Percentage Problems
a) The Concept of Percentage:
Let’s explain the concept of percentage with an example.
Example:
A student preparing for the university entrance exam was able to summarize 220 pages of a 550-page Social Sciences book in one month. Let’s find what percentage of the book this student summarized in one month.
If we find the ratio of the number of pages summarized by the student to the total number of pages in the book,
\[\frac{220}{550} = \frac{22}{55} = \frac{2 }{5} \cdot \frac{20}{20} = \frac{40}{100} \] This ratio
\[ \frac{40}{100} = \frac{1}{100} \cdot 40 = 0.40 \; (0.01 \cdot 40 ) = \% \;40 \] can be written in this form.
The \(\% \) symbol is used instead of \( \Large \frac{1}{100} \) or \( 0.01 \).
\(\% \;40 \) is read as forty percent.
The expression \(\% \;40 \) is called the percentage rate.
It expresses that forty pages out of every one hundred pages have been summarized.
Warning:
In order to write rational numbers with the percentage symbol, the fraction must be expanded or simplified to an equivalent number such that its denominator is 100.
When decimal numbers are expressed as percentages, the decimal point is moved two places to the right, and then written with the percentage symbol.
Examples:
\(\bullet \quad \Large \frac{28}{40} = \frac{7}{10} \cdot \frac{10}{10} = \frac{70}{100} \) \( = \%70 \)
\(\bullet \quad \Large \frac{120}{600} = \frac{20}{100} \) \( = \%20 \)
\(\bullet \quad 0.55= 0.01 \cdot 55 = \%55 \) or \(0.55 = \) \( \Large \frac{55}{100} \) \( = \%55 \)
Let’s explain the concepts of **Base Amount (Base Number)**, **Percentage Rate**, and **Percentage Portion (Percentage Value)** with an example.
Example:
In a class of 30 students, \(\%70 \) of the students are successful in the Turkish course. Let’s find the number of students who are successful in the Turkish course in this class.
\[
\begin{aligned}
\text{If out of } 100 \; \text{students} \quad \quad &70 \; \text{are successful,} \\
\\
\text{then out of } 30 \; \text{students} \quad \quad &x \; \text{are successful.} \\
\\
\hline \text{Direct Proportion}
\end{aligned}
\]
\[ \underbrace{x}_{\Large \text{percentage portion} } = \underbrace{ 30}_{\Large \text{base number} } \cdot \quad \underbrace{\frac{70}{100} }_{\Large \text{percentage rate} } \]
\[ \underbrace{21}_{\Large \text{percentage portion} } = \underbrace{ 30}_{\Large \text{base number} } \cdot \quad \underbrace{\frac{70}{100} }_{\Large \text{percentage rate} } \]
From here, the equality
\[ \textbf{Percentage portion} = \textbf{Base number} \times \textbf{Percentage rate} \]
can be written.
Results:
\(1) \) The expression “percentage a” can be shown in three ways:
\( \bullet \) \(\% a \)
\( \bullet \) \(\Large \frac{a}{100} \)
\( \bullet \) \(0.01 \cdot a \)
\[ \%a = \frac{a}{100} = 0.01 \cdot a \]
\(2) \) To find \(\%a \) of a number, this number is multiplied by \(\Large \frac{a}{100} \).
Example:
Let’s find \( \% 15 \) of 400.
\[ \Rightarrow 400 \cdot \frac{15}{100} = 60 \; \text{is found.} \]
Example:
Let’s find of which number \(0.008 \) is \(\% 40 \).
Let’s call the number we are looking for \(x \). Since \( \% \;40 \) of \(x \) is \(0.008\), if the equation \[ x \cdot \frac{40}{100} = 0.008 \] is written, \[ x= 0.02 \; \text{is found.} \]
Example:
In a 100-question exam, a student correctly answered 90% of the first 30 questions. Given that the overall correct answer rate across all questions is 80%, let’s find how many of the remaining questions the student answered correctly.
Since they answered 80% of the 100 questions correctly: \[ 100 \cdot \frac{80}{100} = 80 \, \text{(correct answers)} \] Since they answered 90% of the first 30 questions correctly: \[ 30 \cdot \frac{90}{100} = 27 \, \text{(correct answers)} \] Since there must be a total of 80 correct answers, the number of correctly answered questions among the remaining ones is: \[ 80 – 27 = 53 \] Consequently, the student answered \(53\) of the other questions correctly.
← Previous Page | Next Page →