Maharashtra Board Class 11 Maths Part 2 Chapter 9 Commercial Mathematics 9.5 Solutions

Std 11 Maths 2 Exercise 9.5 Solutions Commerce Maths

Question 1.Three partners shared the profit in a business in the ratio 5 : 6 : 7. They had partnered for 12 months, 10 months, and 8 months respectively. What was the ratio of their investments?
Answer:Solution: Let the ratio of investments of the three partners be p: q : r. They partnered for 12 months, 10 months, and 8 months respectively.
\( \therefore \) The profit shared by the partners will be in proportion to the product of capital invested and their respective time periods.
\( \therefore \) \( 12 \times p : 10 \times q : 8 \times r = 5 : 6 : 7 \)
Now, \( \frac{12p}{10q} = \frac{5}{6} \)
\( \implies \frac{p}{q} = \frac{5 \times 10}{6 \times 12} = \frac{50}{72} \) .......(i)
and \( \frac{10q}{8r} = \frac{6}{7} \)
\( \implies \frac{q}{r} = \frac{6 \times 8}{7 \times 10} = \frac{48}{70} = \frac{24}{35} \)
\( \implies \frac{q}{r} = \frac{24 \times 3}{35 \times 3} = \frac{72}{105} \) ..........(ii)
From (i) & (ii), we have
\( p : q : r = 50 : 72 : 105 \)
\( \therefore \) The ratio of their investments was 50 : 72 : 105.
In simple words: The profit ratio is directly proportional to the product of capital and time. By setting up equations based on given profit and time ratios, we can determine the investment ratio for each partner.

🎯 Exam Tip: Remember that profit sharing in a partnership is based on the product of investment and the duration for which it was invested. Pay close attention to unit consistency (e.g., months).

 

Question 2.Kamala, Vimala and Pramila enter into a partnership. They invest Rs. 40,000, Rs. 80,000 and Rs. 1,20,000 respectively. At the end of the first year, Vimala withdraws Rs. 40,000, while at the end of the second year, Pramila withdraws Rs. 80,000. In what ratio will the profit be shared at the end of 3 years?
Answer:Solution: Given that, Kamala, Vimala, and Pramila invest Rs. 40,000, Rs. 80,000, and Rs. 1,20,000 respectively. The ratio of profits is to be calculated at the end of 3 years. Vimala withdraws Rs. 40,000 at the end of the first year.
\( \therefore \) Vimala invested Rs. 80,000 for one year and 40,000 for 2 years. Pramila withdraws Rs. 80,000 at the end of the second year.
\( \therefore \) Pramila invested Rs. 1,20,000 for two years and 40,000 for one year. Kamala invested 40,000 for all the 3 years.
\( \therefore \) The ratio of profits to be shared at the end of 3 years will be
\( = 40,000 \times 3 : (80,000 \times 1 + 40,000 \times 2) : (1,20,000 \times 2 + 40,000 \times 1) \)
\( = 1,20,000 : 1,60,000 : 2,80,000 \)
\( = 12 : 16 : 28 \)
\( = 3 : 4 : 7 \)
Alternate Method: Given that, Kamala, Vimala and Pramila invest Rs. 40,000, Rs. 80,000 & Rs. 1,20,000 respectively. Given, information can be tabulated as:

	Kamala	Vimala	Pramila
Year 1	40,000	80,000	1,20,000
Year 2	40,000	40,000	1,20,000
Year 3	40,000	40,000	40,000
Total	1,20,000	1,60,000	2,80,000

\( \therefore \) The profits to be shared at the end of 3 years will be
\( = 1,20,000 : 1,60,000 : 2,80,000 \)
\( = 12 : 16 : 28 \)
\( = 3 : 4 : 7 \)
In simple words: The total investment for each partner over three years is calculated by summing their capital for each year. The final profit ratio is then derived by simplifying the ratio of their total investments.

🎯 Exam Tip: When investments change over time, calculate the 'capital-time product' for each partner by summing (investment x duration) for all periods. This total product then determines the profit-sharing ratio.

 

Question 3.Sanjeev started a business investing 25,000 in 1999. In 2000, he invested an additional amount of Rs. 10,000 and Rajeev joined him with an amount of Rs. 35,000. In 2001, Sanjeev invested another additional amount of 10,000 and Pawan joined them with an amount of 35,000. What will be Rajeev's share in the profit of 1,50,000 earned at the end of 3rd year from the start of the business in 1999?
Answer:Solution: The given information can be tabulated as:

Year\( \downarrow \)	Investment in Rs.
	Sanjeev	Rajeev	Pawan
1999	25,000/-	0	0
2000	(25,000+10,000)35,000/-	35,000/-	0
2001	(35,000+10,000)45,000/-	35,000/-	35,000/-
Total	1,05,000/-	70,000/-	35,000/-

\( \therefore \) The ratio of profits to be shared at the end of 3 years will be 1,05,000 : 70,000 : 35,000 i.e. in the proportion 3 : 2 : 1 Given, profit earned Rs. 1,50,000/-
\( \therefore \) Rajeev's share in the profit \( = \frac{2}{6} \times 1,50,000 = \text{Rs. } 50,000/- \)
In simple words: First, calculate each partner's total investment over the three years. Then, simplify these total investments to get their profit-sharing ratio. Finally, apply this ratio to the total profit to find Rajeev's share.

🎯 Exam Tip: For problems with changing investments over several years, create a clear year-wise breakdown of each partner's capital contribution to accurately sum their total effective investment.

 

Question 4.Teena, Leena, and Meena invest in a partnership in the ratio: 7/2, 4/3, 6/5. After 4 months, Teena increases her share by 50%. If the total profit at the end of one year is Rs. 21,600, then what is Leena's share in the profit?
Answer:Solution: Investment of Teena, Leena and Meena are in the ratio \( \frac{7}{2} : \frac{4}{3} : \frac{6}{5} \) After 4 months, Teena's share increases by 50%. i.e. \( \frac{7}{2} + \left(\frac{7}{2} \times \frac{50}{100}\right) = \frac{7}{2} + \frac{7}{4} \) i.e. \( \frac{21}{4} \) The profit will be shared in the proportion of product of capitals and respective time periods in months. i.e. \( \left(\frac{7}{2} \times 4 + \frac{21}{4} \times 8\right) : \left(\frac{4}{3} \times 12\right) : \left(\frac{6}{5} \times 12\right) \) i.e. \( (14 + 42) : 16 : \frac{72}{5} \) i.e. \( 56 : 16 : \frac{72}{5} \) i.e. \( 7 : 2 : \frac{9}{5} \) i.e. in the proportion 35 : 10 : 9 .....[Multiplying throughout by 5] Given that profit at the end of one year = Rs. 21,600/-
\( \therefore \) Leena's share in the profit \( = \frac{10}{54} \times 21,600 \)
\( = 5 \times 800 \)
\( = 4000 \)
\( \therefore \) Leena's share in the profit is Rs. 4000/-.
In simple words: Calculate the effective investment of each partner over the year, considering Teena's capital change. Then, determine the profit-sharing ratio based on these effective investments. Finally, use this ratio to find Leena's share of the total profit.

🎯 Exam Tip: When investments change mid-year, calculate the effective capital by summing (initial capital × initial duration) + (changed capital × remaining duration) for each partner.

 

Question 5.Dilip and Pradeep invested amounts in the ratio 2 : 1, whereas the ratio between amounts invested by Dilip and Sudip was 3 : 2. If Rs. 1,49,500 was their profit, how much amount did Sudip receive?
Answer:Solution: Let the amounts invested by Dilip, Pradeep and Sudip be 'd', 'p' and 's' respectively. Given that, d : p = 2 : 1 To make the ratio consistent, we find a common multiple for 'd'. Multiply the first ratio by 3:
\( \therefore \) d : p = 6 : 3 .....(i) and d : s = 3 : 2 Multiply the second ratio by 2:
\( \therefore \) d : s = 6 : 4 .....(ii) From (i) and (ii),
\( d : p : s = 6 : 3 : 4 \)
\( \therefore \) The ratio of profits to be shared among Dilip, Pradeep and Sudip will be 6 : 3 : 4. Given, profit earned = Rs. 1,49,500/- Total parts in ratio \( = 6 + 3 + 4 = 13 \)
\( \therefore \) Sudip's share in the profit \( = \frac{4}{13} \times 1,49,500 \)
\( = 4 \times 11,500 \)
\( = \text{Rs. } 46,000/- \)
In simple words: First, combine the two given ratios to find a single, consistent investment ratio for all three partners. Then, calculate Sudip's share by dividing his ratio part by the total ratio parts and multiplying by the total profit.

🎯 Exam Tip: When given multiple ratios involving common elements, find a common value for the shared element to combine them into a single, comprehensive ratio before calculating individual shares.

 

Question 6.The ratio of investments of two partners Jatin and Lalit is 11 : 12 and the ratio of their profits is 2: 3. If Jatin invested the money for 8 months, find for how much time Lalit invested his money.
Answer:Solution: Let 'x' be the time in months for which Lalit invested his money Jatin and Lalit invested their money in the ratio 11 : 12. Jatin invested his money for 8 months and the ratio of their profits is 2 : 3.
\( \therefore \) \( 11 \times 8 : 12 \times x = 2 : 3 \)
\( \implies \frac{88}{12x} = \frac{2}{3} \)
\( \implies 2 \times 12x = 88 \times 3 \)
\( \implies x = \frac{88 \times 3}{2 \times 12} \)
\( \implies x = 11 \)
\( \therefore \) Lalit invested his money for 11 months.
In simple words: Profit ratio is proportional to (investment x time). Set up an equation using the given investment ratios, time durations, and profit ratios, then solve for the unknown time Lalit invested his money.

🎯 Exam Tip: This type of problem often involves setting up a proportion where the ratio of (capital x time) for each partner equals their profit ratio.

 

Question 7.Three friends had dinner at a restaurant. When the bill was received, Alpana paid \( \frac{2}{3} \) as much as Beena paid and Beena paid \( \frac{1}{2} \) as much as Catherin paid. What fraction of the bill did Beena pay?
Answer:Solution: Let 'T' be the total bill amount at the restaurant and 'a', 'b', and 'c' be the share of Alpana, Beena, and Catherin respectively. Given, that Alpana paid \( \frac{2}{3} \) as much as Beena paid
\( \therefore a = \frac{2}{3} b \) ....(i) Also, Beena paid \( \frac{1}{2} \) as much as Catherin paid.
\( \implies b = \frac{1}{2} c \)
\( \therefore c = 2b \) .......(ii)
\( \therefore \) Three friends paid the total bill amount.
\( \therefore a + b + c = T \) .....(iii) Using (i) and (ii) in (iii), we get
\( \frac{2}{3} b + b + 2b = T \)
\( \implies b \left(\frac{2}{3} + 1 + 2\right) = T \)
\( \implies b \left(\frac{2+3+6}{3}\right) = T \)
\( \implies b \left(\frac{11}{3}\right) = T \)
\( \therefore b = \frac{3}{11} T \) Thus, Beena paid \( \left(\frac{3}{11}\right)^{\text{th}} \) fraction of the total bill amount.
In simple words: Express Alpana's and Catherin's payments in terms of Beena's payment. Then, substitute these expressions into the total bill equation and solve for Beena's payment as a fraction of the total bill.

🎯 Exam Tip: When dealing with inter-dependent fractions, express all quantities in terms of a single common variable to simplify the calculation of proportions.

 

Question 8.Roy starts a business with Rs. 10,000, Shikha joins him after 2 months with 20% more investment than Roy, after 2 months Tariq joins him with 40% less than Shikha. If the profit earned by them at the end of the year is equal to twice the difference between the investment of Roy and ten times the investment of Tariq. Find the profit of Roy?
Answer:Solution: Given that, Roy starts the business with Rs. 10,000. Shikha joins him after 2 months with 20% more investment than Roy.
\( \therefore \) Shikha's investment \( = 10,000 + \left(10,000 \times \frac{20}{100}\right) = 10,000 + 2,000 = \text{Rs. } 12,000 \) Tariq joins after two more months (total 4 months from start) with an investment 40% less than Shikha.
\( \therefore \) Tariq's investment \( = 12,000 - \left(12,000 \times \frac{40}{100}\right) = 12,000 - 4,800 = \text{Rs. } 7,200 \) Now, the profit will be shared in the proportion of product of capitals and respective periods in months. Roy's capital-time product: \( 10,000 \times 12 \) (12 months) Shikha's capital-time product: \( 12,000 \times 10 \) (joined after 2 months, so 10 months) Tariq's capital-time product: \( 7,200 \times 8 \) (joined after 4 months, so 8 months) i.e. \( 10,000 \times 12 : 12,000 \times 10 : 7,200 \times 8 \)
\( = 1,20,000 : 1,20,000 : 57,600 \) Divide throughout by 4,800: i.e. in the proportion, \( \frac{1,20,000}{4,800} : \frac{1,20,000}{4,800} : \frac{57,600}{4,800} = 25 : 25 : 12 \) .....(i) Given that, profit at the end of the year = twice of the difference between investment of Roy and ten times the investment of Tariq. Total profit \( = 2 \times [(10 \times \text{Tariq's Investment}) - \text{Roy's Investment}] \) (Assuming "investment of Roy" here refers to his initial capital as per context of difference)
\( \therefore \) Profit \( = 2 [(10 \times 7,200) - 10,000] \)
\( = 2[72,000 - 10,000] \)
\( = 2 \times 62,000 \)
\( = \text{Rs. } 1,24,000 \) Total parts in the profit ratio \( = 25 + 25 + 12 = 62 \)
\( \therefore \) Roy's share of profit \( = \frac{25}{62} \times 1,24,000 \) .....[From (i)]
\( = 25 \times 2,000 \)
\( = \text{Rs. } 50,000/- \)
In simple words: Calculate each partner's effective investment over the year. Determine the total profit using the given formula. Finally, allocate Roy's share based on the capital-time product ratio.

🎯 Exam Tip: Carefully track changing investments and the duration for which each capital amount is employed. The final profit distribution depends on the cumulative (investment × time) product for each partner.

 

Question 9.If 4(P's Capital) = 6(Q's Capital) = 10 (R's Capital), then out of the total profit of Rs. 5,580, what is R's share?
Answer:Solution: Let 'p', 'q' and 'r' be P, Q and R's Capital for the business respectively.
\( \therefore 4p = 6q = 10r \) L.C.M of 4, 6, 10 = 60 To find the ratios, we can set \( 4p = 6q = 10r = K \) (where K is a common multiple). For simplicity, let \( 4p = 6q = 10r = 60x \)
\( \implies 4p = 60x \implies p = 15x \)
\( \implies 6q = 60x \implies q = 10x \)
\( \implies 10r = 60x \implies r = 6x \)
\( \therefore p : q : r = 15x : 10x : 6x \)
\( \therefore p : q : r = 15 : 10 : 6 \) Given that total profit = Rs. 5580 Total parts in ratio \( = 15 + 10 + 6 = 31 \) R's share in the profit \( = \frac{6}{31} \times 5580 \)
\( = 6 \times 180 \)
\( = \text{Rs. } 1080/- \)
In simple words: First, use the given equality \( 4p = 6q = 10r \) to determine the ratio of capitals (p:q:r). Then, calculate R's share of the total profit based on this derived capital ratio.

🎯 Exam Tip: To find a ratio from an equality like \( Ax = By = Cz \), divide by the LCM of A, B, C or assume it equals a constant (e.g., K) to easily express x, y, z in terms of that constant.

 

Question 10.A and B start a business, with A investing the total capital of Rs. 50,000, on the condition that B pays interest at the rate of 10% per annum on his half of the capital. A is a working partner and receives Rs. 1,500 per month from the total profit and any profit remaining is equally shared by both of them. At the end of the year, it was found that the income of A is twice that of B. Find the total profit for the year?
Answer:Solution: Let 'x' and 'y' be the profits earned by A and B respectively and let 'z' be the total profit for the year. A is the working partner and receives Rs. 1500 per month from the total profit. i.e. \( 12 \times 1500 = \text{Rs. } 18,000 \) at the end of the year. The remaining profit is shared between A and B equally. Remaining profit \( = z - 18,000 \) B's share from remaining profit \( = \frac{z - 18,000}{2} \)
\( \therefore y = \frac{z - 18,000}{2} \) .....(i) Thus, profit earned by A at the end of that year is given by A's share from remaining profit \( = \frac{z - 18,000}{2} \) A's total profit component \( x = 18000 + \left(\frac{z - 18,000}{2}\right) \)
\( \therefore x = \frac{36000 + z - 18000}{2} \)
\( \implies x = \frac{z + 18000}{2} \) .....(ii) A invests the entire capital of Rs. 50,000. Half of the capital is Rs. 25,000. B pays A interest at the rate of 10% per annum on his half of the capital.
\( \therefore \) At the end of the first year, A will receive \( \frac{10}{100} \times 25,000 \) i.e. Rs. 2500/- over and above his share of profit.
\( \therefore \) A's income = Profit of A + 2500 \( = x + 2500 \) Given that, income of A = twice the income of B Note: B's income is just the profit 'y' because he pays interest, not receives it.
\( \therefore x + 2500 = 2y \) .....(iii) Using (i) and (ii) in (iii), we get
\( \frac{z + 18000}{2} + 2500 = 2 \left(\frac{z - 18000}{2}\right) \)
\( \frac{z + 18000}{2} + 2500 = z - 18000 \) Multiply by 2:
\( z + 18000 + 5000 = 2(z - 18000) \)
\( z + 23000 = 2z - 36000 \)
\( 23000 + 36000 = 2z - z \)
\( 59000 = z \)
\( \therefore z = 59,000 \)
\( \therefore \) The total profit for the year = Rs. 59,000/-
In simple words: First, calculate A's fixed income and B's share from the remaining profit in terms of total profit 'z'. Then, account for the interest A receives from B. Finally, set up an equation where A's total income is twice B's total income and solve for the total profit 'z'.

🎯 Exam Tip: Break down partnership problems into individual income components: fixed salaries, profit shares, and interest adjustments. Ensure each partner's final income is correctly calculated before applying the given relationships.