Partnership and Discount

Question: Three Hikers A, B and C start on a trip with Tk. 50 each and agree to share the expenses equally. If at the end of the trip. A has Tk. 20 left with him, B Tk.30 and C Tk. 40 how must they settle their accounts?

Solution:
Expenses of A = 50 - 20 = 30
Expenses of B = 50 - 30 = 20
Expenses of C = 50 - 40 = 10

Total expenses = 30 + 20 + 10 = 60
Each person's share = 60/3 = 20
Since A overpaid by Tk. 10 and C underpaid by Tk. 10, C should pay Tk. 10 to A to balance the expenses.

Question: A girl bought a Nepenthes plant for Tk. 1600 after getting a 20% discount. What was the original catalog price of the plant?

Solution:
20% discount,
If the catalog price is Tk. 100 then the purchased price = (100 - 20)
= Tk. 80

Now,
If the purchase price is Tk. 80, the catalog price = 100 taka.
∴ If the purchase price is Tk. 1, the catalog price = 100/80 taka
∴ If the purchase price is Tk. 1600, the catalog price = (100 × 1600)/80
= Tk. 2000

∴ the original catalog price of the plant is  Tk. 2000

Question: A, B, C enter into a partnership investing Tk. 40,000, Tk. 50,000 and Tk. 60,000 respectively. Total profit of A, B, C is Tk. 45,000. So B's profit is-

Solution:
A : B : C = 40000 : 50000 : 60000
A : B : C = 4 : 5 : 6

∴ Sum of the ratio = 4 + 5 + 6 = 15

∴ B's profit = 45000 × (5/15)
= 45000 × (1/3)
= 15,000

∴ B's profit is Tk. 15,000

Question: A shopkeeper offers two successive discounts of 10% and 20% on an item marked at Tk. 1,200. What is the final selling price?

Solution: 
New Price after First Discount : 
1200 - 10% of 1200
= 1200 - 120
= 1080 Taka

Final Selling Price after Second Discount:
1080 - 20% of 1080
= 1080 - 216
= 864

So, Final Selling Price = Tk. 864

Let the total capital be Tk. x.
Then, Rahul's share = Tk. x/2
Amin's share = Tk. x/3
Akash's share = [x - {(x/2) + (x/3)}]
= Tk. x/6
∴ Required ratio = x/2 : x/3 : x/6 = 1/2 : 1/3 : 1/6
= 3 : 2 : 1

Question: Karim started a business with Tk. 80,000. After some months, Rahim joined with Tk. 60,000. At the end of the year, the profit was divided in the ratio 8 : 3. For how many months was Rahim in the business?

Solution:
Let, Rahim joined for x months.

ATQ,
(80,000 × 12)/(60,000 × x) = 8/3
⇒ (80,000 × 12 × 3) = (60,000 × x × 8)
⇒ 2,880,000 = 480,000x
⇒ x = 2,880,000/480,000
⇒ x = 6

∴ Rahim joined for 6 months.

Question: A watch priced at Tk. 1200 discount two times and bought at Tk. 600. The first discount was 20%. What is the second discount?

 Solution:
Marked Price = 1200 taka
First discount = 20%
Second discounted price = 600

Suppose,
Second discount % = x

First discounted price = 1200 - (20% of 1200)
= 1200 - 240
= 960

Second discounted price = 960 - (x% of 960)
600 = 960 - (x% of 960) 
960(1 - x / 100) = 600

Question: A and B started a business investing in the ratio 3 : 4. After 5 years, A withdrew his capital. B continued alone for 5 more years. If the total profit after 10 years is Tk. 88,000, what is A's share of the profit?

Solution:
Let A's capital = 3x
B's capital = 4x

A's investment for 5 years = 3x × 5 = 15x
B's investment for 10 years = 4x × 10 = 40x

Ratio of profits = 15x : 40x
= 15 : 40
= 3 : 8

Total contribution units = 3 + 8 = 11

Total profit = Tk. 88,000
∴ A's share = (3/11) × 88,000
= 3 × 8,000
= 24,000

∴ A's share = Tk. 24,000

Question: If 2 (P's capital) = 5 (Q's capital) = 6 (R's capital), then out of the total profit of TK 6500, Q will receive-

solution:
Let,
P's capital = p
Q's capital = q
R's capital = r

Then,
2p = 5q = 6r

⇒ q = 2p/5

⇒ r = 2p/6 = p/3

P : Q : R = p : 2p/5 : p/3
= 15p : 6p : 5p [multiply by 15]
= 15 : 6 : 5

∴ sum of ratio = (15 + 6 + 5) = 26

so, Q's share = 6500 × 6/26 = 250 × 6 = 1500 TK

Question: A and B are partners in a business. A invests Taka 25,000 for 8 months and B invests Taka 30,000 for 6 months. If the profit at the end of the year is Taka 15,200, what is B's share in the profit?

Solution: 
A’s Contribution = 25000×8 = 200000 Taka
B’s Contribution = 30000×6 = 180000 Taka

The ratio of A's and B's investment = 200000 : 180000
= 10 : 9

So, B's share in profit = {9/(10+9)} × 15200
= 7200 Taka

Question: A sum of money is to be divided among P, Q, R, S in the ratio 7 : 3 : 5 : 2. If R gets Tk. 2000 more than S, what is Q's share?

Solution:
Let their shares be 7x, 3x, 5x, and 2x respectively.

ATC,
5x - 2x = 2000
⇒ 3x = 2000
⇒ x = 2000/3

Therefore,
Q's share = 3x
= 3 × (2000/3)
= 2000 taka.

Question: The marked price of a t-shirt was Tk. 800. A man bought the same for Tk. 420 after getting two successive discounts. the first being 25%. then the second discount rate is-

Solution:
Marked price = 800
Actual price = 420
First discount = 25%
Let the second discount be x%

Then, we can write

after 25% discount,
discounted amount = 800 × (25/100)
= 200

New price = 800 - 200
= 600 Tk

Again, second discount,
discounted amount = 600 × (x/100)
= 6x

New price = 600 - 6x

ATQ,
600 - 6x = 420
⇒ 600 - 420 = 6x
⇒ 6x = 180
⇒ x = 30

∴ Second discount = 30%

Question: Two partners invest Tk. 1,20,000 and Tk. 80,000. After 6 months, they admit a new partner with Tk. 1,00,000. What is the ratio of their profits after one year?

Solution:
Profit sharing ratio depends on: Capital × Time.
Let the partners be A, B, and C.

A’s investment: Tk. 1,20,000 for 12 months
⇒ 1,20,000 × 12 = 14,40,000

B’s investment: Tk. 80,000 for 12 months
⇒ 80,000 × 12 = 9,60,000

C’s investment: Tk. 1,00,000 for 6 months
⇒ 1,00,000 × 6 = 6,00,000

Now, the ratio of profits:
14,40,000 : 9,60,000 : 6,00,000

Simplify = 12 : 8 : 5

∴ Ratio = 12 : 8 : 5

Let selling price each oranges is TK 1
so selling price of 36 oranges is Tk. 36
loses on oranges is Tk. 4
so cost price = 36+4=40
therefore, Loss percent= (loss/cost price) × 100
= (4/40) × 100
= 10%

Question: A, B, and C are partners in a business. Their investments are in the ratio 1/3 : 1/4 : 1/6 respectively. B withdraws half of his capital after 4 months. The business runs for 12 months. If the total profit is Tk. 6,300, what is C’s share of the profit?

Solution:
Ratio of initial investments = 1/3 : 1/4 : 1/6
= 4 : 3 : 2 (Multiplying by LCM 12)

Let the initial investments be 4x, 3x, and 2x respectively.
A : B : C = (4x × 12) : (3x × 4) + (3x × 1/2 × 8) : (2x × 12) [B withdraws half of his capital after the first 4 months]
= 48x : (12x + 12x) : 24x
= 48x : 24x : 24x
= 2 : 1 : 1

Sum of the ratio = 2 + 1 + 1 = 4
Total profit = Tk. 6,300

∴ C’s share = 6,300 × (1/4)
= 1,575

∴ C’s share of the profit is Tk. 1,575.

Question: P, Q, and R share a total of Tk. 2100 such that P receives twice as much as Q, and Q receives three times as much as R. How much money does Q get?

Solution:
Given,
P = 2Q
Q = 3R

∴ P = 2 × 3R = 6R

So, the ratio of P : Q : R = 6 : 3 : 1

∴ Q receives = (2100 × 3/10) Tk
= 630 Tk

∴ Q receives Tk. 630.

Question: Find the rate of discount being given on a shirt whose selling price is Tk. 420 after deducting a discount of Tk. 80 on its marked price.

Solution:
দেওয়া আছে,
বিক্রয়মূল্য = 420 টাকা
কমিশন = 80 টাকা

∴ গায়ে লিখা দাম = (420 + 80) টাকা 
= 500 টাকা

500 টাকায় কমিশন দেয় 80 টাকা
∴ 1 টাকায় কমিশন দেয় 80/500 টাকা
∴ 100 টাকায় কমিশন দেয় (80 × 100)/500 টাকা
= 16 টাকা

Price after 1^stdiscount of 20% = (100-20)%
= 80% of Marked Price
∴ Price = (80/100) × 12800
= Tk. 10240

SP = (100 - Discount) % of Price

∴ 9216 = (100 - Discount) % x 10240
⇒ (100 - Discount)/100 = 9216/10240
⇒ 100 - Discount = (9216 × 100)/10240
⇒ 100 - Discount = 90
⇒ Discount = 100 - 90 = 10%

10% is the percent of 2^nddiscount offered.

Question: A, B, and C invest a total of TK. 80000 in a business. A invests TK. 7000 more than B, and B invests TK. 5000 more than C. Out of a total profit of TK. 48000. How much profit does A receive?

Solution:
Let C = x.
Then, B = x + 5000
and A = x + 5000 + 7000
= x + 12000

So,
x + x + 5000 + x + 12000 = 80000
⇒ 3x + 17000 = 80000
⇒ 3x = 80000 - 17000
⇒ 3x = 63000
⇒ x = 21000
∴ x = 21000

so, C = 21000, B = 21000 + 5000 = 26000, A = 21000 + 12000 = 33000

A : B : C = 33000 : 26000 : 21000
= 33 : 26 : 21

∴ Total ratio = 33 + 26 + 21 = 80

So A's Share
= TK. 48000 × (33/80)
= TK. 19800

Question: The marked price of a shirt was Tk. 2400. A customer bought it for Tk. 1620 after getting two successive discounts. The first discount was 25%. What was the second discount rate?

Solution:
Price after first discount (25%),
= 2400 - (25% of 2400)
= 2400 - (2400 × 0.25)
= 2400 - 600
= Tk. 1800

Amount of second discount = 1800 - 1620 = Tk. 180

∴ Second discount rate = (Second discount amount/Price after first discount) × 100
= (180/1800) × 100
= 0.10 × 100
= 10%

∴ The second discount rate is 10%.

Question: A, B and C together invest Tk. 20000 in a business. A invests Tk. 2000 more than B and B invests Tk. 3000 more than C. Out of a total profit of Tk. 6000, find the share of A.

Solution:
Let C's investment = x
Then, B invests Tk. 3000 more than C
∴ B = x + 3000

A invests Tk. 2000 more than B
∴ A = (x + 3000) + 2000 = x + 5000

Total investment,
A + B + C = 20000
⇒ (x + 5000) + (x + 3000) + x = 20000
⇒ 3x + 8000 = 20000
⇒ 3x = 20000 - 8000
⇒ 3x = 12000
∴ x = 4000

Now,
C = x = 4000
B = 4000 + 3000 = 7000
A = 4000 + 5000 = 9000

∴ Profit sharing ratio is,
A : B : C = 9000 : 7000 : 4000 = 9 : 7 : 4

Sum of the ratios = 9 + 7 + 4 = 20

∴ Share of A = 6000 × (9/20)
= 2700

So, the share of A is Tk. 2700.

When P scores 80, Q scores 64.
When R scores 80, P scores 60
Hence, when R scores 150, Q scores (60×64×150)/(80×80) = 90
Therefore, in a game of 150, R can give 60 points to Q.

Question: A and B started a business with investments in the ratio 2:3. After 4 years, A withdrew his capital. B continued alone for 2 more years. If the total profit after 6 years is Tk. 39,000, what is A's share of the profit?

Solution:
Let A's capital = 2x
Let B's capital = 3x

A's investment for 4 years = 2x × 4 = 8x
B's investment for 6 years = 3x × 6 = 18x

Ratio of profits = 8x : 18x = 4 : 9

∴ Total contribution units = 4 + 9 = 13

Here, total profit = Tk. 39,000
∴ A's share = (4/13) × 39000
= (4 × 39000)/13
= Tk. 12,000

Question: Three partners shared the profit in a business in the ratio 5 : 7 : 8. They had partnered for 14 months, 8 months and 7 months respectively. What was the ratio of their investments?

Solution:
Let their investments be Tk. x for 14 months, Tk. y for 8 months and Tk. z for 7 months respectively.
Then, 14x : 8y : 7z = 5 : 7 : 8.
Now,
14x/8y = 5/7
⇒ 98x = 40y
∴ y = (49/20) x

And,
14x/7z = 5/8
⇒ 112x = 35z
∴ z = (112/35) x = (16/5) x.

x : y : z = x : (49/20) x : (16/5) x = 20 : 49 : 64.

Question: A shop offers a ‘Buy 2, Get 1 Free’ deal. What is the equivalent percentage discount?

Solution:
Assume the price of each item = 1 Taka.
Then,
Cost of 2 items = 2 Taka
Since 1 item is received for free, total items = 2 + 1 = 3

After the discount,
Cost of 3 items = 2 Taka
∴ Cost per item = 2/3 Taka

Actual price per item = 1 Taka
∴ Discount per item = 1 − (2/3) = 1/3 Taka

So, discount on 1 Taka = 1/3 Taka
∴ Discount on 100 Taka = 100 × (1/3) = 33.33 Taka

Therefore, in a “Buy 2, Get 1 Free” offer, the equivalent discount is 33.33%.

ধরি, y বেতন পায় 100 টাকা
x পায় (133.33% of 100) টাকা = 133.33 টাকা
এবং, z পায় (100 × 100 /75) = 400/3 = 133.33 টাকা
x : y : z = 133.33 : 100 : 133.33 = 400 : 300 : 400 = 4 : 3 : 4
∴ z পায় = 5500 এর 4/(4 + 3 + 4) টাকা = 2000 টাকা

The seller manages to make a 10% profit.
Value of this 10% profit = Tk. 77
% Profit = Profit/Cost price × 100
∴ 10 = 77/C.P. × 100
∴ CP = Tk. 770
And S.P. = 770+77 = Tk. 847

Price after Commission = Selling Price
= (100-23)% of L.P.
= 77% of L.P.
∴ 847 = 77/100 × L.P.
∴ L.P. = List Price = Tk. 1100

Question: An article has a marked price of Tk. 500. If two successive discounts of x% and 5% reduce the selling price to Tk. 427.50, determine the value of x.

Solution:
Marked Price = Tk. 500
Final Selling Price after two successive discounts = Tk. 427.50
Let the first discount = x%
Second discount = 5%

First discount be:
500 × (1 - x/100​) × (1 - 0.05) = 427.50
⇒ 500 × (1 - x/100​) × 0.95 = 427.50
⇒ 475 × (1 - x/100​) = 427.50 
⇒ (1 - x/100​) = 427.50 / 475
⇒ 1 - x/100 = 0.9
⇒ - x/100 = 0.9 - 1
⇒ - x/100 = - 0.1
⇒ x = 0.1 × 100
⇒ x = 10%

∴ First discount = 10%

Question: X and Y share profits in the ratio 3 : 2. If 10% of the total profit is donated to a fund and X's share is Tk. 5400, find the total profit.

Solution:
মনে করি, মোট লাভ = Tk. x
10% ফান্ডে দেওয়ার পর বাকি থাকে = 100% - 10% = 90% of x
= 90x/100

X এবং Y এর লাভের অনুপাত 3 : 2
অর্থাৎ, X এর অংশ = 90x/100 এর 3/(3 + 2)
= 90x/100 এর 3/5

প্রশ্নমতে,
90x/100 × (3/5) = 5400
⇒ 9x/10 × (3/5) = 5400
⇒ 27x/50 = 5400
⇒ 27x = 5400 × 50
⇒ x = 270000/27
∴ x = 10000

সুতরাং, মোট লাভ হলো Tk. 10000

Given,
Q = 2/3(P + R)
Or, 3Q = 2(P + R)
Again,
P = (Q + R)/2
Or, 2P = Q + R
Now,
P + Q + R = 366
Or, P + 2P = 366
Or, 3P = 366
∴ P = 122

মনে করি,
z এর আয় = p টাকা
y এর আয় = 4p টাকা
x এর আয় = 2.5×4p টাকা = 10p টাকা

∴ x : y : z = 10p : 4p : p
= 10 : 4 : 1
তাহলে, z এর আয় = 1/(10 + 4 + 1) × 1800
= 1/15 × 1800
= 120 টাকা

Question: The investment ratio of two partners, P and Q, is 7 : 9, while their profit ratio is 14 : 27. Given that P invested his funds for 6 months, determine the duration of Q's investment.

Solution:
Let P invested Tk 7x for 6 months
Q invested Tk 9x for y months

Now,
(7x × 6) : (9x × y) = 14 : 27
⇒ (7x × 6)/(9x × y) = 14/27
⇒ 42x/9xy = 14/27
⇒ 42 × 27 = 14 × 9y
⇒ 1134 = 126y
⇒ y = 1134/126
∴ y = 9

So, Q invested for 9 months.

Given total sum = Tk. 427
And given that 3 times A’s share, 4 times B’s share and 7 times C’s share are all equal.
=> 3A = 4B = 7C
But given
=> A + B + C = 427
Now express A and B in terms of C i.e
=> (7C/3) + (7C/4) + C = 427
=> C = 84

ধরি, মোট বিনিয়োগ x টাকা
A বিনিয়োগ করে x/3 টাকা
B + C = (x - x/3) = 2x/3
কিন্তু, B এর বিনিয়োগ পরিমান A ও C এর মোট বিনিয়োগের সমান।
∴ x/3 + C + C = 2x/3
⇒ 2C = 2x/3 - x/3
⇒ 2C = x/3
⇒ C = x/6 এবং B = 2x/3 - x/6 = 3x/6 = x/2
A : B : C = x/3 : x/2 : x/6 = 2 : 3 : 1
∴ C এর লাভের পরিমাণ { 900 এর 1/(3 + 2 + 1)} = (900 এর 1/6) = 150 টাকা

Let the cost price of goods = Tk. 100
Mark price of goods = Tk.130 (30% discount)
Selling price of goods 
= 90% of 130 = Tk 117
Gain = 117 - 100 = Tk. 17
Gain% = 17/100×100 = 17%

Question: If the list price of a mobile phone is Tk. 10000, and a Tk. 1500 discount is offered on the mobile phone, then what is the discount percentage?

Solution:
Marked Price = Tk. 10000
Discount = Tk. 1500

∴ Discount (%) = (Discount/marked Price) × 100%
∴ Discount (%) = (1500/10000) × 100%
= 15%

∴ So the discount percentage is 15%.

Ratio of the shares of A,B,C and D = (24 × 3) : (10 × 5) : (35 × 4) : (21 × 3)
= 72:50:140:63

Total rent be X then A’s share = x × (72/325)
= 72x/325

A/Q,

72x/325 = 720
Or, X = (720 × 325)/72
= 3250

Question: A trader, while selling a shirt, was asking for such a price that would enable him to offer a 20% discount and still make a profit of 25% on cost. If the cost of the shirt was Tk. 400, what was his asking price?

Solution:
প্রথম ধাপে, 25% লাভে শার্টটির বিক্রয় মূল্য নির্ণয় করতে হবে।
ক্রয়মূল্য = 400 টাকা
বিক্রয়মূল্য = ক্রয়মূল্য + ক্রয়মূল্যের 25%
= 400 + (400 × 25/100) = 400 + 100 = 500 টাকা

এখন, দ্বিতীয় ধাপে, 20% ছাড়ের উপর ভিত্তি করে ধার্য মূল্য নির্ণয় করতে হবে।
যদি ধার্য মূল্য 100 টাকা হয়,
তাহলে 20% ছাড়ে বিক্রয়মূল্য হবে 100 - 20 = 80 টাকা।

বিক্রয়মূল্য 80 টাকা হলে, ধার্য মূল্য = 100 টাকা
বিক্রয়মূল্য 1 টাকা হলে, ধার্য মূল্য = 100/80 টাকা
বিক্রয়মূল্য 500 টাকা হলে, ধার্য মূল্য = (100/80) × 500 = 625 টাকা
সুতরাং, শার্টটির ধার্য মূল্য ছিল 625 টাকা।

Question: Find the difference of amount if 40% discount is given on Tk. 1000 and two consecutive discounts 30% and 10% are given on the same amount.

Solution:
40% discount on 1000 = 1000 × 40% = 400
 
Two consecutive discounts on 1000.
30% discount on 1000 = 30% of 1000
= 300

After 30% discount on 1000 = 1000 - 300
= 700

Again,
After 10% discount on 700 = 10% of 700
= 70

Total discount = 300 + 70
= Tk. 370
So, the difference = 400 - 370
= Tk. 30

Question: The investment ratio of two partners, P and Q, is 7 : 9, while their profit ratio is 14 : 27. Given that P invested his funds for 6 months, determine the duration of Q's investment.

Solution:
Let P invested Tk 7x for 6 months
Q invested Tk 9x for y months

Now,
(7x × 6) : (9x × y) = 14 : 27
⇒ (7x × 6)/(9x × y) = 14/27
⇒ 42x/9xy = 14/27
⇒ 42 × 27 = 14 × 9y
⇒ 1134 = 126y
⇒ y = 1134/126
∴ y = 9

So, Q invested for 9 months.

Question: Rafsan and Karim entered into a business partnership by investing Tk. 20,000 and Tk. 30,000 respectively. If the total profit earned is Tk. 7,000, what is Rafsan’s share of the profit?

Solution:
Given,
Investment ratio = 20000 : 30000
= 2 : 3
Sum of the ratio's = 2 + 3 = 5

∴ Rafsan's share = 7000 × (2/5)
= Tk. 2800