Percentage and Profit & Loss

Marked Price = 2000
SP after first Discount of 20% = 2000 - 20% of 2000 = 1600
SP after second Discount of 10% = 1600 - 10% of 1600 = 1440
Now, the final selling price at cash = 1440 - 5% of 1440 = Tk. 1368

Question: A machine is sold at a profit of 20%. Had it been sold for Tk. 60 less, there would have been a loss of 20%. What was the cost price?

Solution:
ধরি, মেশিনটির ক্রয়মূল্য = x টাকা

20% লাভে বিক্রয়মূল্য = x + x এর 20%
= x + 20x/100
= 12x/10

20% ক্ষতিতে বিক্রয়মূল্য = x - x এর 20%
= x - 20x/100
= 8x/10

প্রশ্নমতে,
(12x/10) - (8x/10)= 60
⇒ 4x/10 = 60
⇒ 4x = 60 × 10
⇒ x = (60 × 10)/4
∴ x = 150

∴ মেশিনটির ক্রয়মূল্য = 150 টাকা

Since the number of books is not given, I could not plug in the values. So this is solved using quadratic equations.
Let the number of books bought initially for Tk. 900 be 'x'. So the original price of the book was 900/x
Now price of the book is up by 3. i.e., (900/X) + 3. and number of books bought is reduced by 15. i.e., (x - 15)
Since new total amount spent is still same, the product of new price and new number of books is still 900

[(900/x) + 3] (x - 15) = 900
Or, (900 + 3x) (x - 15) = 900x
Or, 3x² + 855 - 13500 = 900x Now use quadratic equation to find the value of x.
Or, x² - 15x - 4500 = 0
Or, x² - 75x + 60x - 4500 = 0
Or, x(x - 75) + 60(x - 75) = 0
Or, (x - 75) (x + 60) = 0
So, x = 75 or x = - 60
since x cannot be negative, so, x = 75
∴ the original price of the book = 900/75 = 12

Let,
Total votes = a.
This means that Votes of candidate 1 + Votes of candidate 2 = a
We know that,
Votes of candidate 1 = 40% of a
= 40a/100
Hence, Votes of candidate 2 = (100% - 40%) of a
= 60% of a
= 60a/100

1^stcandidate lost by 1000 votes = difference of votes between both candidates
60a/100 - 40a/100 = 2000
∴ a = 10,000.

Question: The population of a town increases every year by 5%. If its present population is 60,000, then after 2 years, what will be the population?

Solution: 
We know, Population after n years = P × [1 + (r/100)]ⁿ

∴ Population after 2 years = 60000 × [1 + (5/100)]²  
= 60000 × (1 + 0.05)² 
= 60000 × 1.1025
= 66150

ধরি, price x টাকা
TV এর price বাড়ার পর = x + x এর 20%
= 1.2x
Computer এর price কমার পর = x - x এর 10%
= 0.9x
পূর্বে 4 টি TV এবং 4 টি computer এর price = 8x
এখন 4 টি TV এবং 4 টি computer এর price = 4.8x + 3.6x
= 8.4x
দাম বেশি হয় = (8.4x - 8x) = 0.4x টাকা

সুতরাং, খরচ বাড়বে = (0.4/8) × 100
= 5%

Question: A man sells a watch at a 5% loss. If he had sold it for Tk. 56.25 more, he would have made a 10% profit. What was the cost price of the watch?

Solution:
Let,
the price of watch x taka

According to the question,
x(100% - 5%) + 56.25 = x(100% + 10 %)
→ 95%x + 56.25 = 110% x
→ 15%x = 56.25
→ x = (5625/15)
∴ x = 375 taka

∴ the cost of the watch is 375 taka.

Remaining apples (100 - 40) % = 60%
60% = 420 
∴ 100% = 420 × 100 / 60 = 700 

Number of games played = 40
Number of won games = 24
Percentage of games played = (24/40)× 100
= 60%

Question: An investment becomes Tk. 6,720 in 2 years and Tk. 7,392 in 3 years at compound interest. Find the rate of interest per annum.

Solution:
Let the principal = P, rate = R%.

Compound Interest:
A = P[1 + 100/R​]^T

Amount after 2 years = 6,720
6,720 = P[1 + 100/R​]²

Amount after 3 years = 7,392
7,392 = P[1 + 100/R​]³

Divide the 3rd year amount by 2nd year amount:
7,392/6,720 = P(1 + R/100)³/ P(1 + R/100)² 
⇒ 1.1 = 1 + R/100 [7,392/6,720 = 1.1]
⇒ 1.1 - 1 = R/100
⇒ 0.1 = R/100
⇒ R = 0.1 × 100
⇒ R = 10%

∴ R = 10%

Question: The price of an article is raised by 30% and then two successive discounts of 10% each are allowed. Ultimately, the price of the article is:

Solution: 
let, the price be 100 taka 

after 30% raise = 100 + 30 = 130 taka

after first 10% discount = 130 - 13 = 117 taka

after second 10% discount = 117 - 117 × 0.1 
= 117 - 11.7
= 105.3

cost price = 80000 + 5000 + 1000
= 86000
profit = 25%
Selling price = 86000 + 86000 × (1/4)
= Tk. 107500

According to the question
The cost price of 12 oranges = The sale price of 9 oranges
So profit% = (12 C.P. - 9 C.P.)/9 C.P × 100 = 33.33%

Then it is said that,
5 S.P. - 5 C.P. = 10 M.P. -10 S.P.
From that
we get the relation between M.P. and S.P., that is,
27 S.P. = 24 M.P.(With help of 12 C.P. = 9 S.P.)
Then Discount%
= M.P. - S.P./M.P
= (27 M.P. - 24 M.P.)/27 M.P × 100
= 11.11%
So, % point discount
= 33.33% - 11.11%
= 22.22%

Question:

Solution: 

Question: Mr. Khan's salary is Tk. 5000.00 and he gets 10% commission of his salary. If his salary increased by 10%, by what percent his commission will increase?

Solution:
পূর্বের কমিশন:
= 5000 এর 10% টাকা
= 5000 × 10/100
= 500 টাকা

১০% বৃদ্ধিতে বর্তমান বেতন = 5000 + 5000 এর 10%
= 5000 + 5000 × 10/100
= 5000 + 500 = 5500 টাকা

নতুন কমিশন = 5500 এর 10% = 5500 × 10/100 = 550 টাকা

কমিশন বৃদ্ধি পায় = 550 - 500 = 50 টাকা

শতকরা কমিশন বৃদ্ধি পায় = (50/500) × 100% = 10%

Question: What percent of 700 is 2.1?

Solution: 
700 এর x% = 2.1
⇒ 700 এর x/100 = 2.1
⇒ 7x = 2.1
⇒ x = 2.1/7
x = 0.3

Question: Sarah sold her bicycle for Taka 5000 while making a profit of Taka 570. Then what is the price at which she bought that cycle?

Solution:
Given,
Selling Price = Taka 5000
Profit = 570 Taka

Now, 
Cost Price = Selling Price - Profit
⇒ Cost Price = 5000 - 570
⇒ Cost Price = 4430 Taka
Thus, cost price of the bicycle is Taka 4430

Question: Carlos & Co. generated revenue of BDT 1,250 in 2006. This was 12.5% of its gross revenue. In 2007, the gross revenue grew by BDT 2,500. What is the percentage increase in the revenue in 2007? 

Solution: 
We are given that Carlos & Co. generated revenue of BDT 1,250 in 2006 and that this was 12.5% of the gross revenue.
Hence, if 1250 is 12.5% of the revenue, then 100% (gross revenue) is (100/12.5) × (1250) = BDT 10,000 
Hence, the total revenue by the end of 2007 is BDT 10,000.

In 2006, revenue grew by BDT 2500. 
∴ Percentage increase in the revenue = (2500/10000) × 100 
= 25%

Let the marked price = 100x
Shop Keeper gives 5% discount, therefore, Price becomes = 95x
Again if shopkeeper gives 6% discount then the price become = 94x
According to the question if he gives 6% discount his earning will Tk.15 less
∴ 95x - 94x = 15
⇒ x = 15
∴ Marked price = 100 × 15 = Tk. 1500

Question: A man spends 75% of his income. If his income increases by 25% and expenditure increases by 15%, find the percentage increase in his savings.

Solution:
Let,
Original income = 100 Taka
Original expenditure = 75% of 100 = 75 Taka
Original savings = Income - Expenditure = 100 - 75 = 25 Taka

After increase,
New income = 100 + 25% of 100 = 100 + 25 = 125 Taka
New expenditure = 75 + 15% of 75 = 75 + 11.25 = 86.25 Taka
New savings = 125 - 86.25 = 38.75 Taka

Percentage increase = [(New savings - Original savings) / Original savings] ​× 100 
= [(38.75 - 25) / 25] × 100
= [13.75/ 25] × 100
= 0.55 × 100
= 55% 

∴ Saving increased = 55%

Number of liters of water in150 liters of the mixture = 20% of 150 = 20/100 × 150 = 30 liters.
P liters of water added to the mixture to make water 25% of the new mixture.
Total amount of water becomes (30 + P) and total volume of mixture is (150 + P).
(30 + P) = 25/100 × (150 + P)
120 + 4P = 150 + P
=> P = 10 liters.

Question: A trader sets his products 50% above cost price and gives a discount of 20%. Find his profit percent.

Solution:
Let the cost price (CP) = Tk. 100

Marked Price (MP)
MP = CP + 50% of CP
= 100 + 50
= 150

Selling Price (SP) after discount:
SP = MP - 20% of MP
= 150 - 30
= 120

Profit = SP - CP
= 120 - 100
= 20

Profit percent:
Profit % = (Profit/CP) × 100
= (20/100) × 100
= 20%

∴ Profit = 20%

Question: In January, the stock price went up by 50%. It then dropped by 20% in February, rose again by 25% in March, and declined by 10% in April. If Tk. 200 was invested initially and sold after April, calculate the net percentage change in price.

Solution:
At the end of January,
The value of the stock is = Tk. 200 + 50% of (Tk. 200)
= Tk. 200 + Tk. 100 = Tk. 300.

At the end of February,
The value of the stock is = Tk. 300 - 20% of (Tk. 300)
= Tk. 300 - Tk. 60 = Tk. 240.

At the end of March,
The value of the stock is = Tk. 240 + 25% of (Tk. 240)
= Tk. 240 + Tk. 60 = Tk. 300.

At the end of April,
The value of the stock is = Tk. 300 - 10% of (Tk. 300)
= Tk. 300 - Tk. 30 = Tk. 270.

Now, the percentage change in price is,
= (Change in price/Original price) × 100%
= (270 - 200)/200 × 100%
= (70/200) × 100%
= 35%

Question: 60% of a number is 30 less than three-fourth of the same number. Find the number.

Solution:
Let,
the number = x.

ATQ,
60% of x = (3/4) of x - 30
⇒ 60x/100 = (3x/4) - 30
⇒ 3x/5 = (3x/4) - 30
⇒ (3x/4) - (3x/5) = 30
⇒ (15x - 12x)/20 = 30
⇒ 3x/20 = 30
∴ x = 200

So, the number is 200.

Let the cost price = 100
Selling price = (100×117)/100 = 117
So, Marked price = (117×100)/90 = 130
∴ Profit Percentage without discount = {(130 - 100)/100} × 100 = 30%

Question: A man's salary was reduced by 50%, again the reduced salary was increased by 50%. Find the loss of in terms of percentage.

Solution: 
Let, The wages of man = 100 Tk.
Then wages decreased by 50%
So wages = (100 - 50) = 50 Tk
The reduced wages were increased by 50%
Then wages = 50 × (150/100) = 75 Tk.
So, the percentage of wages loss = (100 - 75) tk
= 25 Tk

Both game is played by = 100 - (35 + 45) = 20% children
So, in all cricket can be played by = 55% of 80 children
= 55/100 × 80 = 44 children

Let cost price = x selling price = y
Then, profit = y − x
If selling price is doubled, selling price = 2y
profit = 2y − x

2y − x = 3 (y − x)
⇒ 2y − x = 3y − 3 x
⇒ y = 2x
profit = (y − x) = (2x − x) = x
∴ profit percent = (x × 100)/x = 100%

Let,
A's salary = Tk. x.
Then, B's salary = Tk. (2000 - x)
According to the question,
(100 - 95)% of A = (100 - 85)% of B
⇒ (5x/100) = (15/100) × (2000 - x)
⇒ x = 1500.

Let the profit be X% and loss be Y% . So,
Net profit or loss% = X + (-Y) + X×(-Y)/100
(Negative sign denotes that their is a loss)
Their is 10% loss and 10% profit then
∴ Net profit or loss% = 10 + (-10) + 10×(-10)/100
= 10 - 10 -100/100
= -1
∴ the net loss is 1%

'A' sells an article, which costs him Tk 400, to B at a profit of 20%.
profit of A = 400 × 20/100 = Tk 80
Cost Price for B = 400 + 80 = Tk 480
B then sells it to C, making a profit of 10% on the price he paid to A
Profit for B = 480 × 10/100 = Tk 48
Cost Price for C = 480 + 48 = Tk 528
Thus C pays Tk 528 to B.

Cost price = Tk 3000
Selling price = [{3600 × 100}/{100 + (10 × 2)}]
= Tk. 3000
Gain = 0%.

দ্রব্যের ক্রয়মূল্য 150 টাকা
অতিরিক্ত ব্যয়ভার (150 এর 12%) = (150 × 12/100) = 18 টাকা
∴ মোট খরচ (150 + 18)  = 168 টাকা
10% লাভে বিক্রয়মূল্য (168 + 168 এর 10%) = 168 + 16.8 = 184.80 টাকা

Question: 5.4 is 45 percent of 20 percent of a certain number. What is the number?

Solution:
ধরি, সংখ্যাটি = x

প্রশ্নমতে,
45% of (20% of x) = 5.4
⇒ (45/100) × {(20/100) × x} = 5.4 
⇒ (9/20) × (1/5) × (x )= 5.4
⇒ 9x/100 = 5.4
⇒ 9x = 5.4 × 100
⇒ 9x = 540
⇒ x = 540/9
∴ x = 60

ATQ, x×10% = y×25%
⇒ x = (25×16)/10 
∴ x = 40

1 থেকে 70 পর্যন্ত মোট সংখ্যা 70 টি।
এর মধ্যে এককের স্থানে 1 অথবা 9 আছে এমন সংখ্যা: 1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69 মোট 14 টি
∴ সংখ্যা হার = (14/70 × 100) % = 20 %