Step 1: Understanding the Concept:
The concept of percentage is central to business mathematics and daily calculations. The word "percent" is derived from the Latin "per centum," meaning "by the hundred."
When we say \(45%\) of a quantity, we are referring to a ratio where \(45\) units represent the portion out of a total "whole" represented by \(100\) units.
In algebraic terms, the phrase "of a number" implies a multiplication between the percentage fraction and an unknown variable.
The word "is" acts as an equality sign. Thus, the problem provides us with a fractional part of a number and asks us to find the original full value (the \(100%\) value).
Step 2: Key Formula or Approach:
Let the unknown number be \(x\).
The mathematical translation of the problem statement is:
\[ \left( \frac{\text{Percentage}}{100} \right) \times x = \text{Resulting Value} \]
To solve for \(x\), we rearrange the formula:
\[ x = \text{Resulting Value} \times \left( \frac{100}{\text{Percentage}} \right) \]
Step 3: Detailed Explanation:
Let the required unknown number be represented by the variable \(x\).
According to the problem description:
\[ 45% \text{ of } x = 180 \]
Converting the percentage into a fraction:
\[ \frac{45}{100} \times x = 180 \]
To isolate the variable \(x\), we multiply both sides of the equation by the reciprocal of the fraction \(\frac{45}{100}\), which is \(\frac{100}{45}\).
\[ x = 180 \times \frac{100}{45} \]
Now, let's simplify the multiplication and division. It is easier to divide \(180\) by \(45\) first rather than multiplying \(180\) by \(100\).
Look for the relationship between \(180\) and \(45\):
\(45 \times 2 = 90\)
\(90 \times 2 = 180\)
Therefore, \(45 \times 4 = 180\).
Dividing \(180\) by \(45\) gives us exactly \(4\).
\[ x = 4 \times 100 \]
\[ x = 400 \]
Verification:
To verify, calculate \(45%\) of \(400\).
\(10%\) of \(400 = 40\).
\(40%\) of \(400 = 40 \times 4 = 160\).
\(5%\) of \(400 = \text{half of } 10% = 20\).
Total \(45% = 160 + 20 = 180\).
This matches the initial problem statement perfectly.
Step 4: Final Answer:
The required number is 400.