Percentage and Profit & Loss

Let price of book = 100tk

Price of pen = 100 + 100 × 25% = 125 tk
Price of pen-holder = 100 + 100 × 50% = 150 tk

Difference is = 150 -125 = 25 tk

∴ Percentage = (25 × 100)/125 = 20%

মনে করি,
প্রত্যেক ধরনের চকলেট কেনার হলো 60 টি করে।
তাহলে,
Total cost = 60 × (1/4) + 60 × (1/6)
= 25।
আবার, selling price = 60 × 2 × (1/5)
= 24।
কাজেই percentage loss = (25 - 24)/25 × 100
= 4%

Question: Rifat sold an article for Tk. 528 alter allowing a discount of 12% on its marked price. What was the marked price of the article?

Solution: 
Marked price of a article be  x
According to the question 
x88/100 = 528
88x = 528 × 100
x = (528 × 100)/88
x= 600

Question: An article is sold at a 25% discount, resulting in a 20% profit. If the discount is reduced to 15%, what will be the new profit percentage?

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

With 20% profit:
Selling Price (SP) = CP + 20% profit
= 100 + 20
= Tk. 120

This SP is after 25% discount on Marked Price (MP).
∴ SP = 75% of MP 
⇒ 120 = 0.75 × MP
⇒ MP = 120 / 0.75
= Tk. 160

Now, with reduced discount of 15%:
New Selling Price = 85% of Marked Price
= 0.85 × 160
= Tk. 136

∴ New profit = New Selling Price – Cost Price 
= 136 – 100
= Tk. 36

∴ New profit percentage = (36 / 100) × 100% = 36%

CP of first tea = Tk. 192 per kg.
CP of Second tea = Tk. 150 per kg.
The mixture is to be sold in Tk. 194.40 per kg, which has included 20% profit. So,
SP of Mixture = Tk. 194.40 per kg.

Let the CP of Mixture be Tk. X per kg. Therefore,
X + 20% of X = SP
6x/5 = 194.40
6X = 194.40 × 5
X = Tk. 162 per kg.

Let N kg of first tea and M kg of second tea be added.

Now, Using Alligation,
We get,
N/M = 12/30
N/M = 2 : 5

Question: The price of the sugar rise by 25%. If a family wants to keep their expenses on sugar the same as earlier, the family will have to decrease its consumption of sugar by-

Solution:
২৫% বৃদ্ধিতে দাম = (১০০ + ২৫) = ১২৫টাকা।
 
১২৫ টাকায় চিনি খাওয়া কমে = ১২৫ - ১০০ = ২৫ টাকা
∴ ১ টাকায় চিনি খাওয়া কমে = ২৫/১২৫ টাকা
∴ ১০০ টাকায় চিনি খাওয়া কমে = (২৫ × ১০০)/১২৫
= ২০ টাকা
 
অর্থাৎ, শতকরা ২০% কমিয়েছিল

Question: A fruit seller purchases guavas at the rate of 2 guavas for 1 taka and sells them at the rate of 5 guavas for 3 taka. What is his gain percentage?

Solution:
2 guavas cost 1 tk
So 1 lemon costs 1/2 tk

For comparing,
let's calculate CP (Cost Price) for 10 guavas
∴ CP of 10 guavas = 10 × (1/2) = 5 tk

Now,
Let's find the selling price (SP) of guavas:
5 guavas cost 3 tk
∴ 10 guavas will cost = (10/5) × 3 = 6 tk

∴ Gain = SP - CP = 6 - 5 = 1 tk
∴ Gain percentage = (Gain/CP) × 100
= (1/5) × 100
= 20%

Question: A mobile phone is sold for Tk. 9,600, and the seller makes a profit of 25% on the cost price. What will be the new selling price if he reduces the profit to 10%?

Solution:
At 25% profit,
Selling Price = 125% of Cost Price
∴ Cost Price = (100/125) × Selling Price
= (100/125) × 9,600
= 7,680 Tk.

Now, if the seller wants 10% profit,
∴ New Selling Price = 7,680 + (10% of 7,680)
= 7,680 + {(10/100) × 7,680}
= 7,680 + 768
= 8,448 Tk.

Question: After paying a 10 percent tax on all income over Tk. 3,000, a person had a net income of Tk. 12,000. What was the income before taxes?

Solution:
প্রশ্নে ৩০০০ টাকা পর্যন্ত আয়ের উপর কোন কর দেয়া লাগে না এবং ৩০০০ টাকার উপর আয় করলে বাকি টাকার উপর কর দিতে হয়।

ধরি,
আয়ের করযুক্ত অংশ ক টাকা 

∴ ক - ক এর ১০% + ৩০০০ = ১২০০০
বা, ক - (১০ক)/১০০ + ৩০০০ = ১২০০০
বা, ক - ক/১০ = ৯০০০
বা, ১০ক - ক = ৯০০০০
বা, ৯ক = ৯০০০০
∴ ক = ১০০০০

∴ করযুক্ত টাকা ১০০০০ এবং করমুক্ত টাকা ৩০০০ 
∴ মোট টাকা (১০০০০ + ৩০০০) = ১৩০০০ টাকা 

∴ যদি কর না দিত তাহলে তাঁর আয় ১৩০০০ টাকা হত। কিন্তু ১০০০ টাকা কর দিয়ে দেয়ায় তার আয় থাকে ১২০০০ টাকা 

We are given selling price = Tk. 580 and expected profit of 15%
Therefore, we can easily solve this numerical, considering basic formulae of profit and loss.
Let cost price = x
Selling price = C.P. + Profit
S.P. = C.P. + (15% of C.P.) [We know that profit is gained on cost price]
580 = x + (0.15 x)
580 = 1.15 x

Therefore,
x = 504.347
Cost Price = Tk. 504.35
Now,
we have the cost price and hence,
Profit = S.P. – C.P. = 580 – 504.35
= Tk. 75.65
The trader gets a profit of Tk. 75.65

Let,
the number be x.
Then, (100 + 37.5)% of x
⇒ 137.5% of x = 33
⇒ 137.5 × (1/100) × x = 33
⇒ x = (33 × 100)/137.5
⇒ x = 24.

He buys 1100 grams instead of 1000 grams.
Suppose he bought 1100 grams for Tk. 1000.

He sells only 900 grams when he takes the money for 1 kg.
Now, according to the problem,
he sells at an 8% profit (20% markup, 10% discount).
Hence, his selling price is Tk. 1080 for 900 grams.

Now,
1100 grams for Tk. 1000
Hence, 1188 grams for Tk. 1080
Selling: 900 grams for Tk. 1080
Hence % profit = (288/900) × 100 = 32%

মনে করি,
ক্রয়মূল্য = 100 টাকা
∴ 10% loss এ বিক্রয়মূল্য = 90 টাকা
এবং 5% profit এ বিক্রয়মূল্য = 105 টাকা
∴ বিক্রয়মূল্য (105 - 90) = 15 টাকা বেশি হলে,
5% profit হতো।
এখন ঐকিক নিয়ম ব্যবহার করে, বিক্রয়মূল্য 15 টাকা বেশি হলে ক্রয়মূল্য 100 টাকা
∴ বিক্রয়মূল্য 1 টাকা বেশি হলে ক্রয়মূল্য (100/15) টাকা
∴ বিক্রয়মূল্য 6 টাকা বেশি ক্রয়মূল্য (100/15) × 6
= 40 টাকা।

Let,
the number be x.
Then, 75% of x + 75 = x
⇒ x - 75x/100 = 75
⇒ x - 3x/4 = 75
⇒ x/4 = 75
⇒ x = 300

Question: If a student of the mathematics department sells all his books for Tk. 7,500, he loses 25%. To gain 10%, at what price should he sell the books?

Solution:
Let,
The cost price of the books x.
Selling price = x - 25% of x
⇒ 7500 = x - (25x/100)
⇒ 7500 = 75x/100
⇒ 75x = 750000
∴ x = 10000

To gain 10% = 10000 + 10% of 10000
= 10000 + 1000
= Tk. 11000

To gain 10%, he should sell the books for Tk. 11000.

y% of x = 29
Or, y/100 × x = 29
So, x = 29×100/y = 2900/y

Question: A reduction of 10% in the price of tea enables a dealer to purchase 25 kg more tea for Tk 22500. What is the reduced price per kg of tea?

Solution:
Let 10% of 22500 = 2250
Now,
25 kg = 2250
⇒ 1kg = 2250/25
∴ 1kg = 90

In the original 125 gallons of mixture, 20% is water.
Hence, no. of gallons of other materials in the mixture: 125 x 80% = 100 gallons
In the new mixture, water makes up 25%, thus 75% is other materials.
As no. of gallons of others is unchanged, 100 gallons = 75% in the new mixture volume.
The total volume of the new mixture is : 100 / 75% = 100/ 0.75 = 133.33 gallons.
∴ Required additional amount of water = 133.3 – 125 = 8.33 = 8(1/3) gallons

Question: What percent of 500 is 1.5?

Solution:
ধরি,
500 এর x% = 1.5
⇒ 500 এর (x/100) = 1.5
⇒ 5x = 1.5
⇒ x = 1.5/5
⇒ x = 0.3

​সুতরাং, 500 এর 0.3% হলো 1.5

At p% profit, selling price = 100 + p
at p% loss, new selling price = (100 + p) - (100 + p) × p/100
= (100 + p) - {(100p + p²)/100}
= 10000 + 100p - 100p - p²
= (10000 - p²)/100

so, (10000 - p²)/100 is equal to 1
1 is equal to 1/(10000 - p²)/100
100 is equal to 1 × (100/ 10000 - p²) × 100
= 10000/(10000 - p²)

অপশন গ) (10000/10000-p^2) হওয়ার কথা ছিলো। সেক্ষেত্রে উত্তর ঠিক হয়।
লাইভ পরীক্ষার প্রশ্নে অপশন ভুল থাকায় উত্তর হিসাবে ঙ) None of above কে ধরা হয়েছে।

Question: At what percentage profit must an article be sold such that selling it at two-thirds of that price results in a 20% loss?
​
​Solution:
Let,
Cost Price be x .
Selling Price be y.
​
​Selling at 2/3 of y, causes 20% loss,
So, 2y/3 = x - 20% of x
⇒ 2y/3 = x - (20x/100)
⇒ 2y/3 = x{(1 - (20/100)}
⇒ 2y/3 = x × (80/100)
∴ y = 6x/5
​
​Profit = Selling Price - Cost Price
= (6x/5) - x
= (6x - 5x)/5
= x/5
​
​∴ Profit Percentage = (Profit/Cost Price) × 100%
= {(x/5)/x} × 100%
= 20%

Question: A man spends 45% of his salary on rent, 25% on food, and saves the rest. If his salary is Tk. 60000. how much does he save?
 
Solution: 
Given that, 
Total salary = Tk. 60000

Now, 
Spends on rent = 45% of 60000
= (45/100) × 60000
= Tk. 27000 

And spends on food = 25% of 60000
= (25/100) × 60000
= Tk. 15000

∴ Total expenditure = rent + food
= 27000 + 15000
= Tk. 42000

∴ Savings = Total salary - Total expenditure
= 60000 - 42000
= Tk. 18000

He saves Tk. 18000

Shortcut method:
Percentage saved = 100% - (45% + 25%)
= 100% - 70%
= 30%

∴ Savings = 30% of 60000
= 0.30 × 60,000
= Tk. 18000

A:: - G:S = 40:60
B:: - G:S:C = 35:40:25
New,
G:S= (1×40 + 4×35) : (40×4 + 1×60)
= 180:220
= 18:22
= 9:11

Question: A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Tk. 1000 more than D, what is B's share?

Solution: 
let, they get 5x, 2x, 4x, 3x 

4x - 3x = 1000 
⇒ x = 1000 

∴ B's share is = 2x 
= 2 × 1000 
= 2000 taka 

Question: If the price of a commodity is decreased by 25% and its consumption is increased by 25%, what will be the increase or decrease in expenditure on the commodity?

Solution:
Let,
The initial expenditure on the commodity be Tk. 100.
Now, the price decreases by 25%,
∴ Current Price = (100 - 25% of 100) = Tk. 75

Same time due to decrements in price 25% consumption has been increased.
So, Current expenses on commodity = (75 + 25% of 75) = Tk. 93.75

Here,
The initial expenditure was Tk. 100 which became Tk. 93.75 at the end, it means there is 6.25% decrements in the expenditure of the commodity.

So the expenditure decreases by 6.25%

Question: A and B are two fixed points 5 cm apart and C is a point on AB such that AC is 3cm. if the length of AC is increased by 6%, the length of CB is decreased by-

Solution:
Given that,
AB = 5 cm (fixed)
Initially, AC = 3 cm
So, CB = AB - AC = 5 - 3 = 2 cm

Now,
Increase in AC = 6%
∴ Increase in AC = (106/100) × 3 = 3.18cm
And,
Decrease in CB = 5 - 3.18 = 1.82 cm
∴ Decrease = 2 - 1.82 = 0.18 cm

So Percentage of decrease = (0.18/2) × 100%
= (18/2)%
= 9%

Thus, the length of CB is decreased by 9%.

Let
Marked price = X
Cost Price(C.P.) = 1440
Sale price(S.P.) = 1440 + 25% of 1440
= Tk. 1800

S.P. = M.P. - 25% of M.P.
S.P. = X - 25% of X
S.P. = X - 0.25X
1800 = 0.75X
X = 1800/.25
= 2400
M.P. = Tk. 2400

Let the Market Price of the product is MP.
Let the Original Cost Price of the product is CP.
Selling Price (Discounted Price) = 100% of MP - 20% of MP = 80% of MP ......... (1)
Profit made by selling at discounted price = 20% of CP ........... (2)
Apply the formula:
Profit = Selling Price - Original Cost Price
⇒ 20% of CP = 80% of MP - 100% CP
⇒ MP = (120×CP)/80 = 3/2 × CP
Now if product is sold without any discount, then,
Profit = Selling Price (without discount) - Original Cost Price
= Market Price - Original Cost Price
= MP - CP
= 3/2 CP - CP
= 1/2 CP
= 50% of CP

1000000 × 0.75 × 0.80 × 0.85=  5,10,000

Question: In January, the value of a stock increased by 50%; and in February, it decreased by 20%. In March, it increased by 25%; and in April, it decreased by 10%. If a person invested Tk. 200 in the stock on January 1 and sold it at the end of April, what was the percentage change in the price of the stock?

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: If a shopkeeper buys eggs at Tk. 42 per sext (six eggs) and sells them at Tk. 56 per two quads (eight eggs), what percentage profit does he make?

Solution:
Cost price of 6 eggs = Tk. 42
Cost price of 1 egg = 42/6
= 7 Tk.

Cost price of 8 eggs = 8 × 7
= 56
Selling price of 8 eggs = Tk. 56

∴ Profit = 56 − 56
= 0 

∴ As the selling and cost prices are the same, there is no profit. 

Question: There is a dishonest shopkeeper whose claim is that he sells a certain product at a cost of Tk 23/kg, which actually costs him Tk 25/kg. The shopkeeper says that he is taking the loss to let his customers get a better deal. When examined thoroughly, a policeman finds that the shopkeeper is actually using an 800 gms weight in place of a 1 kg weight. How much does he gain or lose?

Solution: 
selling price of 0.8 kg = 23 taka

selling price of 1 kg = 23/0.8 taka 
= 115/4 taka
= 28.75 taka 

percentage profit = {(28.75 - 25)/25} × 100%
= (3.75/25) × 100%
= 15%