Question

Explain the mistake, then solve the problem correctly. What percent of 60 is \(15 ?\) $$ \begin{aligned} 60 p &=15 \\ \frac{60 p}{60} &=\frac{15}{60} \\ p &=0.25 \end{aligned} $$

Short answer

Answer: 25%

Step-by-step solution

Write the formula

We will use the formula $$percentage = \frac{part}{whole} \times 100$$. Here, the part is 15 and the whole is 60.

Substitute the values

Substitute the part and the whole into the formula: $$percentage = \frac{15}{60} \times 100$$.

Simplify the fraction

Simplify the fraction \(\frac{15}{60}\) by dividing the numerator and denominator by 15: $$\frac{15\div15}{60\div15} = \frac{1}{4}$$.

Calculate the percentage

Multiply the simplified fraction by 100 to find the percentage: $$percentage = \frac{1}{4} \times 100$$.

Evaluate the expression

Calculate the value of the expression: $$percentage = 25$$. The correct solution is that 15 is 25% of 60.

Key concepts

Percentage Formula

Understand the percentage formula is essential when dealing with comparative quantities. Essentially, a percentage represents a fraction out of 100. To convert a number into a percentage, the formula you should keep at hand is
\[ percentage = \frac{part}{whole} \times 100 \].
By plugging in the values for 'part' and 'whole,' you simplify a complex relationship into a more digestible format. For instance, if you want to know what percent of 60 is 15, you identify 15 as the 'part' and 60 as the 'whole'. Then, substitute them into the formula:
\[ percentage = \frac{15}{60} \times 100 \].
Remember, percentages are incredibly versatile and used in various situations, from calculating discounts during shopping to understanding statistical data in academic research.

Simplifying Fractions

Simplifying fractions is a fundamental step in mathematics that aids in making numbers more manageable and comprehensible. It involves reducing the numerator (top number) and the denominator (bottom number) to their smallest common terms. To simplify fractions, find the greatest common divisor (GCD) for both numbers and divide them by it.
For example, in the fraction \[ \frac{15}{60} \], both the numerator and the denominator can be divided by 15, the GCD for these two numbers. This process will simplify \[ \frac{15}{60} \] to \[ \frac{1}{4} \].
Simplified fractions are easier to work with and especially useful when calculating percentages, as you've seen in the earlier step.

Basic Algebra

Basic algebra is the backbone of problem-solving in mathematics. It involves using symbols and letters to represent numbers and quantities in formula and equations. Critical to solving algebraic equations is understanding the concept of variables and constants.
In our example, the percentage we're trying to calculate is represented by the variable 'p'. The mistake in the original exercise comes from not using the percentage formula correctly and not simplifying the fraction before multiplying by 100. By applying the correct steps and using basic algebra—setting up an equation, isolating the variable, and solving for it—you can find correct solutions to such problems.
When you come across the variable 'p' in an equation, it's just asking you to find 'p' in terms of other numerical values present in the equation. Algebraic techniques are not just abstract skills; they are practical tools that help us manage and interpret the world of numbers around us.