Exam - 43 Math: Topics: Ratio&Proportion, Partnership, Allegation or Mixture, Stock & Share

Question: 500 grams of salt solution has 20% salt in it. How much salt should be added to make 60% in the solution?

Solution:
Amount of salt = 500 × 20/100
= 100 grams
Let,
x gram salt to be added

ATQ,
(100 + X)/(500 + X) = 60%
⇒ (100 + X)/(500 + X) = 60/100
⇒ (100 + X)/(500 + X) = 3/5
⇒ 5(100 + X) = 3(500 + X)
⇒ 5X + 500 = 3X + 1500
⇒ 5X - 3X = 1500 - 500
⇒ 2X = 1000
∴ X = 500 grams

Question: The ratio of the present ages of Adib and his father is 3 : 8. The father's age at the time of Adib's birth was 25 years. Find the father's present age.

Solution:
Present ratio is 8 : 3
Let actual ages are 8x and 3x.

∴ 8x - 3x = 25
⇒ 5x = 25
∴ x = 5

Hence the father's present age = 8 × 5 = 40 years

Question: A began a business with Tk. 60000. He was joined afterwards by B with Tk. 45000. For how much period does B join, if the profits at the end of the year are divided in the ratio of 2 : 1?

Solution:
Suppose B joined for x months.

Then,
(60000 × 12)/(45000 × x) = 2/1
⇒ 2(45000 × x) = (60000 × 12)
⇒ x = (60000 × 12)/(45000 × 2)
∴ x = 8

So, B joined for 8 months.

Question: A grocer mixes two types of rice, one costing 40 taka per kg and the other 55 taka per kg. If the mixture is sold at 46 taka per kg, what is the ratio of two types of rice in the mixture?

Solution:
মনে করি,
প্রথম প্রকার চালের পরিমাণ = x কেজি
দ্বিতীয় প্রকার চালের পরিমাণ = y কেজি

প্রথম প্রকার চালের x কেজির মূল্য = 40x টাকা
দ্বিতীয় প্রকার চালের y কেজির মূল্য = 55y টাকা

প্রশ্নমতে,
40x + 55y = 46(x + y)
⇒ 40x + 55y = 46x + 46y
⇒ 46x - 40x = 55y - 46y
⇒ 6x = 9y
∴ x/y = 9/6 = 3 : 2

Question: Two numbers are 25% and 40% less than a third number respectively. The ratio of first two numbers is-

Solution:
Given that,
Two numbers are 25% and 40% less than a third number respectively.
Let the third number be 100.

Now,
First number = 100 - 25% of 100
= 100 - 25
∴ First number = 75
And
Second number = 100 - 40% of 100
= 100 - 40
∴ Second number = 60

∴ Ratio of the first two numbers = First number/Second number
= 75/60
= 5/4

∴ The ratio of the first two numbers is 5 : 4.

Question: If A : B = 2 : 5, B : C = 3 : 4, and C : D = 6 : 7 then A : D is equal to-

Solution:
Given that,
A : B = 2 : 5, B : C = 3 : 4, and C : D = 6 : 7
Then,
A/D = (A/B) × (B/C) × (C/D)
= (2/5) × (3/4) × (6/7)
= 36/140
= 9/35
= 9 : 35
∴ A : D = 9 : 35

Question: X, Y, and Z are partners in a business. Their investments are in the ratio 1/2 : 1/3 : 1/6 respectively. Y withdraws half of his capital after 4 months. The business runs for 12 months. If the total profit is Tk 9,600, what is Z's share of the profit?

Solution:
Ratio of initial investments = 1/2 : 1/3 : 1/6
= 3 : 2 : 1

Let their initial investments be 3x, 2x, and x respectively.

X : Y : Z = (3x × 12) : (2x × 4) + (2x × 1/2 × 8) : (x × 12)
= 36x : (8x + 8x) : 12x
= 36x : 16x : 12x
= 9 : 4 : 3

Sum of the ratio = 9 + 4 + 3 = 16

Total profit = Tk 9,600
∴ Z's share = 9600 × (3/16)
= 28800/16
= 1800
∴ Z's share of the profit is Tk 1,800

Question: If 0.4 : x :: 5 : 8, then x is equal to:

Solution:
We have the proportion 0.4 : x :: 5 : 8.
Product of means = Product of extremes
x × 5 = 0.4 × 8
5x = 3.2
x = 3.2 / 5
x = 0.64.

Question: The incomes of Ram and Shyam are in the ratio of 4 : 3 and their expenditures are in the ratio of 3 : 2. If each saves Tk. 5000, what is Ram's income?

Solution:
Let, Ram's income be 4x and Shyam's be 3x.
 Ram's expenditure be 3y and Shyam's be 2y.

∴ Savings = Income - Expenditure.
For Ram: 4x - 3y = 5000 ... (1)
For Shyam: 3x - 2y = 5000 ... (2)

Multiply (1) by 2 and (2) by 3 and subtract the first from the second
9x - 6y - 8x + 6y = 15000 - 10000 
⇒ x = 5000
Ram's income = 4x = 4 × 5000 = Tk. 20000.

Question: What annual income can be earned by investing Tk. 10,800 in 15% stock at 90?

Solution:
The market price of the stock is Tk. 90 for every Tk. 100 of face value.

 ∴ face value
= (100/90) × 10,800
= Tk. 12,000

The stock pays a dividend of 15% on its face value.

 ∴ annual income
= 15% of Tk. 12,000
= (15/100) × 12,000
= Tk. 1,800

Therefore, the annual income is Tk. 1,800.

Question: The ratio of alcohol and water in a mixture is 4 : 1. The percentage of water in the mixture is:

Solution:
Given that, 
Ratio of alcohol and water = 4 : 1.
Total parts = 4 + 1 = 5.

Percentage of water = (Part of water/Total parts) × 100
= (1/5) × 100
= 20%.

Question: Coffee worth Tk. 200 per kg and Tk. 240 per kg are mixed with a third variety in the ratio 1 : 2 : 2. If the mixture is worth Tk. 260 per kg, the price of the third variety per kg will be:

Solution:
Let,
the quantities of the three varieties be 1 kg, 2 kg, and 2 kg respectively.
Total mixture = 1 + 2 + 2 = 5 kg.
1 kg mixture value = Tk. 260
∴ 5 kg or Total value of the mixture = 5  × 260 = Tk. 1300.

Value of first variety = 1 kg × Tk. 200 = Tk. 200.
Value of second variety = 2 kg × Tk. 240 = Tk. 480.
Total value of the first two varieties = Tk. 200 + Tk. 480 = Tk. 680.

Value of the third variety (2 kg) = 1300 - 680 = Tk. 620.
∴ Price per kg of the third variety = Tk. 620/2 kg = Tk. 310.

Question: The ratio of a person’s expenditure to savings is 3 : 2. If his income increases by 15% and his savings increase by 6%, by what percentage will his expenditure increase?

Solution:
Let the original expenditure be 3x and the original savings be 2x.

∴ the original income
= 3x + 2x
= 5x

After a 15% increase, the new income
= 5x × (115/100)
= 5.75x

After a 6% increase, the new savings
= 2x × (106/100)
= 2.12x

∴ new expenditure
= New income - New savings
= 5.75x - 2.12x
= 3.63x

Increase in expenditure
= 3.63x - 3x
= 0.63x

Therefore, the percentage increase in expenditure
= (0.63x/3x) × 100%
= 21%

Question: Two numbers are respectively 25% and 40% more than a third number. The ratio of the two numbers is:

Solution:
Let,
the third number be 100.

∴ First number = 100 + 25% of 100 
= 100 + 25
= 125

∴ Second number = 100 + 40% of 100 
= 100 + 40
= 140

∴ Ratio of first to second = 125 : 140 = 25 : 28.

Question: The ratio of the present ages of a daughter and her mother is 3 : 7. The mother's age at the time of the daughter's birth was 20 years. Find the mother's present age.

Solution:
Let,
the present ages be 3x and 7x.

At the time of daughter's birth (0 years old), the mother's age = mother's age - daughter's age.
∴ Difference in ages = 7x - 3x = 4x.

Given that,
The difference in ages is 20 years.
∴ 4x = 20 
⇒ x = 5.

Mother's present age = 7x = 35 years.

Question: Sayed started a software business by investing Tk. 20,000. After 6 months, Rafi joined him with a capital of Tk. 30,000. After 3 years, they earned a profit of Tk. 13,950. What was Sayed's share in the profit?
A) Tk. 6200
B) Tk. 6400
C) Tk. 7750
D) Tk. 9300

Solution:
Total time = 3 years
= 3 × 12 months
= 36 months.

Sayed's capital-time = 20,000 × 36 = 720,000.

Rafi joined after 6 months.
Rafi's time = 30 months.
∴ Rafi's capital-time = 30,000 × 30 = 900,000.

∴ Profit ratio = 720,000 : 900,000 = 72 : 90 = 4 : 5.
Total profit = 13,950.
Total ratio parts = 9.

∴ Sayed's share = (4/9) × 13,950
= 4 × 1550
= Tk. 6200.

Question: A, B, C started a business investing Tk. 250000, Tk. 300000, and Tk. 350000 respectively. Find the share of B, out of an annual profit of Tk. 187200.

Solution:
Given that,
A invests = Tk. 250000
B invests = Tk. 300000
C invests = Tk. 350000

∴ Ratio of investments = 250000 : 300000 : 350000 
= 25 : 30 : 35
= 5 : 6 : 7.
Total profit = 187200.

Total ratio parts
= 5 + 6 + 7
= 18.

∴ B's share = (6/18) × 187200
= (1/3) × 187200
= Tk. 62400.

Question: In a wallet, there are Tk 500, Tk 200, and Tk 100 notes in the ratio 2 : 5 : 7. The total amount of money in the wallet is Tk 13,500. Find the number of each note respectively.

Solution:
Let,
Number of Tk 500 notes = 2x
Number of Tk 200 notes = 5x
Number of Tk 100 notes = 7x

According to the question,
(500 × 2x) + (200 × 5x) + (100 × 7x) = 13500
⇒ 1000x + 1000x + 700x = 13500
⇒ 2700x = 13500
⇒ x = 13500/2700
∴ x = 5

∴ Number of Tk 500 notes = 2 × 5 = 10
∴ Number of Tk 200 notes = 5 × 5 = 25
∴ Number of Tk 100 notes = 7 × 5 = 35

Question: If 3 (A's capital) = 4 (B's capital) = 6 (C's capital), then out of the total profit of Tk 8,100, B will receive:

Solution:
Let,
A's capital = a
B's capital = b
C's capital = c

Then,
3a = 4b = 6c
Now,
3a = 4b
⇒ b = 3a/4
Again,
3a = 6c
⇒ c = 3a/6 = a/2

A : B : C = a : (3a/4) : (a/2)
= 4a : 3a : 2a [multiply by 4]
= 4 : 3 : 2

∴ Sum of ratio = (4 + 3 + 2) = 9
So, B's share = 8100 × (3/9)
= 900 × 3
= Tk 2,700

Question: How many kilograms of sugar costing Tk 12 per kg must be mixed with 12 kg of sugar costing Tk 6 per kg so that there may be a gain of 20% by selling the mixture at Tk 10.80 per kg?

Solution:
Let, 
sugar of Tk 12/kg = x kg
Total mixture = (x + 12) kg

Total Cost Price(CP)
= 12x + (6 × 12) 
= 12x + 72

Selling Price(SP)
= 10.80
= 54/5 Tk/kg
Profit = 20%

CP/kg = (54/5) × (100/120)
= (54/5) × (5/6)
= 54/6
= 9 

Now, 
(12x + 72)/(x + 12) = 9
⇒ 2x + 72 = 9x + 108
⇒ 3x = 36
∴ x = 12 

Question: A jar contains a milk-water mixture in which water is 50%. A part of the mixture is removed and replaced with another mixture containing 20% water. If the final mixture contains 32% water, what fraction of the original mixture was replaced?

Solution:
Let the total quantity of the mixture be 1 litre.
Suppose x litre of the mixture is removed and replaced.

Water remaining from the original mixture
= 50% of (1 - x)
= 0.50(1 - x) litre

Water added with the new mixture
= 20% of x
= 0.20x litre

∴ the total quantity of water in the final mixture
= 0.50(1 - x) + 0.20x

According to the question,
0.50(1 - x) + 0.20x = 0.32
⇒ 0.50 - 0.50x + 0.20x = 0.32
⇒ 0.50 - 0.30x = 0.32
⇒ 0.30x = 0.18
⇒ x = 0.18/0.30
⇒ x = 3/5

Therefore, 3/5 of the original mixture was replaced.

Question: Three partners shared the profit in a business in the ratio 6 : 8 : 9. They had partnered for 12 months, 10 months, and 9 months respectively. What was the ratio of their investments?

Solution:
Given,
Profit ratio of the three partners
= 6 : 8 : 9
Time of investment
= 12 months : 10 months : 9 months

We know,
Investment × Time = Profit
⇒ Investment = Profit/Time
So, the ratio of their investments will be = (6/12) : (8/10) : (9/9)
= (1/2) : (4/5) : 1

The denominators are 2 and 5.
LCM of 2 and 5 = 10.
Multiplying each term by 10:
Investment ratio = 5 : 8 : 10

Question: A man buys Tk. 50 shares paying 8% dividend. The man wants to have an interest of 10% on his money. The market value of each share is:

Solution: 
Here, the face value of each share = Tk. 50
Dividend rate = 8%
dividend per share
= 8% of Tk. 50
= 50 × (8/100)
= Tk. 4

Now, the man wants 10% interest on his money.
That means Tk. 4 should be 10% of the market value.
Let,
Market value of each share = Tk. x
According to the question,
10% of x = 4
⇒ x × (10/100) = 4
⇒ x/10 = 4
⇒ x = 40

∴ The market value of each share is Tk. 40.

Question: A and B invest in a business in the ratio 4 : 5. If 10% of the total profit goes to charity and A's share is Tk. 1080, the total profit is:

Solution: 
Given that, 
A and B invest in the ratio 4 : 5.

Total ratio = 4 + 5 = 9
A’s share = 4 parts

Given,
A’s share = Tk. 1080
∴ 4 parts = Tk. 1080
∴ 1 part = 1080/4
= Tk. 270

∴ total distributed profit
= 9 parts
= 270 × 9
= Tk. 2430

Now,
10% of the total profit goes to charity.
So, A and B receive only 90% of the total profit.

Therefore,
90% of total profit = Tk. 2430
∴ Total profit = 2430 × (100/90)
= 2700

∴ The total profit is Tk. 2700.

Question: A 15% stock yields 10%. What is the market value of the stock?

Solution:
Given that, 
A 15% stock gives an annual dividend of Tk. 15 on a face value of Tk. 100.

We know,
Yield% = (Annual dividend/Market value) × 100
∴ 10 = (15/Market value) × 100
⇒ Market value = (15 × 100)/10
⇒ Market value = Tk. 150

∴ the market value of the stock is Tk. 150.