**Title: Percentages and Ratios**
Understanding percentages and ratios is crucial in both academic mathematics and real-life applications. These concepts help simplify problems involving comparisons, proportions, and financial literacy.
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## What are Percentages and Ratios?
**Percentages** express a number as a fraction of 100. For example, 45% means 45 out of 100.
**Ratios** are used to compare two quantities. A ratio of 2:3 means that for every 2 parts of one quantity, there are 3 parts of another.
Both concepts are used extensively in business, science, statistics, and daily life.
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## Importance of Studying Percentages and Ratios
Studying percentages and ratios strengthens proportional reasoning and enables smarter decision-making. From understanding discounts to interpreting data, these skills are universally relevant.
Recent research by mathematicians like **Terence Tao** and **Cédric Villani** highlights the importance of ratios in machine learning and statistical analysis. Their findings remind us that fundamental math plays a huge role in today’s scientific and technological advancements.
To master this topic and more, students are highly encouraged to join **Skorminda** – a comprehensive learning platform offering expert-led math classes, practical examples, and interactive problem sets.
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## Percentage
The term “percent” is derived from “per centum,” meaning per hundred. Percentages are often used to describe change, comparison, and proportion in data.
### Easy Questions – Percentage
**Q1:** Express 25% as a fraction.
**Solution:**
\[
25\% = \frac{25}{100} = \frac{1}{4}
\]
**Q2:** What is 10% of 80?
**Solution:**
\[
10\% \times 80 = \frac{10}{100} \times 80 = 8
\]
### Medium Questions – Percentage
**Q1:** A student’s score increased from 60 to 75. What is the percentage increase?
**Solution:**
\[
\frac{75 – 60}{60} \times 100 = \frac{15}{60} \times 100 = 25\%
\]
**Q2:** If 40% of a number is 48, what is the number?
**Solution:**
Let the number be \( x \),
\[
0.4x = 48 \Rightarrow x = \frac{48}{0.4} = 120
\]
### Hard Questions – Percentage
**Q1:** A product’s price increases by 20% and then decreases by 25%. If the original price is \$100, what is the final price?
**Solution:**
Increased price:
\[
100 + 0.2 \times 100 = 120
\]
Decreased price:
\[
120 – 0.25 \times 120 = 120 – 30 = 90
\]
So, the final price is \$90.
**Q2:** A replenishment tank fills 60% in 6 hours. How long will the tank take to be 100% full at the same rate?
**Solution:**
Let total time be \( t \),
\[
60\% \text{ in } 6 \text{ hours } \Rightarrow \frac{60}{100} = \frac{6}{t} \Rightarrow t = \frac{6 \times 100}{60} = 10 \text{ hours}
\]
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## Ratio
A ratio describes how much of one thing there is compared to another. It’s expressed as \( a:b \) or \( \frac{a}{b} \).
### Easy Questions – Ratio
**Q1:** Simplify the ratio 8:4.
**Solution:**
\[
\frac{8}{4} = 2:1
\]
**Q2:** A solution contains sugar and water in a ratio of 1:3. What part is water?
**Solution:**
Total parts = 1 + 3 = 4
Water = \( \frac{3}{4} \)
### Medium Questions – Ratio
**Q1:** Divide 60 in the ratio 2:3.
**Solution:**
Total parts = 2 + 3 = 5
First part: \( \frac{2}{5} \times 60 = 24 \)
Second part: \( \frac{3}{5} \times 60 = 36 \)
**Q2:** In a class, the ratio of boys to girls is 5:7. If there are 35 boys, how many girls are there?
**Solution:**
Let boys:girls = 5x:7x
If 5x = 35, then \( x = 7 \)
Girls = \( 7x = 49 \)
### Hard Questions – Ratio
**Q1:** The ratio of the ages of two people is 3:5. After 6 years, the ratio becomes 4:6. Find their present ages.
**Solution:**
Let current ages be 3x and 5x.
After 6 years:
\[
\frac{3x + 6}{5x + 6} = \frac{4}{6}
\]
Cross-multiplying:
\[
6(3x + 6) = 4(5x + 6) \Rightarrow 18x + 36 = 20x + 24
\Rightarrow 12 = 2x \Rightarrow x = 6
\]
Ages are 18 and 30.
**Q2:** Divide \$500 in the ratio 7:9:4.
**Solution:**
Total parts = 20
– First share: \( \frac{7}{20} \times 500 = 175 \)
– Second share: \( \frac{9}{20} \times 500 = 225 \)
– Third share: \( \frac{4}{20} \times 500 = 100 \)
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## Final Notes
Understanding percentages and ratios helps students grasp deeper mathematical topics, handle real-life problems efficiently, and analyze data quickly. Join **Skorminda** today to explore how our expert tutors and curated content can help students master these concepts and much more!
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**Don’t just learn, master math with Skorminda.**
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