Maharashtra Board Class 11 Maths Part 2 Chapter 9 Commercial Mathematics 9.1 Solutions

Std 11 Maths 2 Exercise 9.1 Solutions Commerce Maths

Question 1. Find 77% of 580 + 34% of 390.
Answer:Solution:
\( 77\% \text{ of } 580 + 34\% \text{ of } 390 \) \( = \frac{77}{100} \times 580 + \frac{34}{100} \times 390 \) \( = \frac{77}{5} \times 29 + \frac{17}{5} \times 39 \) \( = \frac{2233 + 663}{5} \) \( = \frac{2896}{5} \) \( = 579.2 \)In simple words: To solve this, convert percentages to decimals (or fractions) and perform the multiplication and addition operations. \(77\% \text{ of } 580\) is \(0.77 \times 580\), and \(34\% \text{ of } 390\) is \(0.34 \times 390\), then add the results.

🎯 Exam Tip: Remember to convert percentages to decimal or fractional form before performing calculations to avoid errors. Simplify fractions where possible to ease computation.

Question 2. 240 candidates appeared for an examination, of which 204 passed. What is the pass percentage?
Answer:Solution: We find the pass percentage using the unitary method
Total number of students = 240
Number of students passed = 204
Percentage passed \( = \frac{204}{240} \times 100 \) \( = \frac{204 \times 100}{240} \) \( = \frac{51 \times 100}{60} \) \( = \frac{17 \times 10}{2} \) \( = 85 \) \( \therefore \) The pass percentage for the examination is 85%.In simple words: The pass percentage is calculated by dividing the number of passed candidates by the total number of candidates and then multiplying by 100. In this case, 204 out of 240 passed, which is 85%.

🎯 Exam Tip: When calculating percentages, ensure the 'part' is divided by the 'whole' before multiplying by 100. Clearly label what each number represents to avoid confusion.

Question 3. What percent of 8.4 kg are 168 grams?
Answer:Solution: Let 168 gms be x% of 8.4 kg
i.e., let 168 gms be \( \frac{x}{100} \) of 8400 gms
\( \therefore 168 = \frac{x}{100} \times 8400 \) \( \therefore x = \frac{168}{84} = 2 \) \( \therefore \) 168 gms is 2% of 8.4 kg.In simple words: To find what percentage 168 grams is of 8.4 kg, first convert both quantities to the same unit (grams). Then, divide 168 grams by 8400 grams and multiply by 100 to get the percentage.

🎯 Exam Tip: Always ensure units are consistent before performing calculations (e.g., both in grams or both in kilograms). A common mistake is to mix units, leading to incorrect results.

Question 4. If the length of a rectangle is decreased by 20%, what should be the increase in the breadth of the rectangle so that the area remains the same?
Answer:Solution: Let x and y represent the length and breadth of the rectangle respectively.
\( \therefore \) The original area of the rectangle = xy
There is a 20% decrease in length.
\( \therefore \) New length \( = x - \frac{20}{100}x = x - \frac{1}{5}x \) \( = x \left(1 - \frac{1}{5}\right) = x \left(\frac{5-1}{5}\right) = \frac{4}{5}x \)
Let k % be the required increase in breadth
\( \therefore \) New breadth \( = y + \frac{k}{100}y \) \( = y \left(1 + \frac{k}{100}\right) \)
Given that the new and old areas should be equal.
\( \therefore \left(\frac{4}{5}x\right) \left(1 + \frac{k}{100}\right)y = xy \) \( \therefore \frac{4}{5} \left(1 + \frac{k}{100}\right) = 1 \) \( \therefore \frac{4}{5} \left(\frac{100+k}{100}\right) = 1 \) \( \therefore \frac{100+k}{100} = \frac{5}{4} \) \( \therefore 100 + k = 125 \) \( \therefore k = 125 - 100 = 25 \)
\( \therefore \) Breadth should be increased by 25% so that the area remains same.In simple words: If the length of a rectangle is reduced by 20%, to keep the area unchanged, the breadth must be increased. This increase compensates for the reduced length. A 20% reduction in length (to 80% of original) requires a 25% increase in breadth to maintain the original area.

🎯 Exam Tip: For problems involving percentage changes and constant area/volume, remember that a decrease in one dimension often requires a disproportionately larger percentage increase in another dimension to compensate. Focus on setting up the 'new area = old area' equation.

Question 5. The price of rice increased by 20%, as a result, a person can have 5kg rice for Rs. 600. What was the initial price of rice per kg?
Answer:Solution: A person can buy 5 kg of rice for Rs. 600 after the increase in price
\( \therefore \) New price of rice \( = \frac{600}{5} = \text{Rs. } 120/\text{kg} .....(\text{i}) \)
Let 'x' be the initial price per kg of rice.
There is a 20% increase in the price of rice.
Thus the new price of the rice will be given as
\( x \left(1 + \frac{20}{100}\right) \)
Equation with (i), we get
\( x \left(1 + \frac{20}{100}\right) = 120 \) \( x \left(\frac{100+20}{100}\right) = 120 \) \( \frac{120x}{100} = 120 \) \( \therefore x = 100 \)
\( \therefore \) The initial price of rice is Rs. 100 per kgIn simple words: First, calculate the new price per kg after the increase. Since the new price is 20% higher than the original price, you can set up an equation where the original price plus 20% of the original price equals the new price to find the initial cost.

🎯 Exam Tip: Clearly define your variables (e.g., 'x' for the initial price). When a quantity increases by a percentage, multiply the original quantity by (1 + percentage/100).

Question 6. What percent is 3% of 5%?
Answer:Solution: Let 3% be x % of 5%.
Then \( \frac{3}{100} = \frac{x}{100} \times \frac{5}{100} \) \( \therefore x = \frac{3 \times 100}{5} = 60 \)
\( \therefore \) 3% is 60% of 5%.In simple words: To find what percentage one percentage is of another, express both as fractions or decimals, then divide the first percentage value by the second and multiply by 100. In this case, 3% divided by 5% is 0.6, which is 60%.

🎯 Exam Tip: Be careful with the phrasing "what percent is A of B". This translates to \( \frac{A}{B} \times 100 \). Ensure you convert percentages to numerical values before performing the division.

Question 7. After availing of two successive discounts of 20% each, Madhavi paid Rs. 64 for a book. If she would have got only one discount of 20%, how much additional amount would she have paid?
Answer:Solution: Let the price of the book be Rs. x.
After the first 20% discount, the price of the book becomes
\( = x \left(1 - \frac{20}{100}\right) \) \( = x \left(1 - \frac{1}{5}\right) \) \( = x \left(\frac{4}{5}\right) \) \( = \frac{4x}{5} .....(\text{i}) \)
After another 20% discount, the price of the book becomes
\( = \left(1 - \frac{20}{100}\right) \left(\frac{4}{5}x\right) \) \( = \left(1 - \frac{1}{5}\right) \left(\frac{4}{5}x\right) \) \( = \left(\frac{4}{5}\right) \left(\frac{4}{5}x\right) \) \( = \frac{16}{25}x \)
This price = Rs. 64 .....[Given]
\( \therefore \frac{16}{25}x = 64 \) \( \therefore x = \frac{64 \times 25}{16} = 4 \times 25 = 100 \)
Thus, Amount of the book after one discount \( = \frac{4}{5}(100) = 80 \) ....[from (i)]
\( \therefore \) The additional amount that Madhavi would have paid \( = 80 - 64 = \text{Rs. } 16 \)In simple words: First, calculate the original price of the book by working backward from the final price after two successive 20% discounts. Then, calculate the price if only one 20% discount was applied. The difference between these two prices is the additional amount she would have paid.

🎯 Exam Tip: When dealing with successive discounts, apply each discount to the *remaining* price, not the original price. Be methodical in your calculations to avoid errors in determining the original value.

Question 8. The price of the table is 40% more than the price of a chair. By what percent price of a chair is less than the price of a table?
Answer:Solution: Let x and y be the price of a table and chair respectively.
The price of the table is 40% more than the price of a chair
\( \therefore \frac{x-y}{y} \times 100 = 40 \) \( \therefore \frac{x-y}{y} = \frac{40}{100} = \frac{2}{5} \) \( \therefore \frac{x}{y} - \frac{y}{y} = \frac{2}{5} \) \( \therefore \frac{x}{y} - 1 = \frac{2}{5} \) \( \therefore \frac{x}{y} = 1 + \frac{2}{5} \) \( \therefore \frac{x}{y} = \frac{7}{5} .....(\text{i}) \)
We need to find by how much percent is the price of a chair less than that of a table.
i.e. \( \left(\frac{x-y}{x}\right) \times 100 \) \( = \left(1 - \frac{y}{x}\right) \times 100 \) \( = \left(1 - \frac{5}{7}\right) \times 100 \quad \left[ \because \frac{x}{y} = \frac{7}{5} \implies \frac{y}{x} = \frac{5}{7} \right] \) \( = \left(\frac{7-5}{7}\right) \times 100 \) \( = \frac{2 \times 100}{7} = 28.57\% \)
\( \therefore \) The price of a chair is 28.57% less than the price of a table.In simple words: If a table's price is 40% more than a chair's, it means the table's price is 140% of the chair's. To find how much less the chair's price is than the table's, calculate the difference in price and divide it by the table's price, then multiply by 100.

🎯 Exam Tip: Be careful to distinguish between "A is x% more than B" and "B is y% less than A". The base for the percentage calculation changes. Always clearly identify the base (denominator) for your percentage calculation.

Question 9. A batsman scored 92 runs which includes 4 boundaries 5 sixes. He scored other runs by running between the wickets. What percent of his total score did he make by running between the wickets?
Answer:Solution: Batsman scores 4 fours (boundaries) and 5 sixes in 92 runs.
Number of runs scored by fours and sixes \( = 4 \times 4 + 5 \times 6 = 16 + 30 = 46 \)
\( \therefore \) Runs scored by running between the wickets \( = 92 - 46 = 46 \)
Let 46 be x% of 92.
Then \( 46 = \frac{x}{100} \times 92 \) \( \therefore x = \frac{46 \times 100}{92} = \frac{100}{2} = 50 \)
\( \therefore \) 50% of the total runs were scored by running between the wickets.In simple words: First, calculate the total runs scored from boundaries (fours and sixes). Subtract these runs from the total score to find the runs scored by running. Then, express these running runs as a percentage of the total score.

🎯 Exam Tip: Break down the problem into smaller steps: calculate runs from fours, then sixes, then total from boundaries. This makes it easier to find the remaining runs and subsequently the percentage. Ensure the percentage is calculated against the total score.