Chapter 1 Commission, Brokerage and Discount : Mathematics and Statistics (Commerce) Part - II STANDARD TWELVE

Mukund Pathak June 18, 2023



Mathematics and Statistics 
(Commerce) Part - II 
STANDARD TWELVE 12th


Chapter 1
Commission, Brokerage and Discount

                                                                    EXERCISE 1.1 


Ans: To calculate the commission earned by the agent and the amount the seller receives, we need to determine the commission amount and the remaining amount after deducting the commission.

Commission amount:
The agent charges a commission of 12% on the total sale amount. To find the commission, we multiply the total sale amount by 12% (or 0.12):

Commission = 48,000 * 0.12
           = Rs. 5,760

Therefore, the agent earns Rs. 5,760 as commission.

Amount received by the seller:
To find the amount received by the seller, we subtract the commission amount from the total sale amount:

Amount received by the seller = Total sale amount - Commission
                            = 48,000 - 5,760
                            = Rs. 42,240

Therefore, the seller receives Rs. 42,240 after deducting the agent's commission.


Ans: To calculate the salesman's total income on a sale of Rs. 200,000, we need to determine the commission earned on the sales up to Rs. 50,000 and the commission earned on the sales over Rs. 50,000, and then sum them up.

Commission on sales up to Rs. 50,000:
The salesman receives a 3% commission on the sales up to Rs. 50,000. To find the commission, we multiply Rs. 50,000 by 3% (or 0.03):

Commission on sales up to Rs. 50,000 = 50,000 * 0.03
                                    = Rs. 1,500

Commission on sales over Rs. 50,000:
The salesman receives a 4% commission on the sales over Rs. 50,000. In this case, the sales over Rs. 50,000 amount to Rs. 200,000 - Rs. 50,000 = Rs. 150,000. To find the commission, we multiply Rs. 150,000 by 4% (or 0.04):

Commission on sales over Rs. 50,000 = 150,000 * 0.04
                                   = Rs. 6,000

Total income of the salesman:
To find the total income of the salesman, we sum up the commission on sales up to Rs. 50,000 and the commission on sales over Rs. 50,000:

Total income = Commission on sales up to Rs. 50,000 + Commission on sales over Rs. 50,000
            = 1,500 + 6,000
            = Rs. 7,500

Therefore, the salesman's total income on the sale of Rs. 200,000 is Rs. 7,500.


Ams: To find the number of computers Ms. Saraswati sold, we need to divide the total commission she earned by the commission rate per computer.

Commission rate per computer:
The commission rate is given as 12.5% or 0.125. To find the commission rate per computer, we multiply the price of each computer (Rs. 32,000) by 0.125:

Commission rate per computer = Rs. 32,000 * 0.125
                            = Rs. 4,000

Number of computers sold:
To find the number of computers sold, we divide the total commission earned (Rs. 88,000) by the commission rate per computer (Rs. 4,000):

Number of computers sold = Rs. 88,000 / Rs. 4,000
                        = 22

Therefore, Ms. Saraswati sold 22 computers.


Ans" Let's assume the total sales made by Anita to be "x" rupees.

Commission on the total sales:
Anita is allowed a commission of 6.5% on the total sales. To find the commission, we multiply the total sales by 6.5% (or 0.065):

Commission on the total sales = x * 0.065

Bonus commission on sales over Rs. 20,000:
Anita receives a bonus of 12% on the sales amount exceeding Rs. 20,000. Therefore, the bonus commission can be calculated by multiplying the amount exceeding Rs. 20,000 by 12% (or 0.12):

Bonus commission = (x - 20,000) * 0.12

Total commission:
The total commission earned by Anita is given as Rs. 3,400. Thus, we can set up the following equation:

Commission on the total sales + Bonus commission = Total commission
(x * 0.065) + ((x - 20,000) * 0.12) = 3,400

Now, we can solve this equation to find the value of "x" (the sales made by Anita).

Solving the equation:
0.065x + 0.12(x - 20,000) = 3,400
0.065x + 0.12x - 2,400 = 3,400
0.185x = 5,800
x = 5,800 / 0.185
x ≈ Rs. 31,351.35

Therefore, Anita made sales of approximately Rs. 31,351.35. 


Ans: To find the sales made by Priya in a certain month, we need to determine the commission earned on sales over Rs. 50,000 and then calculate the total sales amount.

Salary:
Priya receives a fixed salary of Rs. 15,000 per month.

Commission on sales over Rs. 50,000:
Priya receives an 8% commission on the sales amount exceeding Rs. 50,000. Let's assume the sales over Rs. 50,000 to be "x" rupees. Therefore, the commission earned on the sales over Rs. 50,000 can be calculated as follows:

Commission on sales over Rs. 50,000 = x * 0.08

Total income in the certain month:
Priya's total income in the certain month is given as Rs. 17,400, which includes both her salary and commission. Thus, we can set up the following equation:

Salary + Commission on sales over Rs. 50,000 = Total income
15,000 + (x * 0.08) = 17,400

Now, we can solve this equation to find the value of "x" (the sales made by Priya over Rs. 50,000).

Solving the equation:
15,000 + 0.08x = 17,400
0.08x = 17,400 - 15,000
0.08x = 2,400
x = 2,400 / 0.08
x = 30,000

Therefore, Priya made sales over Rs. 50,000 amounting to Rs. 30,000 in that certain month.



Ans: Let's assume Mr. Pavan's weekly salary to be "S" rupees and the commission rate to be "C" (expressed as a decimal).

In the first week:
Total sales = Rs. 68,000
Total income (salary + commission) = Rs. 9,880

The income can be expressed as:
Income = Salary + Commission

Substituting the values:
9,880 = S + (0.01C * 68,000)

In the second week:
Total sales = Rs. 73,000
Total income (salary + commission) = Rs. 10,180

Again, the income can be expressed as:
Income = Salary + Commission

Substituting the values:
10,180 = S + (0.01C * 73,000)

We now have a system of two equations with two variables (S and C).

Solving the system of equations will give us the values of the weekly salary (S) and the commission rate (C).

Subtracting the first equation from the second equation:

10,180 - 9,880 = S + (0.01C * 73,000) - (S + 0.01C * 68,000)
300 = 0.01C * 73,000 - 0.01C * 68,000
300 = 0.01C * (73,000 - 68,000)
300 = 0.01C * 5,000
C = 300 / 50
C = 6

Substituting the value of C back into the first equation:

9,880 = S + (0.01 * 6 * 68,000)
9,880 = S + 4,080
S = 9,880 - 4,080
S = 5,800

Therefore, Mr. Pavan's weekly salary is Rs. 5,800, and the rate of commission paid to him is 6%.


Ans: Let's assume Deepak's sales to be "S" rupees.

Initially, Deepak's salary was Rs. 4,000, and the commission rate was 3%. Therefore, his income can be calculated as follows:

Income = Salary + Commission
Income = 4,000 + (0.03 * S)
Income = 4,000 + 0.03S

After the salary increase to Rs. 5,000 and the reduction in the commission rate to 2%, his income remained unchanged. Therefore, his income can still be calculated as follows:

Income = Salary + Commission
Income = 5,000 + (0.02 * S)
Income = 5,000 + 0.02S

Since his income remained unchanged, we can equate the two expressions for income:

4,000 + 0.03S = 5,000 + 0.02S

Now, let's solve this equation to find the value of "S" (his sales).

0.03S - 0.02S = 5,000 - 4,000
0.01S = 1,000
S = 1,000 / 0.01
S = 100,000

Therefore, Deepak's sales amount to Rs. 100,000.

10. Three cars were sold through an agent for Rs.2,40,000 , Rs.2,22,000 and Rs.2,25,000 respectively. The rates of commission were 17.5% on the first, 12.5% on the second. If the agent overall received 14% commission on the total sales, find the rate of commission paid on the third car. 

Ans: Let's assume the rate of commission paid on the third car to be "R" (expressed as a decimal).

The total sales amount is the sum of the sales amounts of the three cars:

Total sales = Rs. 2,40,000 + Rs. 2,22,000 + Rs. 2,25,000
            = Rs. 6,87,000

The total commission received by the agent is 14% of the total sales amount:

Total commission = 0.14 * Rs. 6,87,000
                 = Rs. 95,580

Now, let's calculate the commission earned on the first and second cars:

Commission on the first car = 17.5% of Rs. 2,40,000
                           = 0.175 * Rs. 2,40,000
                           = Rs. 42,000

Commission on the second car = 12.5% of Rs. 2,22,000
                            = 0.125 * Rs. 2,22,000
                            = Rs. 27,750

To find the commission on the third car, we subtract the sum of the commissions on the first two cars from the total commission:

Commission on the third car = Total commission - (Commission on the first car + Commission on the second car)
                          = Rs. 95,580 - (Rs. 42,000 + Rs. 27,750)
                          = Rs. 25,830

Now, we can determine the rate of commission paid on the third car by dividing the commission by the sales amount of the third car:

Rate of commission on the third car = Commission on the third car / Rs. 2,25,000
                                   = Rs. 25,830 / Rs. 2,25,000
                                   ≈ 0.1148 (or 11.48%)

Therefore, the rate of commission paid on the third car is approximately 11.48%.

11. Swatantra Distributors allows 15% discount on the list price of washing machine. Further 5% discount is given for cash payment. Find the list price of the washing machine if it was sold for the net amount of Rs. 38356.25.

Ans: Let's assume the list price of the washing machine to be "L" rupees.

First, we calculate the price after the 15% discount:

Price after 15% discount = L - (0.15 * L)
                        = L - 0.15L
                        = 0.85L

Next, we calculate the price after the additional 5% discount for cash payment:

Price after additional 5% discount = 0.85L - (0.05 * 0.85L)
                                  = 0.85L - 0.0425L
                                  = 0.8075L

According to the given information, the net amount the washing machine was sold for is Rs. 38,356.25. Therefore, we can set up the following equation:

0.8075L = 38,356.25

Now, we can solve this equation to find the value of "L" (the list price of the washing machine).

Solving the equation:
L = 38,356.25 / 0.8075
L ≈ Rs. 47,550

Therefore, the list price of the washing machine is approximately Rs. 47,550. 

12. A book seller received Rs.1,530 as 15% commission on list price. Find list price of the books. 

Ans: To find the list price of the books, we need to calculate the total commission earned by the book seller and then divide it by the commission rate.

Let's assume the list price of the books to be "L" rupees.

The commission earned by the book seller is given as Rs. 1,530, and it represents 15% of the list price. Therefore, we can set up the following equation:

Commission = 0.15 * List price

Substituting the given values:

1,530 = 0.15 * L

To find the value of L (the list price), we can divide both sides of the equation by 0.15:

L = 1,530 / 0.15

L = 10,200

Therefore, the list price of the books is Rs. 10,200.

13. A retailer sold a suit for Rs.8,832 after allowing 8% discount on marked price and further 4% cash discount. If he made 38% profit, find the cost price and the marked price of the suit. 

Ans: Let's assume the cost price of the suit to be "C" rupees and the marked price to be "M" rupees.

Discount on marked price:
The retailer allowed an 8% discount on the marked price. Therefore, the selling price after the discount can be calculated as follows:

Selling price after 8% discount = M - (0.08 * M)
                               = M - 0.08M
                               = 0.92M

Cash discount:
Further, a 4% cash discount is given. The selling price after the cash discount can be calculated as follows:

Selling price after 4% cash discount = 0.92M - (0.04 * 0.92M)
                                    = 0.92M - 0.0368M
                                    = 0.8832M

According to the given information, the selling price of the suit is Rs. 8,832. Therefore, we can set up the following equation:

0.8832M = 8,832

Now, we can solve this equation to find the value of "M" (the marked price).

Solving the equation:
M = 8,832 / 0.8832
M ≈ Rs. 10,000

The marked price of the suit is approximately Rs. 10,000.

Profit percentage:
The profit percentage is given as 38%. Therefore, the profit earned can be calculated as a percentage of the cost price:

Profit = 0.38 * C

Selling price:
The selling price is Rs. 8,832, which is equal to the cost price plus the profit:

Selling price = Cost price + Profit
8,832 = C + 0.38C
8,832 = 1.38C

Now, we can solve this equation to find the value of "C" (the cost price).

Solving the equation:
C = 8,832 / 1.38
C ≈ Rs. 6,400

Therefore, the cost price of the suit is approximately Rs. 6,400.

In summary:
- The marked price of the suit is Rs. 10,000.
- The cost price of the suit is approximately Rs. 6,400.

14. An agent charges 10% commission plus 2% delcreder. If he sells goods worth Rs.37,200, find his total earnings. 

Ans: To calculate the agent's total earnings, we need to find the commission amount and the del credere amount separately, and then add them together.

Given:
Total sales amount = Rs. 37,200

Commission rate = 10%
Del credere rate = 2%

Commission amount = Commission rate * Total sales amount
Commission amount = 0.10 * Rs. 37,200

Del credere amount = Del credere rate * Total sales amount
Del credere amount = 0.02 * Rs. 37,200

Now, we can calculate the agent's total earnings by adding the commission amount and the del credere amount:

Total earnings = Commission amount + Del credere amount

Substituting the values:

Total earnings = (0.10 * Rs. 37,200) + (0.02 * Rs. 37,200)

Total earnings = Rs. (3,720 + 744)

Total earnings = Rs. 4,464

Therefore, the agent's total earnings are Rs. 4,464.

15. A whole seller allows 25% trade discount and 5% cash discount. What will be the net price of an article marked at Rs. 1600.

Ans: To calculate the net price of the article, we need to apply the trade discount and the cash discount to the marked price.

Given:
Marked price = Rs. 1600
Trade discount rate = 25%
Cash discount rate = 5%

Trade discount amount = Trade discount rate * Marked price
Trade discount amount = 0.25 * Rs. 1600

Cash discount amount = Cash discount rate * (Marked price - Trade discount amount)
Cash discount amount = 0.05 * (Rs. 1600 - Trade discount amount)

Now, we can calculate the net price of the article by subtracting the trade discount amount and the cash discount amount from the marked price:

Net price = Marked price - Trade discount amount - Cash discount amount

Substituting the values:

Net price = Rs. 1600 - (0.25 * Rs. 1600) - (0.05 * (Rs. 1600 - Trade discount amount))

Simplifying the expression:

Net price = Rs. 1600 - 0.25 * Rs. 1600 - 0.05 * Rs. 1600 + 0.05 * Trade discount amount

Net price = Rs. 1600 - Rs. 400 - Rs. 80 + 0.05 * Trade discount amount

Net price = Rs. 1120 + 0.05 * Trade discount amount

To calculate the net price, we need to know the value of the trade discount amount. However, it is not provided in the given information. If the trade discount amount is known, we can substitute its value into the equation to calculate the net price.

Abbreviations: 
Present Worth : P.W. or P 
Sum Due/Face Value : S.D. or F.V. 
True Discount : T.D.
Banker’s Gain : B.G. 
Banker’s Discount : B.D. 
Cash Value : C.V. 
Notation Period (in Years) : n Rate of Interest (p.a.) : r 

List of Formula: 
(1) S.D. = P.W. + T.D. 
(2) T.D. = P.W.× ×n r 100 
(3) B.D. = S.D.× ×n r 100 
(4) B.G. = B.D. − T.D. 
(5) B.G. = T.D.× ×n r 100 
(6) Cash value = S.D. − B.D

EXERCISE 1.2 

Ans: To calculate the present worth of a sum due in the future, we can use the formula for simple interest:

Simple Interest = Principal * Interest Rate * Time

Given:
Principal (sum due in the future) = Rs. 10,920
Interest Rate = 8% p.a.
Time = 6 months

First, let's convert the time from months to years:
Time in years = 6 months / 12 months = 0.5 years

Now, we can calculate the simple interest:
Simple Interest = Principal * Interest Rate * Time
Simple Interest = Rs. 10,920 * 0.08 * 0.5

Next, we can calculate the present worth by subtracting the simple interest from the principal (sum due in the future):
Present Worth = Principal - Simple Interest
Present Worth = Rs. 10,920 - (Rs. 10,920 * 0.08 * 0.5)

Simplifying the expression:
Present Worth = Rs. 10,920 - (Rs. 436.80)

Present Worth = Rs. 10,483.20

Therefore, the present worth of a sum of Rs. 10,920 due six months hence at 8% p.a. simple interest is Rs. 10,483.20.


Ans: To calculate the sum due in the future, we can use the formula for simple interest:

Simple Interest = Principal * Interest Rate * Time

Given:
Principal (initial sum) = Rs. 8,000
Interest Rate = 12.5% p.a.
Time = 4 months

First, let's convert the time from months to years:
Time in years = 4 months / 12 months = 1/3 years (or approximately 0.333 years)

Now, we can calculate the simple interest:
Simple Interest = Principal * Interest Rate * Time
Simple Interest = Rs. 8,000 * 0.125 * 0.333

Next, we can calculate the sum due in the future by adding the simple interest to the principal:
Sum Due = Principal + Simple Interest
Sum Due = Rs. 8,000 + (Rs. 8,000 * 0.125 * 0.333)

Simplifying the expression:
Sum Due = Rs. 8,000 + (Rs. 333.33)

Sum Due = Rs. 8,333.33

Therefore, the sum due of Rs. 8,000 due 4 months hence at 12.5% simple interest is Rs. 8,333.33.


Ans: To find the sum due and present worth of the bill, we need to calculate the true discount and use the formulas related to it.

Given:
True Discount = Rs. 560
Time = 8 months
Interest Rate = 12% p.a.

Let's denote the sum due as "S" and the present worth as "P".

The formula for true discount is given by:
True Discount = S - P

Since the interest rate and time are given, we can use the formula for simple interest to relate the sum due and present worth:
Simple Interest = Principal * Interest Rate * Time

Therefore, we have:
True Discount = Principal * Interest Rate * Time
Rs. 560 = S * 0.12 * (8/12)

Simplifying the expression:
560 = 0.08S

Dividing both sides by 0.08:
S = 560 / 0.08
S = Rs. 7,000

Now, we can calculate the present worth using the formula:
Present Worth = Sum Due - True Discount
P = S - Rs. 560
P = Rs. 7,000 - Rs. 560
P = Rs. 6,440

Therefore, the sum due on the bill is Rs. 7,000 and the present worth of the bill is Rs. 6,440.


Ans: To find the period of the bill, we can use the formula for true discount and manipulate it to isolate the time variable.

Given:
True Discount = 3/8 of the sum due
Interest Rate = 12% p.a.

Let's denote the sum due as "S" and the period of the bill as "t" (in years).

The formula for true discount is given by:
True Discount = S * Interest Rate * Time

Since the true discount is given as 3/8 of the sum due, we can write:
True Discount = (3/8) * S

Setting these two equations equal to each other, we get:
(3/8) * S = S * Interest Rate * Time

Simplifying the equation, we can cancel out the "S" term on both sides:
3/8 = Interest Rate * Time

Now, we can isolate the time variable:
Time = (3/8) / Interest Rate

Substituting the given interest rate (12% = 0.12), we have:
Time = (3/8) / 0.12

Calculating the value:
Time = (3/8) / 0.12
Time = (3/8) * (1/0.12)
Time = (3/8) * (25/3)
Time = 25/8
Time = 3.125

Therefore, the period of the bill is approximately 3.125 years or 3 years and 1.5 months.


Ans: Let's assume the cost per copy of the book is "x" and the rate of interest per annum is "r".

According to the given information:
For purchasing 20 copies payable at the end of 6 months, the total amount payable is 20x.
For purchasing 21 copies in ready cash, the total amount payable is 21x.

We can set up the following equation based on the concept of simple interest:

21x = 20x + (20x * r * (6/12))

Simplifying the equation:
21x = 20x + 10xr
21x - 20x = 10xr
x = 10xr

Dividing both sides by "xr" (assuming r is not equal to zero):
1 = 10

The equation 1 = 10 is not possible and leads to a contradiction. This suggests that there is no consistent solution for the given information.

It seems there might be an error in the problem statement or some crucial information might be missing. Please double-check the information provided or provide any additional details if available.


Ans: To find the true discount, banker's discount, and banker's gain on a bill, we need to use the formulas and concepts related to simple interest and discounts.

Given:
Principal (bill amount) = Rs. 4,240
Time = 6 months
Interest Rate = 9% p.a.

First, let's calculate the true discount using the formula:
True Discount = Principal * Interest Rate * Time
True Discount = Rs. 4,240 * 0.09 * (6/12)
True Discount = Rs. 191.40

Next, we can calculate the banker's discount using the formula:
Banker's Discount = (True Discount * 100) / (100 + (Interest Rate * Time))
Banker's Discount = (191.40 * 100) / (100 + (0.09 * 6))
Banker's Discount = Rs. 180

Now, we can calculate the banker's gain using the formula:
Banker's Gain = Banker's Discount - True Discount
Banker's Gain = Rs. 180 - Rs. 191.40
Banker's Gain = -Rs. 11.40

The negative value of the banker's gain indicates a loss for the banker.

Therefore, the true discount on the bill of Rs. 4,240 due 6 months hence at 9% p.a. is Rs. 191.40, the banker's discount is Rs. 180, and the banker's gain is -Rs. 11.40.


Ans: To find the rate of interest, we can use the formulas and concepts related to true discount and banker's discount.

Given:
True Discount = Rs. 2,200
Banker's Discount = Rs. 2,310
Time = 10 months

Let's denote the rate of interest per annum as "r".

The formula for true discount is given by:
True Discount = Principal * Interest Rate * Time

Substituting the given values, we have:
Rs. 2,200 = Principal * r * (10/12)

Simplifying the equation, we get:
Principal * r = (2,200 * 12) / 10
Principal * r = Rs. 2,640

Similarly, the formula for banker's discount is given by:
Banker's Discount = (True Discount * 100) / (100 + (Interest Rate * Time))

Substituting the given values, we have:
Rs. 2,310 = (2,200 * 100) / (100 + (r * 10))

Simplifying the equation, we get:
(2,200 * 100) / (100 + 10r) = 2,310

Cross-multiplying and simplifying further, we have:
220,000 = 2,310 * (100 + 10r)
220,000 = 231,000 + 23,100r

Rearranging the terms, we get:
23,100r = 231,000 - 220,000
23,100r = 11,000

Dividing both sides by 23,100:
r = 11,000 / 23,100
r ≈ 0.4762

Therefore, the approximate rate of interest is 0.4762, which is equivalent to 47.62%.


Ans: To calculate the banker's discount and the cash value of the bill, we need to consider the time between the discounting date and the maturity date.

Given:
Principal (bill amount) = Rs. 6,935
Discounting date = 28th February 2015
Maturity date = 19th September 2015
Interest Rate = 8% p.a.

First, let's calculate the time between the discounting date and the maturity date:
Time = 19th September 2015 - 28th February 2015

To calculate the time in days, we consider each month to have 30 days:
Time = (7 months * 30 days/month) + (22 days)
Time = 210 days + 22 days
Time = 232 days

Now, we can calculate the banker's discount using the formula:
Banker's Discount = (Principal * Interest Rate * Time) / (365 days)
Banker's Discount = (Rs. 6,935 * 0.08 * 232) / 365
Banker's Discount ≈ Rs. 3,502.90

The cash value of the bill is calculated by subtracting the banker's discount from the principal:
Cash Value = Principal - Banker's Discount
Cash Value = Rs. 6,935 - Rs. 3,502.90
Cash Value ≈ Rs. 3,432.10

Therefore, the banker's discount is approximately Rs. 3,502.90, and the cash value of the bill is approximately Rs. 3,432.10.

9. A bill of Rs.8,000 drawn on 5th January 1998 for 8 months was discounted for Rs.7,680 on a certain date. Find the date on which it was discounted at 10% p.a. 

Ans: To find the date on which the bill was discounted, we can use the formula for banker's discount and work backward to determine the date.

Given:
Principal (bill amount) = Rs. 8,000
Banker's Discount = Rs. 7,680
Time = 8 months
Interest Rate = 10% p.a.

Let's denote the discounting date as "D" and the maturity date as "M".

Using the formula for banker's discount:
Banker's Discount = Principal * Interest Rate * Time
Rs. 7,680 = Rs. 8,000 * 0.10 * (8/12)

Simplifying the equation, we have:
Rs. 7,680 = Rs. 6,666.67

Since the banker's discount is less than the principal, it indicates that the bill was discounted before the maturity date.

Now, we need to calculate the number of days between the discounting date and the maturity date.

Using the concept that each month has approximately 30.44 days, we can estimate the time in days as:
Time = 8 months * 30.44 days/month
Time ≈ 243.52 days

To determine the discounting date, we need to subtract the time in days from the maturity date.

Discounting date = Maturity date - Time
Discounting date = 5th January 1998 - 243.52 days

To perform the subtraction, we need to consider the number of days in each month and the leap years.

Calculating the discounting date requires a more detailed calculation, taking into account the number of days in each month, leap years, and the specific dates involved. However, without additional information, it is not possible to provide an exact date for the discounting.

10. A bill drawn on 5th June for 6 months was discounted at the rate of 5% p.a. on 19th October. If the cash value of the bill is Rs 43,500, find face value of the bill. 

Ans: To find the face value of the bill, we can use the formula for cash value and work backward to determine the original amount.

Given:
Cash Value of the bill = Rs. 43,500
Discounting Rate = 5% p.a.
Discounting Date = 19th October
Time = 6 months

Let's denote the face value of the bill as "FV".

Using the formula for cash value:
Cash Value = Face Value - (Face Value * Discounting Rate * Time)
Rs. 43,500 = FV - (FV * 0.05 * (6/12))

Simplifying the equation, we have:
Rs. 43,500 = FV - (FV * 0.025)

Rearranging the terms, we get:
Rs. 43,500 = FV * (1 - 0.025)

Dividing both sides by (1 - 0.025):
FV = Rs. 43,500 / (1 - 0.025)
FV = Rs. 43,500 / 0.975
FV ≈ Rs. 44,615.38

Therefore, the face value of the bill is approximately Rs. 44,615.38.

11. A bill was drawn on 14th April for Rs.7,000 and was discounted on 6th July at 5% p.a. The Banker paid Rs.6,930 for the bill. Find period of the bill. 

Ans: To find the period of the bill, we can use the formula for banker's discount and rearrange it to solve for time.

Given:
Principal (bill amount) = Rs. 7,000
Banker's Discount = Rs. 6,930
Interest Rate = 5% p.a.

Let's denote the time (period of the bill) as "t".

Using the formula for banker's discount:
Banker's Discount = Principal * Interest Rate * Time
Rs. 6,930 = Rs. 7,000 * 0.05 * (t/12)

Simplifying the equation, we have:
Rs. 6,930 = Rs. 350 * t/12

Multiplying both sides by 12 and dividing by 350, we get:
t = (6,930 * 12) / 350
t ≈ 238.29

Therefore, the period of the bill is approximately 238.29 days.

12. If difference between true discount and banker’s discount on a sum due 4 months hence is Rs 20. Find true discount, banker’s discount and amount of bill, the rate of simple interest charged being 5%p.a. 

Ans: To find the true discount, banker's discount, and the amount of the bill, we need to consider the relationship between true discount, banker's discount, and simple interest.

Given:
Difference between true discount and banker's discount = Rs. 20
Time = 4 months
Rate of simple interest = 5% p.a.

Let's denote the true discount as "TD," the banker's discount as "BD," and the amount of the bill as "AB."

The relationship between true discount, banker's discount, and simple interest is as follows:
True Discount = Banker's Discount + (Principal * Interest Rate * Time)

We are given the difference between the true discount and the banker's discount, which is Rs. 20. Therefore, we can write the equation as:
TD - BD = 20

Substituting the given values, we have:
TD - BD = 20
TD - (BD + (Principal * Interest Rate * Time)) = 20
TD - (BD + (Principal * 0.05 * (4/12))) = 20

Simplifying the equation, we have:
TD - (BD + (Principal * 0.05 * (1/3))) = 20
TD - BD - (Principal * 0.05 * (1/3)) = 20

Since the rate of interest is 5% p.a., the rate for 4 months (1/3 of a year) is 1.67%.

Now, let's solve for TD, BD, and AB simultaneously.

We know that:
TD = BD + (Principal * 0.05 * (1/3))

Since TD - BD = 20, we can substitute the value of TD - BD into the equation:
20 = Principal * 0.05 * (1/3)

Simplifying the equation, we have:
20 = Principal * 0.0167
Principal = 20 / 0.0167

Calculating the value of Principal, we have:
Principal ≈ 1,197.60

Now, let's substitute the value of Principal back into the equation for TD:
TD = BD + (Principal * 0.05 * (1/3))
TD = BD + (1,197.60 * 0.05 * (1/3))

Simplifying the equation, we have:
TD = BD + 19.96

Since TD - BD = 20, we can write:
TD = BD + 20

Substituting this equation into the previous equation, we have:
BD + 20 = BD + 19.96

This equation is true, so we have a consistent solution.

Therefore, the true discount is approximately Rs. 20, the banker's discount is approximately Rs. 0.04, and the amount of the bill is approximately Rs. 1,197.60. 

13. A bill of Rs.51,000 was drawn on 18th February 2010 for 9 months. It was encashed on 28th June 2010 at 5% p.a. Calculate the banker’s gain and true discount. 

Ans: To calculate the banker's gain and true discount, we need to know the discounting rate and the time between the drawing date and the encashment date. 

Given:
Principal (bill amount) = Rs. 51,000
Drawing date = 18th February 2010
Encashment date = 28th June 2010
Rate of interest = 5% p.a.

First, let's calculate the time between the drawing date and the encashment date:

18th February 2010 to 18th May 2010 = 3 months
18th May 2010 to 18th June 2010 = 1 month
18th June 2010 to 28th June 2010 = 10 days

Total time = 3 months + 1 month + 10 days = 4 months + 10 days

Next, let's calculate the banker's gain:

Banker's Gain = Principal * Rate of interest * Time
Banker's Gain = Rs. 51,000 * 0.05 * (4 + 10/30)
Banker's Gain = Rs. 51,000 * 0.05 * (4.33)
Banker's Gain ≈ Rs. 931.05

Finally, let's calculate the true discount:

True Discount = Principal - Banker's Gain
True Discount = Rs. 51,000 - Rs. 931.05
True Discount ≈ Rs. 50,068.95

Therefore, the banker's gain is approximately Rs. 931.05 and the true discount is approximately Rs. 50,068.95.

14. A certain sum due 3 months hence is 21 20 of the present worth, what is the rate of interest? 

Ans: To find the rate of interest, we can use the formula for calculating the present worth of a sum due in the future.

Given:
Sum due = 21/20 times the present worth
Time = 3 months

Let's denote the present worth as "PW" and the rate of interest as "R".

We know that:
Sum due = PW + (PW * R * Time)

We are given that the sum due is 21/20 times the present worth. Therefore, we can write the equation as:
Sum due = (21/20) * PW

Substituting the given values, we have:
(21/20) * PW = PW + (PW * R * (3/12))

Simplifying the equation, we have:
(21/20) * PW = PW + (PW * R * (1/4))

Now, let's solve for the rate of interest (R).

Dividing both sides of the equation by PW and rearranging terms, we get:
(21/20) = 1 + (R * (1/4))
(21/20) - 1 = R/4
(21/20) - (20/20) = R/4
1/20 = R/4

Multiplying both sides by 4, we have:
4/20 = R
R = 0.20

Therefore, the rate of interest is 0.20 or 20%.

15. A bill of a certain sum drawn on 28th February 2007 for 8 months was encashed on 26th March 2007 for Rs. 10,992 at 14% p.a. Find the face value of the bill. 

Ans: To find the face value of the bill, we can use the formula for calculating the present worth of a bill.

Given:
Encashment date: 26th March 2007
Face value of the bill: Unknown
Rate of interest: 14% p.a.
Time: 8 months

Let's denote the face value of the bill as "FV".

Using the formula for present worth:
Present Worth = Face Value / (1 + Rate of Interest * Time)

We are given that the present worth (encashment amount) is Rs. 10,992, and the rate of interest is 14% p.a., and the time is 8 months. We can write the equation as follows:
10,992 = FV / (1 + 0.14 * (8/12))

Simplifying the equation, we have:
10,992 = FV / (1 + 0.09333)

Multiplying both sides of the equation by (1 + 0.09333), we get:
10,992 * (1 + 0.09333) = FV

Calculating the value, we have:
10,992 * 1.09333 = FV
FV ≈ 12,013.96

Therefore, the face value of the bill is approximately Rs. 12,013.96.


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