Partnership and Discount

Question: The marked price of a ceiling fan is Tk. 1250 and the shopkeeper allows a discount of 6% on it. Find the selling price of the fan.

Solution:
Marked price = Tk. 1250 and discount = 6%. 

Discount = 6% of Marked Price 
= (6% of Tk. 1250) 
= Tk. {1250 × (6/100)} 
= Tk. 75 

Selling price = (Marked Price) - (discount) 
= Tk. (1250 - 75) 
= Tk. 1175. 

∴ Hence, the selling price of the fan is Tk. 1175. 

Question: A, B, and C started a business by investing Tk. 24,000, Tk. 32,000 and Tk. 40,000 respectively. If the total profit at the end of the year is Tk. 18,900, what is B's share of the profit?

Solution:
A, B এবং C এর বিনিয়োগের অনুপাত,
A : B : C = 24000 : 32000 : 40000
= 24 : 32 : 40
= 3 : 4 : 5

অনুপাতগুলোর যোগফল = 3 + 4 + 5 = 12

মোট লাভ = 18900 টাকা
B এর লভ্যাংশ = (B এর অনুপাত/অনুপাতগুলোর যোগফল) × মোট লাভ
= (4/12) × 18900
= (1/3) × 18900
= 6300 টাকা

সুতরাং, B এর লভ্যাংশ হল 6300 টাকা।

Dividend on tk. 20 = tk.9/100x 20= 9/5
tk.12 is an income on tk.100.
Tk. 9/5 is an income on tk. 100/12 x 9/5 = tk.15

Question: The marked price of a watch was Tk. 1500. A customer bought it for Tk. 1080 after getting two successive discounts. The first discount was 20%. What was the second discount rate?

Solution:
Price after first discount (20%),
= 1500 - (20% of 1500)
= 1500 - (1500 × 0.20)
= 1500 - 300
= Tk. 1200

Amount of second discount
= 1200 − 1080
= Tk. 120

∴ Second discount rate
= (Second discount amount/Price after first discount) × 100
= (120/1200) × 100
= 0.10 × 100
= 10%

∴ The second discount rate is 10%.

Question: Find the rate of discount being given on a shirt whose selling price is Tk.546 after deducting a discount of Tk.104 on its marked price.

Solution:
The price written on the item = 546 + 104 Tk.
= 650 Tk.

On 650 Taka, the commission is 104 Taka.
∴ Therefore, the commission on 100 Taka is (104 × 100)/650 Tk.
= 16 Tk.

প্রশ্ন: একটি কোম্পানি 'ক' মোট p টাকা লাভ করেছিল। লাভের অর্ধেক কোম্পানির প্রতিষ্ঠাতা পাওয়ার পর বাকি অর্ধেক সমানভাবে তাঁর চারজন অংশীদারের মধ্যে ভাগ করে দেয়া হয়। p-এর হিসেবে, প্রতিটি অংশীদার কত টাকা পেয়েছিলো?

সমাধান:
এখানে, কোম্পানি 'ক' এর মোট লাভ = p টাকা

প্রতিষ্ঠাতা পায় লাভের অর্ধেক  = p/2

∴ বাকি লাভ = (p − p/2) টাকা
= p/2 টাকা

বাকি অংশ 4 জন অংশীদারের মধ্যে সমানভাবে ভাগ করা হয়,
∴ প্রতিজন অংশীদার পায় = (p/2)/4
= (p/2) × (1/4)
= p/8

Question: A shop is offering a 15% discount on a television whose original price is Tk. 40,000. What will be the selling price of the television after applying the discount?

Solution:
The original price of the television = Tk. 40000
And the discount percentage = 15%

We know,
Discount Amount = (Discount Percentage ÷ 100) × Original Price
= (15 ÷ 100) × 40000
= 6000

∴ Selling Price = 40000 - 6000 = 34000

The selling price after the discount is Tk. 34000.

Question: A, B, and C invest in a business in the ratio 3 : 4 : 5. If the total profit is Tk 24,000, how much more did C get than A?

Solution:
→ Profit shares = 3x : 4x : 5x

Now
3x + 4x + 5x = 24000
12x = 24,000
→ x = 2,000

→ A’s share = 3 × 2,000 = 6,000
→ C’s share = 5 × 2,000 = 10,000
→ Difference = 10,000 - 6,000 = Tk 4,000

Question: X, Y, and Z share Tk. 1680 in such a way that X has 3.5 times as much as Y, and Y has 2 times as much as Z. How much money does Z receive?

Solution:
Given,
X = 3.5Y
Y = 2Z

∴ X = 3.5 × 2Z = 7Z

So, the ratio of X, Y, Z = 7 : 2 : 1

∴ Z receive = [1680 × (1/10)] Tk.
= 168 Tk.

Question: The contributions made by Rasel and Akash are in the ratio of 6 : 4. If 10% of the total profit is donated and Rasel gets 9000 as his share of profit, what is the total profit?

Solution:
Let,
the total profit after donated = p
The ratio of contribution by Rasel and Akash = 6 : 4
sum of ratios = 10

Rasel's share = (6/10) × p = 9000
⇒ p = (10 × 9000)/6
∴ p = 15000
so, total profit after donated = 15000

ATQ,
Rasel gets 9000 after 10% donated
Now,
90% = 15000
1% = 15000/90
And, 100% = (15000 × 100)/90 = 16,666.66

Question: : On a 12% discount sale, an article costs Tk. 704. What was the original price of the article?

Solution:
Let the original price be Tk. x.

According to the question,
x - 12% of x = 704
⇒ x - (12x/100) = 704
⇒ (100x - 12x)/100 = 704
⇒ 88x = 70400
∴ x = 800

So, the original price of the article was Tk. 800.

Cost price of motor-car = Tk. 17000
Mark price of motor-car
= Tk 17000 × 100/85 = Tk. 20000
After successive discount,Cost price
= Tk. 20000 × 95/100×90/100
= Tk. 17100 

Sourav invests Tk. 4000 for 24 months, Fahim invests Tk. 3000 for 24 months and Arif invests
Tk. 4000 for 18 months.
Then,
Sourav:Fahim:Arif = (4000x24):(3000x24):(4000x18)
= 4x24 : 3x24 : 4x18
= 4 : 3 : 3
Therefore, Fahim's share = Tk. 5000 x 3/10
= Tk. 1500.

Question: Rina ,Ratul and Ayesha started a business. Rina invested 1/3 part, Ratul 1/4 part and rest of the capital was invested by Ayesha. The ratio of their profits will be-

Solution:
Let,
Total capital be TK x
Then,
Rina's share = x/3 TK
Ratul's share = x/4 TK
Ayesha's share = x - {(x/3) + (x/4)}
= x - {(4x + 3x)/12}
= x - (7x/12)
= (12x - 7x)/12
= 5x/12 TK

∴ Required ratio
= (x/3) : (x/4) : (5x/12)
= 4x : 3x : 5x     [multiply by 12]
= 4 : 3 : 5

Question: Yesterday, a shop made Tk. 5,000 in total sales. One-fifth of these sales came from items that were sold at a 20% discount. What would the shop’s total sales have been yesterday if all items had been sold at full price?

Solution:
Total sales = Tk. 5,000
Discounted sales = 1/5 of 5,000 = 1,000

Since these items were sold at 20% discount,
The full price of these items = 1,000 ÷ (1 - 0.20) = 1,000 ÷ 0.8 = Tk. 1,250
Sales from items not discounted = 5,000 - 1,000 = Tk. 4,000

If all items were sold at full price, total sales
= 4,000 + 1250
= Tk. 5250

Let, B invested the money for t months
Then the ratio of investment = (12×11 : 11× t) = 12 : t
So, 12/t = 4/1
⇒ t = 3 months

Question: Two partners invest TK 1,00,000 and TK 60,000 .After 6 months, they admit a new partner with TK 80,000. What is the ratio of their profits after one year?

Solution:
Profit sharing ratio depends on : Capital × Time
Let,
the partners be P, Q and R

P's invest : TK 1,00,000 for 12 months
⇒ 1,00,000 × 12 = 12,00,000

Q's invest : TK 60,000 for 12 months

⇒ 60,000 × 12 =7,20,000

R's invest: TK 80,000 for 6 months

⇒ 80,000 × 6 = 4,80,000

Now, the ratio of profits:
12,00,000 : 7,20,000 : 4,80,000

Simplify = 5 : 3 : 2

∴ Ratio = 5 : 3 : 2

Question: The investment ratio of two partners, A and B, is 11 : 12, while their profit ratio is 2:3. Given that A invested his funds for 8 months, determine the duration of B's investment.

Solution:
Let, A invested Tk 11x for 8 months
and B invested Tk 12x for y months

Now,
(11x × 8) : (12x × y) = 2 : 3
⇒ (11x × 8) / (12x × y) = 2/3
⇒ 88x/12xy = 2/3
⇒ 24y = 264
⇒ y = 11

So, B invested for 11 months. 

Present Worth (P.W.) = 840 - 105 
= Tk. 735
Therefore, S.I. on Tk. 735 = Tk. 105
S.I. on Tk. 8400
= 105 × 840/735
= Tk. 120

Dividend on 1 share = (12.5 x 50)/100 = tk.6.25
tk.25 is income on an investment of tk.100
tk.6.25 is income on an investment of tk. (6.25 x 100)/25 = tk.25

Question: The true discount on Tk. 2562 due 4 months hence is Tk. 122. What is the rate percent?

Solution: 
Given,
Amount (A) = 2562 Tk
True Discount (TD) = 122 Tk
Time (T) = 4 months = 1/3 of a year

Present Worth, P.W. = (2562−122) Tk
= 2440 Tk

True Discount (TD) = Simple Interest (SI) on the Present Worth (PW) for the given time period.
∴ S.I. on 2440 Tk for 4 months is 122 Tk.

We know, SI = PRT/100
∴ R = (SI × 100)/(P × T)
= [(100 × 122)/(2440 × (1/3)]
= 15

∴ Rate is 15%

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: Kamal and Rajon enter into a partnership with their capitals in the ratio 5 : 6. At the end of 6 months Kamal withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long was Rajon's capital used?

Solution: 
Let,
Kamal invested Tk. 5x for 6 months
And
Rajon invested Tk. 6x for y months
Then,
(5x × 6)/(6x × y) = 5/9
⇒ 30x/6xy = 5/9
⇒ 5/y = 5/9
∴ y = 9

So Rajon capital was used for 9 months

Question: Rana and Rahim invest TK 30,000 and 40,000 respectively in a partnership. They earned a profit of TK 7000. Find Rana's share of the profit.

Solution:
Given that,
Investment ratio = 30,000 : 40,000
= 3 : 4

sum of  the ratio's = 3 + 4 = 7
∴ Rana's share of the profit = 7000 × (3/7)
= 1000 × 3
= 3000 tk

Investment = Tk.[20 × (50 − 5)] = Tk. 900.
FaceValue = Tk. (50 × 20) = Tk.1000.
Dividend = Tk. (27/2 × 1000/100) = Tk.135.
Interest Obtained = (135/900 × 100)% = 15%

Question: A “Buy 3, Get 1 Free” deal is available at a store. How much discount does this equal in percentage terms?

Solution:
Assume the price of each item = 1 Taka.
Then,
Cost of 3 items = 3 Taka
Since 1 item is received for free, total items = 3 + 1 = 4

After the discount,
Cost of 4 items = 3 Taka
∴ Cost per item = 3/4 Taka

Actual price per item = 1 Taka
∴ Discount per item = 1 − ( 3/4 ) = 1/4 Taka

So, discount on 1 Taka = 1/4 Taka
∴ Discount on 100 Taka = 100 × ( 1/4 ) = 25 Taka

Therefore, in a “Buy 3, Get 1 Free” offer, the equivalent discount is 25%.

Question: Two friends invested TK.1500 and TK. 3500 in a business. They earned a profit of TK. 1000. One-half of the profit was divided equally between them and the other half was divided in proportion to their capitals. How much did each of them receive ?

Solution:
Ratio of shares
= 1500 : 3500
= 3 : 7

Share of first friend
= TK. [(500/2) + {500 × (3/10)}]
= TK.[250 + 150]
= TK. 400

Share of second friend
= TK. [(500/2) + {500 × (7/10)}]
= TK.[250 + 350]
=  TK. 600

Question: B's income is 60% of A's, and the ratio of their expenditures is 9 : 5. If each saves Tk. 4,000, find A’s income.

Solution:
Suppose,
A's income = 100,
B's income 60% of 100 = 60
A : B = 100 : 60 = 5 : 3

So,
A’s income = 5x, B’s income = 3x
A’s expense = 9y, B’s expense = 5y

Then their savings are:
A’s savings = Income - Expense = 5x - 9y.......(1)
B’s savings = Income - Expense = 3x - 5y........(2)

Given that each saves Tk. 4000:
5x - 9y = 4000
3x - 5y = 4000

Subtract equation (2) from (1):
(5x - 9y) - (3x - 5y) = 0 
 ⇒ 2x - 4y = 0 
⇒ x = 2y

Substitute x = 2y into equation (2):
3(2y) - 5y = 4000
⇒ 6y - 5y = 4000
⇒  y = 4000

Then x = 2y = 8000

Finally, A’s income = 5x = 5 × 8000 = 40,000 Taka

∴ A's income = 40,000 Taka

Question: Last year, Company X made p dollars in profit. Half of the profit went to the company's founder. The rest was split evenly among his four other partners. In terms of p, how much did each of the other partners receive?

Solution:
Here,
Company X's profit is p dollars.

Half of the profit went to the founder, so the founder received,
1/2 of p = p/2

Remaining Profit = p - p/2 = p/2

The remaining profit was split among 4 partners.
∴ Each partner gets = (p/2) ÷ 4
= (p/2) × (1/4)
= p/8