ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: A vessel contains 140 litres of milk and water in the ratio of 4 : 3. If 20 litres of milk and 30 litres of water are added to the mixture, the difference between milk and water in the final mixture is Y. Find the value of 7Y?

Solution:
Given,
Total mixture = 140 litres
Ratio of milk and water = 4 : 3
Sum of the ratios = 4 + 3 = 7

Initially,
Quantity of milk = 140 × (4/7) = 80 litres
Quantity of water = 140 × (3/7) = 60 litres

After adding 20 litres of milk and 30 litres of water:
New quantity of milk = 80 + 20 = 100 litres
New quantity of water = 60 + 30 = 90 litres

According to the question, the difference between milk and water is Y:
∴ Y = |100 - 90| = 10 litres

∴ The value of 7Y = 7 × 10 = 70

Question: A 15% stock yields 12.5%. What is the market value of the stock?

Solution:
Earn Tk. 12.5 when market value = Tk. 100
Earn Tk. 1 when market value = Tk. (100 / 12.5)
Earn Tk. 15 when market value = Tk. (100 × 15)/12.5
= 1500/12.5
= 120

∴ The market value of the stock is Tk. 120.

Question: A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. The quantity sold at 18% profit is:

Solution:
Total quantity of sugar = 1000 kg

Let x kg be sold at 18% profit
Then (1000 - x) kg be sold at 8% profit

Profit from x kg at 18% = 18x/100
Profit from (1000 - x) kg at 8% = 8(1000 - x)/100

Total profit = 14% of 1000 = 14000/100 = 140

According to the question,
18x/100 + 8(1000 - x)/100 = 140
⇒ 18x + 8(1000 - x) = 14000
⇒ 18x + 8000 - 8x = 14000
⇒ 10x = 14000 - 8000
⇒ 10x = 6000
⇒ x = 600

∴ 600 kg of sugar was sold at 18% profit.

বিকল্প সমাধান:
By the rule of alligation,

Ration of 1^st and 2^nd parts = 4 : 6 = 2 : 3
∴ Quantity of 2^nd kind = (3/5) × 1000 = 600 kg

Question: P, Q, and R started a business by investing Tk. 120,000, Tk. 180,000 and Tk. 200,000 respectively. Find the share of Q out of an annual profit of Tk. 125,000.

Solution:
The ratio of investments of P, Q, and R is:
P : Q : R = 120,000 : 180,000 : 200,000
= 12 : 18 : 20
= 6 : 9 : 10

Sum of the ratios = 6 + 9 + 10 = 25
Total annual profit = Tk. 125,000

Q's share of profit = (9/25) × 125,000
= 9 × 5,000
= 45,000 Tk.

Question: How much water be mixed in 45 litre of milk worth Tk. 6.40 per litre, so that value of mixture is Tk. 4.80 per litre?

Solution:
মনে করি, x লিটার পানি মেশাতে হবে।

মিশ্রণের মোট পরিমাণ = (45 + x) লিটার
45 লিটার দুধের দাম = 45 × 6.40 = 288 টাকা
পানির দাম = 0 টাকা

প্রশ্নমতে, মিশ্রণের মোট দাম = (45 + x) × 4.80
যেহেতু পানি বিনামূল্যে পাওয়া যায়, তাই মোট দাম অপরিবর্তিত থাকবে:
288 = (45 + x) × 4.80
⇒ 288/4.80 = 45 + x [উভয় পক্ষকে 4.80 দ্বারা ভাগ করে পাই]
⇒ 60 = 45 + x
⇒ x = 60 - 45
⇒ x = 15 litres 

∴ 15 লিটার পানি মেশাতে হবে।

Question: The capital stock of a company is Tk. 500,000 and is divided into 5,000 shares. If the company declares a total dividend of Tk. 75,000, how much will Rahim receive for his 80 shares?

Solution:
5,000 shares income Tk. 75,000
∴ 1 share income = Tk. 75,000 / 5,000
∴ 80 shares income = Tk. (75,000 × 80)/5,000
= Tk. 1,200

∴ Rahim will receive Tk. 1,200 as his share of the dividend.

Question: A mixture of 200 liters of milk and water contains 15% water. How much more water should be added so that water becomes 20% of the new mixture?

Solution:
Amount of water in the 200-liter mixture = 15% of 200
= (15/100) × 200
= 30 liters

Let,
P liters of water are added.
So, new amount of water = (30 + P)
and new total mixture = (200 + P)

ATQ,
(30 + P) = 20% of (200 + P)
⇒ 30 + P = (20/100) × (200 + P)
⇒ 30 + P = (1/5) × (200 + P)
⇒ 150 + 5P = 200 + P
⇒ 5P - P = 200 - 150
⇒ 4P = 50
∴ P = 12.5

∴ 12.5 liters more water should be added.

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: Tea worth Tk. 240 per kg and Tk. 280 per kg are mixed with a third variety in the ratio 3 : 2 : 5. If the mixture is worth Tk. 300 per kg, the price of the third variety per kg will be:

Solution:
Let the price of the third variety be x Tk. per kg.
The given ratio of the three varieties is 3 : 2 : 5.
For calculation, let the quantities be 3 kg, 2 kg, and 5 kg respectively.

Total weight of the mixture = (3 + 2 + 5) = 10 kg
Total value of the mixture = 10 × 300 = Tk. 3000

According to the question (ATQ),
(3 × 240) + (2 × 280) + (5 × x) = 3000
⇒ 720 + 560 + 5x = 3000
⇒ 1280 + 5x = 3000
⇒ 5x = 3000 − 1280
⇒ 5x = 1720
⇒ x = 1720 / 5
∴ x = 344

∴ The price of the third variety is Tk. 344 per kg.

Question: A woman purchased 500 shares of the face value of Tk. 100 each from the market at Tk. 160 per share. If a dividend of 20% is declared, find her earning percent on the investment.

Solution:
Given,
Price of 1 share (Market Value) = Tk. 160
∴ Price of 500 shares (Total Investment) = Tk. (160 × 500) = Tk. 80,000

Dividend per share = 20% of 100 = Tk. 20
∴ Total dividend income = 500 × 20 = Tk. 10,000

∴ Earning % = (Total Dividend/Total Investment) × 100
= (10,000/80,000) × 100
= (1/8) × 100
= 12.5%

Question: The milk and water in a mixture are in the ratio 5 : 4. When 15 litres of water are added to it, the ratio of milk and water in the new mixture becomes 5 : 7. The total quantity of water in the new mixture is:

Solution:
Let the initial quantity of milk = 5x litres
and initial quantity of water = 4x litres

According to the question,15 litres of water is added.

New amount of water = (4x + 15)
The amount of milk remains the same = 5x

As per the new ratio,
5x/(4x + 15) = 5/7
⇒ 7(5x) = 5(4x + 15)
⇒ 35x = 20x + 75
⇒ 35x − 20x = 75
⇒ 15x = 75
⇒ x = 5

The total quantity of water in the new mixture,
= (4x + 15)
= (4 × 5) + 15
= 20 + 15
= 35 litres

∴ Total quantity of water in new mixture 35 litres.

Question: What yearly income can be earned by investing Tk. 18000 in 12% stock at Tk. 90?

Solution:
12% stock at Tk. 90 বলতে বুঝায় স্টকটির ফেস ভ্যালু 100 টাকায় 12 টাকা লাভ হয় এবং স্টকটির বাজার মূল্য 90 টাকা।

এখন, 
90 টাকা বিনিয়োগ করে আয় হয় 12 টাকা
∴ 1 টাকা বিনিয়োগ করে আয় হয় 12/90 টাকা
∴ 18000 টাকা বিনিয়োগ করে আয় হয় (18000 × 12)/90 টাকা
= 216000/90 টাকা = 2400 টাকা

Question: In what ratio must a mixture of 30% alcohol strength be mixed with that of 50% alcohol strength so as to get a mixture of 45% alcohol strength?

Solution:
মনে করি, প্রথম প্রকারের x একক এবং দ্বিতীয় প্রকারের y একক মিশ্রিত করতে হবে।

প্রশ্নমতে,
প্রথম প্রকারের অ্যালকোহলের পরিমাণ = 30% of x = 0.30x
দ্বিতীয় প্রকারের অ্যালকোহলের পরিমাণ = 50% of y = 0.50y

মিশ্রণের মোট পরিমাণ = (x + y)
মিশ্রণে অ্যালকোহলের মোট পরিমাণ = 45% of (x + y) = 0.45(x + y)

শর্তমতে,
0.30x + 0.50y = 0.45(x + y)
⇒ 30x + 50y = 45(x + y)  [উভয় পক্ষকে 100 দ্বারা গুণ করে]
⇒ 30x + 50y = 45x + 45y
⇒ 50y - 45y = 45x - 30x
⇒ 5y = 15x
⇒ x/y = 5/15
⇒ x/y = 1/3
∴ x : y = 1 : 3

∴ The ratio is 1 : 3

Question: A man invested Tk. 36,000 when he bought Tk. 100 shares at Tk. 144. If 16% dividend is declared, find his annual income.

Solution:
For Tk. 144 he gets Tk. 16
∴ For Tk. 1 he gets Tk. 16/144
∴ For Tk. 36,000 he gets Tk. (16 × 36,000)/144
= Tk. (1/9) × 36,000
= Tk. 4,000

∴ The man's annual income is Tk. 4000.

Question: In what ratio must a grocer mix two varieties of pulses costing Tk. 15 and Tk. 20 per kg respectively so as to get a mixture worth Tk. 16.50 per kg?

Solution:
Let the ratio be x : y

Cost of first variety = Tk. 15 per kg
Cost of second variety = Tk. 20 per kg 
Cost of mixture = Tk. 16.50 per kg

For x kg of first variety and y kg of second variety:
Total cost of mixture = (x × 15) + (y × 20) = 15x + 20y
Total quantity of mixture = (x + y) kg

According to the question,
(15x + 20y)/(x + y) = 16.50
⇒ 15x + 20y = 16.50(x + y) 
⇒ 15x + 20y = 16.50x + 16.50y 
⇒ 20y - 16.50y = 16.50x - 15x 
⇒ 3.50y = 1.50x 
⇒ y/x = 1.50/3.50 
⇒ y/x = 150/350 
⇒ y/x = 3/7

Therefore, x : y = 7 : 3

বিকল্প সমাধান:
By the rule of alligation,

 ∴ Required rate = 3.50 : 1.50 = 7 : 3