in not just academic exercises but financial literacy and everyday decision-making as well.