Percentage (சதவீதம்)

Overview: Percentage (சதவீதம்)

tip

This category involves problems related to percentages. Percentage means "per hundred" and is represented by the symbol %. The general approach is to express one quantity as a fraction of another and then multiply by 100. Key concepts include converting fractions to percentages, calculating percentage increase/decrease, and comparing values using percentages. These problems are fundamental in mathematical reasoning.

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Definitions and Formulas (வரையறை மற்றும் சூத்திரங்கள்)

Here are some common formulas used in percentage problems:

1. Converting a Percentage to a Fraction (சதவீதத்தை பின்னமாக மாற்றுதல்): To convert X% to a fraction, divide X by 100.

   $$X\% = \frac{X}{100}$$

2. Converting a Fraction to a Percentage (பின்னத்தை சதவீதமாக மாற்றுதல்): To convert a fraction a/b to a percentage, multiply it by 100.

   $$\left(\frac{a}{b}\right) \times 100\%$$

3. Effect of Price Change on Consumption (விலை மாற்றத்தின் விளைவு):

   • If the price of a commodity increases by R%, the reduction in consumption so as not to increase the expenditure is: $$\text{Reduction} \% = \left(\frac{R}{100 + R}\right) \times 100\%$$
   • If the price of a commodity decreases by R%, the increase in consumption so as not to decrease the expenditure is: $$\text{Increase} \% = \left(\frac{R}{100 - R}\right) \times 100\%$$

4. Population Growth (மக்கள்தொகை வளர்ச்சி): If the population of a town is P and it increases at the rate of R% per annum (similar to compound interest calculations):

   • Population after 'n' years (n ஆண்டுகளுக்குப் பிறகு மக்கள்தொகை): $$P \left(1 + \frac{R}{100}\right)^n$$
   • Population 'n' years ago (n ஆண்டுகளுக்கு முன்பு மக்கள்தொகை): $$\frac{P}{\left(1 + \frac{R}{100}\right)^n}$$

5. Depreciation (தேய்மானம்): If the value of a machine is P and it depreciates at the rate of R% per annum:

   • Value after 'n' years (n ஆண்டுகளுக்குப் பிறகு மதிப்பு): $$P \left(1 - \frac{R}{100}\right)^n$$
   • Value 'n' years ago (n ஆண்டுகளுக்கு முன்பு மதிப்பு): $$\frac{P}{\left(1 - \frac{R}{100}\right)^n}$$

Example Problem

Question: If 80% of A = 50% of B and B = X% of A, then what is the value of X?

Solution:

1. Write the given equation: $$80\% \text{ of } A = 50\% \text{ of } B$$
2. Convert percentages to fractions: $$\frac{80}{100} \times A = \frac{50}{100} \times B$$
3. Solve for B in terms of A: $$B = \frac{80}{100} \times \frac{100}{50} \times A$$ $$B = \frac{80}{50} A = \frac{8}{5} A$$
4. Convert the fraction to a decimal: $$B = 1.6 A$$
5. Use the second given equation B = X% of A: $$B = \frac{X}{100} \times A$$
6. Equate the two expressions for B and solve for X: $$1.6 A = \frac{X}{100} A$$ Cancel A from both sides: $$1.6 = \frac{X}{100}$$ $$X = 1.6 \times 100 = 160$$ Therefore, the value of X is 160.

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Practice Questions

1. A -ன் 80 % = B -ன் 50 % மற்றும் B = A-ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 400 b) 300 c) 160 d) 150

   Answer: c) 160

   Solution

   A-ன் 80 % = B-ன் 50 %

   $$\frac{80}{100} \times A = \frac{50}{100} \times B$$

   To find B in terms of A:

   $$B = A \times \frac{80}{100} \times \frac{100}{50}$$ $$B = \frac{8}{5} A \implies B = 1.6 A$$

   We are given B = X% of A, which is $B = \frac{X}{100} \times A$.

   $$1.6 A = \frac{X}{100} A$$ $$X = 1.6 \times 100 = 160$$

   Therefore, the required value X = 160.

   Why this question belongs to Percentage

   This question involves keywords like %, A-ன் 80 % (80% of A), and B = A-ன் X% which directly relate to the concept of finding a percentage value by comparing two quantities.

2. A -ன் 90 % = B -ன் 30 % மற்றும் B = A -ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 500 b) 350 c) 300 d) 700

   Answer: c) 300

   Solution

   A-ன் 90 % = B-ன் 30 %

   $$\frac{90}{100} \times A = \frac{30}{100} \times B$$

   Solving for B:

   $$B = A \times \frac{90}{100} \times \frac{100}{30}$$ $$B = \frac{90}{30} A \implies B = 3A$$

   Now, using B = X% of A:

   $$3A = \frac{X}{100} \times A$$ $$X = 3 \times 100 = 300$$

   Therefore, the required value X = 300.

   Why this question belongs to Percentage

   This question uses percentage terms like 90 %, 30 %, and asks to find X%, making it a core percentage problem focused on algebraic comparison.

3. A -ன் 70 % = B -ன் 40 % மற்றும் B=A -ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 175 b) 195 c) 225 d) 255

   Answer: a) 175

   Solution

   A-ன் 70 % = B-ன் 40 %

   $$\frac{70}{100} \times A = \frac{40}{100} \times B$$

   Solving for B:

   $$B = A \times \frac{70}{100} \times \frac{100}{40}$$ $$B = \frac{7}{4} A$$

   Now, using B = X% of A:

   $$\frac{7}{4} A = \frac{X}{100} \times A$$ $$X = \frac{7}{4} \times 100 = 7 \times 25 = 175$$

   Therefore, the required value X = 175.

   Why this question belongs to Percentage

   This question requires converting relationships given in percentages (70%, 40%) into an equation to find an unknown percentage (X%).

4. A -ன் 80 % = B -ன் 60 % மற்றும் B = A -ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) $156 \frac{1}{3}$ b) $143 \frac{1}{3}$ c) $133 \frac{1}{3}$ d) $174 \frac{1}{3}$

   Answer: c) $133 \frac{1}{3}$

   Solution

   A-ன் 80 % = B-ன் 60 %

   $$\frac{80}{100} \times A = \frac{60}{100} \times B$$

   Solving for B:

   $$B = A \times \frac{80}{100} \times \frac{100}{60}$$ $$B = \frac{80}{60} A \implies B = \frac{4}{3} A$$

   Now, using B = X% of A:

   $$\frac{4}{3} A = \frac{X}{100} \times A$$ $$X = \frac{4}{3} \times 100 = \frac{400}{3}$$ $$X = 133 \frac{1}{3}$$

   Therefore, the required value $X = 133 \frac{1}{3}$.

   Why this question belongs to Percentage

   This problem is a percentage comparison question that results in a fractional percentage, testing the ability to handle division and mixed fractions within a percentage context.

5. A -ன் 30 % = B -ன் 20 % மற்றும் B=A -ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 190 b) 120 c) 150 d) 260

   Answer: c) 150

   Solution

   A-ன் 30 % = B-ன் 20 %

   $$\frac{30}{100} \times A = \frac{20}{100} \times B$$

   Given B = X % of A. First, let's solve for B:

   $$B = A \times \frac{30}{100} \times \frac{100}{20}$$ $$B = \frac{3}{2} A$$

   Now, using B = X% of A:

   $$\frac{3}{2} A = \frac{X}{100} \times A$$ $$X = \frac{3}{2} \times 100 = 3 \times 50 = 150$$

   Therefore, the required value X = 150.

   Why this question belongs to Percentage

   This question involves the core skill of equating two percentage-based expressions (30% of A, 20% of B) and then expressing one variable as a percentage of another.

6. A -ன் 60 % = B -ன் 30 %, B = C -ன் 40%, மற்றும் C = A -ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 200 b) 500 c) 800 d) 700

   Answer: b) 500

   Solution

   Step 1: Relate A and B.

   $$\frac{60}{100} A = \frac{30}{100} B \implies 2A = B$$

   Step 2: Relate B and C using the result from Step 1. We are given $B = \frac{40}{100} C$. Substitute $B=2A$:

   $$2A = \frac{40}{100} C$$

   Step 3: Solve for C in terms of A.

   $$C = 2A \times \frac{100}{40} = 2A \times \frac{10}{4} = 2A \times \frac{5}{2}$$ $$C = 5A$$

   Step 4: Use the final condition C = X% of A.

   $$5A = \frac{X}{100} \times A$$ $$X = 5 \times 100 = 500$$

   Therefore, the required value X = 500.

   Why this question belongs to Percentage

   This problem is a multi-step percentage question involving three variables (A, B, C). It requires chaining percentage relationships (% of A = % of B, % of B = % of C) to find the final percentage relationship between C and A.

7. A-ன் 30 % = B-ன் 10 % மற்றும் B = C-ன் 30%, C = A-ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 1000 b) 700 c) 900 d) 650

   Answer: a) 1000

   Solution

   Step 1: Relate A and B.

   $$\frac{30}{100} A = \frac{10}{100} B \implies 3A = B$$

   Step 2: Relate B and C using the result from Step 1. We are given $B = \frac{30}{100} C$. Substitute $B=3A$:

   $$3A = \frac{30}{100} C$$

   Step 3: Solve for C in terms of A.

   $$C = 3A \times \frac{100}{30} = A \times \frac{100}{10}$$ $$C = 10A$$

   Step 4: Use the final condition C = X% of A.

   $$10A = \frac{X}{100} \times A$$ $$X = 10 \times 100 = 1000$$

   Therefore, the required value X = 1000.

   Why this question belongs to Percentage

   This is a chained percentage problem. The solution requires expressing B in terms of A, then C in terms of B, and finally C in terms of A to find the value of X%.

8. A -ன் 90 % = B -ன் 40 % மற்றும் B = C -ன் 60%, C = A -ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) 155 b) 375 c) 215 d) 380

   Answer: b) 375

   Solution

   Step 1: Relate A and B.

   $$\frac{90}{100} A = \frac{40}{100} B \implies \frac{9}{4} A = B$$

   Step 2: Relate B and C using the result from Step 1. We are given $B = \frac{60}{100} C$. Substitute $B = \frac{9}{4} A$:

   $$\frac{9}{4} A = \frac{60}{100} C$$

   Step 3: Solve for C in terms of A.

   $$C = \frac{9}{4} A \times \frac{100}{60} = \frac{9}{4} A \times \frac{10}{6} = \frac{9}{4} A \times \frac{5}{3}$$ $$C = \frac{3 \times 5}{4} A = \frac{15}{4} A$$

   Step 4: Use the final condition C = X% of A.

   $$\frac{15}{4} A = \frac{X}{100} A$$ $$X = \frac{15}{4} \times 100 = 15 \times 25 = 375$$

   Therefore, the required value X = 375.

   Why this question belongs to Percentage

   This question involves multiple percentage relationships (90%, 40%, 60%) that must be solved sequentially to find the final percentage X%.

9. A -ன் 50 % = B -ன் 30 % மற்றும் B = C -ன் 70%, C = A-ன் X% எனில் X-ன் மதிப்பு என்ன?

   a) $235 \frac{4}{21}$ b) $216 \frac{10}{21}$ c) $238 \frac{2}{21}$ d) $211 \frac{4}{17}$

   Answer: c) $238 \frac{2}{21}$

   Solution

   Step 1: Relate A and B.

   $$\frac{50}{100} A = \frac{30}{100} B \implies \frac{5}{3} A = B$$

   Step 2: Relate B and C using the result from Step 1. We are given $B = \frac{70}{100} C$. Substitute $B = \frac{5}{3} A$:

   $$\frac{5}{3} A = \frac{70}{100} C$$

   Step 3: Solve for C in terms of A.

   $$C = \frac{5}{3} A \times \frac{100}{70} = \frac{5}{3} A \times \frac{10}{7}$$ $$C = \frac{50}{21} A$$

   Step 4: Use the final condition C = X% of A.

   $$\frac{50}{21} A = \frac{X}{100} A$$ $$X = \frac{50}{21} \times 100 = \frac{5000}{21}$$

   Converting to a mixed fraction: $5000 \div 21 = 238$ with a remainder of $2$.

   $$X = 238 \frac{2}{21}$$

   Therefore, the required value $X = 238 \frac{2}{21}$.

   Why this question belongs to Percentage

   This is a complex chained percentage problem that requires careful fraction manipulation and conversion to a mixed fraction, testing a deeper understanding of percentage calculations.

10. A -ன் 70 % = B -ன் 30 % மற்றும் B = C -ன் 50%, C = A -ன் X% எனில் X-ன் மதிப்பு என்ன?

    a) $464 \frac{2}{3}$ b) $466 \frac{2}{3}$ c) $472 \frac{2}{3}$ d) $475 \frac{2}{3}$

    Answer: b) $466 \frac{2}{3}$

    Solution

    Step 1: Relate A and B.

    $$\frac{70}{100} A = \frac{30}{100} B \implies \frac{7}{3} A = B$$

    Step 2: Relate B and C using the result from Step 1. We are given $B = \frac{50}{100} C$. Substitute $B = \frac{7}{3} A$:

    $$\frac{7}{3} A = \frac{50}{100} C$$

    Step 3: Solve for C in terms of A.

    $$C = \frac{7}{3} A \times \frac{100}{50} = \frac{7}{3} A \times 2$$ $$C = \frac{14}{3} A$$

    Step 4: Use the final condition C = X% of A.

    $$\frac{14}{3} A = \frac{X}{100} A$$ $$X = \frac{14}{3} \times 100 = \frac{1400}{3}$$

    Converting to a mixed fraction: $1400 \div 3 = 466$ with a remainder of $2$.

    $$X = 466 \frac{2}{3}$$

    Therefore, the required value $X = 466 \frac{2}{3}$.

    Why this question belongs to Percentage

    This question uses keywords like 70 % of A, 30 % of B, 50 % of C to create a chain of dependencies. Solving it requires isolating variables through multiple percentage equations.

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