Ratio & Proportion, Alligation or Mixture

PrepBank · পাতা ৭ / ১১ · ৬০১–৭০০ / ১,০৮৬

৬০১.

A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk respectively-

ক
ক) 4 : 5
খ
খ) 1 : 5
গ
গ) 1 : 4
ঘ
ঘ) 2 : 5

ব্যাখ্যা

Question: A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk respectively-

Solution:
Let the milkman has the milk of Tk 100
After mixing the water the mixture sold for Tk 100 + 25 = Tk 125

In Tk 125, Milk is of Tk 100, and water is of Tk 25

So, the ratio of water and milk in the mixture = 25 : 100 = 1 : 4

৬০২.

Rafi, Nabil, and Hasan started a business together. Rafi invested one-fourth of the total capital, Nabil invested one-sixth, and the remaining capital was invested by Hasan. What is the ratio of their profits?

ক
3 : 2 : 5
খ
3 : 5 : 7
গ
1 : 2 : 7
ঘ
3 : 2 : 7

ব্যাখ্যা

Question: Rafi, Nabil, and Hasan started a business together. Rafi invested one-fourth of the total capital, Nabil invested one-sixth, and the remaining capital was invested by Hasan. What is the ratio of their profits?

Solution:
Let the total capital be 12x
Then, Rafi's share = 12x × (1/4) = 3x
Nabil's share = 12x × (1/6) = 2x
Hasan's share = 12x - (3x + 2x) = 7x

So, required ratio = 3x : 2x : 7x = 3 : 2 : 7

৬০৩.

k:l = 4:3 and l:m = 5:3, then find k:l:m?

ক
ক) 20 : 15 : 9
খ
খ) 18 : 24 : 11
গ
গ) 9 : 15 : 1
ঘ
ঘ) 21 : 7 : 3

ব্যাখ্যা

Given k : l = 4 : 3 = 20 : 15
and, l : m = 5 : 3 = 15 : 9
∴ k : l : m = 20 : 15 : 9

৬০৪.

In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 5 : 7. If the shop carries 48 cats and bunnies total in stock, how many dogs are there?

ক
9
খ
12
গ
15
ঘ
21

ব্যাখ্যা

Question: In a certain pet shop, the ratio of dogs to cats to bunnies in stock is 3 : 5 : 7. If the shop carries 48 cats and bunnies total in stock, how many dogs are there?

Solution:
5x + 7x = 48
⇒ 12x = 48
⇒ x = 48/12 = 4

number of dogs = 3x
= 3 × 4
= 12

৬০৫.

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?

ক
35
খ
70
গ
84
ঘ
105

ব্যাখ্যা

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

৬০৬.

A mixture contains two liquids 'A' and 'B' are in the ratio 4 : 1. If 10 litres of mixture is withdrawn and replaced with 10 litres of 'B', then the ratio becomes 2 : 3. What was the initial quantity of A?

ক
24 ltr.
খ
16 ltr.
গ
20 ltr.
ঘ
12 ltr.

ব্যাখ্যা

Question: A mixture contains two liquids 'A' and 'B' are in the ratio 4 : 1. If 10 litres of mixture is withdrawn and replaced with 10 litres of 'B', then the ratio becomes 2 : 3. What was the initial quantity of A?

Solution:
ধরি, মিশ্রণের প্রাথমিক পরিমাণ = 5x লিটার

A এর পরিমাণ = 4x লিটার
B এর পরিমাণ = x লিটার

∴ 10 লিটার মিশ্রণ তুলে নেওয়ার পর,
A এর পরিমাণ = 4x - (4/5) × 10 = 4x - 8 লিটার
B এর পরিমাণ = x - (1/5) × 10 = x - 2 লিটার

আবার,
B তে 10 লিটার যোগ করার পর,
B এর পরিমাণ = x - 2 + 10 = x + 8 লিটার

∴ প্রদত্ত অনুপাত,
⇒ (4x - 8)/(x + 8) = 2/3
⇒ 12x - 24 = 2x + 16
⇒ 10x = 16 + 24
⇒ x = 40/10
⇒ x = 4

∴ A এর পরিমাণ = 4 × 4 = 16 লিটার

৬০৭.

The sides of a triangle are in the ratio 1/2 : 1/3 : 1/4 and its perimeter is 208 cm. The length of the long side is -

ক
ক) 92 cm
খ
খ) 96 cm
গ
গ) 72 cm
ঘ
ঘ) 78 cm

ব্যাখ্যা

Question: The sides of a triangle are in the ratio 1/2 : 1/3 : 1/4 and its perimeter is 208 cm. The length of the long side is -

Solution:
Ratio of sides = 1/2 : 1/3 : 1/4
= (1/2) × 12 : (1/3) × 12 : (1/4) × 12
= 6 : 4 : 3

Let, the sides be 6x, 4x, and 3x

Then,
6x + 4x + 3x = 208
⇒ 13x = 208
⇒ x = 208/16
⇒ x = 16

The length of the long side is = 16 × 6 = 96 cm

৬০৮.

What is the ratio of one-fourth of

and three-fourths of the same number?

ক
1 : 2
খ
3 : 2
গ
1 : 3
ঘ
2 : 5

ব্যাখ্যা

Question: What is the ratio of one-fourth of

and three-fourths of the same number?

Solution:
one-fourth of 9/20 is {(1/4) × (9/20)} = 9/80
three-fourth of 9/20 is {(3/4) × (9/20)} = 27/80

∴ ratio = 9/80 : 27/80 = 9 : 27 = 1 : 3

৬০৯.

The ratio of pens and books in a shop is 5 : 2 respectively. The average number of pens and books is 322. What is the number of books in the shop?

ক
ক) 92 Pieces
খ
খ) 276 Pieces
গ
গ) 460 Pieces
ঘ
ঘ) 184 Pieces

ব্যাখ্যা

Question: The ratio of pens and books in a shop is 5 : 2 respectively. The average number of pens and books is 322. What is the number of books in the shop?

Solution:
Let,
There are pens in the shop = 5x
There are books in the shop = 2x

ATQ,
(5x + 2x)/2 = 322
⇒ 7x/2 = 322
⇒ 7x = 644
∴ x = 92

∴ There are books in the shop = (2 × 92) = 184 Pieces.

৬১০.

Two numbers X and Y are in the ratio 5 : 8 and their sum is 52. Then Y is-

ক
ক) 24
খ
খ) 32
গ
গ) 36
ঘ
ঘ) 40

ব্যাখ্যা

Question: Two numbers X and Y are in the ratio 5 : 8 and their sum is 52. Then Y is-

Solution:
let X = 5a and Y = 8a

∴ 5a + 8a = 52
13a = 52
a = 4

X = 20
Y = 32

৬১১.

A started a business with tk 21,000 and is joined afterwards by B with tk 36,000. After how many months did B join if the profits at the end of the year are divided equally?

ক
ক) 4
খ
খ) 5
গ
গ) 6
ঘ
ঘ) 7

ব্যাখ্যা

Suppose B joined after x months
21000 × 12 = 36000 × (12 - x)
⇒ 36x = 180
⇒ x = 5

৬১২.

A vessel contains a mixture of P and Q in the ratio of 5 : 3. 16 liters of this mixture is taken out and 5 liters of P is poured in. The new mixture has a ratio of P to Q as 11 : 6. Find the total original quantity of mixture.

ক
98 liters
খ
96 liters
গ
94 liters
ঘ
92 liters

ব্যাখ্যা

Question: A vessel contains a mixture of P and Q in the ratio of 5 : 3. 16 liters of this mixture is taken out and 5 liters of P is poured in. The new mixture has a ratio of P to Q as 11 : 6. Find the total original quantity of mixture.

Solution:
Let,
Original Quantity of P = 5x
Original Quantity of Q = 3x

The quantity of P and Q in 16 liters of the mixture:
Quantity of P = (16 × 5x)/8x = 10 liters
Quantity of Q = (16 × 3x)/8x = 6 liters

Now,
5 liters of P poured in so the Quantity of P will be = 5x - 10 + 5 liters
= 5x - 5 liters

ATQ,
(5x - 5)/(3x - 6) = 11/6
⇒ 6(5x - 5) = 11(3x - 6)
⇒ 30x - 30 = 33x - 66
⇒ 3x = 36
∴ x = 12

So total mixture originally = 8x = 8 × 12 = 96 liters

৬১৩.

If a rectangular photograph that Is 10 inches wide by 15 inches long Is to be enlarged so that the width will be 22 inches and the ratio of width to length will be unchanged, then the length, in inches, of the enlarged photograph will be-

ক
33
খ
32
গ
30
ঘ
27
ঙ
25

ব্যাখ্যা

Question: If a rectangular photograph that is 10 inches wide by 15 inches long is to be enlarged so that the width will be 22 inches and the ratio of width to length will be unchanged, then the length, in inches, of the enlarged photograph will be-

Solution:
We can use the ratio width/length
Let x = the length of the enlarged photograph

ATQ,
10/15 = 22/x
⇒ 2/3 = 22/x
⇒ 2x = 66
∴ x = 33

৬১৪.

Raju and Saju were carrying some money such that their money was in the ratio 3 : 8. A friend gives each of them Tk. 5 and their money is now in the ratio 2 : 5. Which is the smaller money of the two?

ক
ক) 45
খ
খ) 64
গ
গ) 105
ঘ
ঘ) 120

ব্যাখ্যা

The ratio of original money numbers = 3:8
Common factor helps in finding actual values easily
So, take 'M' as a common factor.
∴ Original numbers will be 3M and 8M
Adding 5 to them, we get (3M + 5) and (8M+5)
∴ (3M + 5)/(8M + 5) = 2/5 ............ (Ratio of new numbers is 2:5)
∴ 15M + 25 = 16M + 10
∴ M = 15
Smaller money value is 3M = 3 x 15 = 45.

৬১৫.

A cat leaps 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog?

ক
ক) 16 : 15
খ
খ) 15 : 16
গ
গ) 15 : 11
ঘ
ঘ) 11 : 15

ব্যাখ্যা

Question: A cat leaps 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog?

Solution:
Given;
3dog = 4 cat
Or, dog/cat = 4/3

Let cat's 1 leap = 3 meter and dogs 1 leap = 4 meter

Then, ratio of speed of cat and dog = (3 × 5) / (4 × 4) = 15 : 16

৬১৬.

The ratio of income and expenditure of a person is 5 : 3 per annum. If he saves Tk 24000 per annum, his monthly income is -

ক
ক) 5000 Tk
খ
খ) 6000 Tk
গ
গ) 7200 Tk
ঘ
ঘ) 8000 Tk

ব্যাখ্যা

Question: The ratio of income and expenditure of a person is 5 : 3 per annum. If he saves Tk 24000 per annum, his monthly income is -

Solution:
Let income be Tk 5x
and expenditure is Tk 3x

ATQ,
5x - 3x = 24000
⇒ 2x = 24000
⇒ x = 12000

His annual income = 5 × 12000 = 60000 Tk
His monthly income = 60000/12 = 5000 Tk

৬১৭.

The speed of three motorcycles are in the ratio 2 : 3 : 4. The ratio of the times taken by three motorcycles to travel the same distance is -

ক
ক) 6 : 5 : 3
খ
খ) 2 : 4 : 3
গ
গ) 6 : 4 : 3
ঘ
ঘ) 6 : 4 : 1

ব্যাখ্যা

Question: The speed of three motorcycles are in the ratio 2 : 3 : 4. The ratio of the times taken by three motorcycles to travel the same distance is -

Solution:
ধরি, তাদের বেগ যথাক্রমে 2x, 3x, 4x

y কিমি যেতে সময় লাগে (y/2x), (y/3x), (y/4x)

সময়ের অনুপাত = (y/2x) : (y/3x) : (y/4x)
= (1/2) : (1/3) : (1/4)
= (12/2) : (12/3) : (12/4)
= 6 : 4 : 3

৬১৮.

A jar contains only marbles of three colour: red, green and yellow. The red and green marbles are in the ratio of 2 : 5 and the yellow and red marbles are in ration of 5 : 6. Which of the following could be the total number of marbles?

ক
52
খ
64
গ
100
ঘ
None of these

ব্যাখ্যা

Question: A jar contains only marbles of three colour: red, green and yellow. The red and green marbles are in the ratio of 2 : 5 and the yellow and red marbles are in ration of 5 : 6. Which of the following could be the total number of marbles?

Solution:
Red : Green = 2 : 5 = 6 : 15
Yellow : Red = 5 : 6 = 5 : 6
Red : Green : Yellow = 6 : 15 : 5

অনুপাতের রাশিগুলোর সমষ্টি = 6 + 15 + 5 = 26
মার্বেল সংখ্যা হবে 26 এর গুণিতক।
মার্বেল সংখ্যা হতে পারে 26, 52, 78, 104,..............

৬১৯.

The ratio of investments of two partners A and B is 11 : 12, and the ratio of their profits is 2 : 3. If A invested the money for 8 months, then for how much time B invested his money?

ক
ক) 9 months
খ
খ) 10 months
গ
গ) 11 months
ঘ
ঘ) 12 months

ব্যাখ্যা

Question: The ratio of investments of two partners A and B is 11 : 12, and the ratio of their profits is 2 : 3. If A invested the money for 8 months, then for how much time B invested his money?

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

৬২০.

If x : y = 5 : 3, then (3x - 2y) : (3x + 2y) =?

ক
ক) 5 : 7
খ
খ) 3 : 1
গ
গ) 3 : 7
ঘ
ঘ) 4 : 7

ব্যাখ্যা

Question: If x : y = 5 : 3, then (3x - 2y) : (3x + 2y) =?

Solution:
x : y = 5 : 3
⇒ x/y = 5/3
⇒ 3x/2y = (5 × 3)/(3 × 2) [ 3/2 দ্বারা গুণ করে ]
⇒ 3x/2y = 15/6
⇒ (3x - 2y)/(3x + 2y) = (15 - 6)/(15 + 6)
⇒ (3x - 2y)/(3x + 2y) = 9/21 [বিয়োজন - যোজন করে]
∴ (3x - 2y):(3x + 2y) = 3/7
= 3 : 7

৬২১.

A juice vendor has two cans of juice. The first contains 30% water and the rest juice. The second contains 60% water. How much juice should he mix from each of the containers so as to get 18 litres of juice such that the ratio of water to juice is 1:2?

ক
4 litre, 12 litre
খ
6 litre, 20 litre
গ
2 litre, 16 litre
ঘ
None of the above

ব্যাখ্যা

Question: A juice vendor has two cans of juice. The first contains 30% water and the rest juice. The second contains 60% water. How much juice should he mix from each of the containers so as to get 18 litres of juice such that the ratio of water to juice is 1:2?

Solution:
Let the cost of 1 litre of pure juice be Tk. 1.

Juice in 1 litre mix from 1st can = 7/10 litre; C.P. of 1 litre mix. in 1st can Tk. 7/10
Juice in 1 litre mix from 2nd can = 4/10 litre; C.P. of 1 litre mix. in 2nd can Tk. 4/10
Juice in 1 litre of the final mix = 2/3 litre; C.P. of 1 litre of the final mix = Tk. 2/3

By the rule of alligation, we have:
Quantity of Cheaper/ Qunatity of Dearer = (CP of Dearer - Mean Price)/ (Mean Price - CP of Cheaper)
⇒ Quantity of 2nd can : Quantity of 1st can = {(7/10) - (2/3)} : {(2/3) - (4/10)}
= (21 - 20)/30 : (20 - 12)/30
= (1/30) : (8/30)
= 1 : 8

Thus, the quantity from the first can is = (1/9) × 18 = 2
The quantity from the second can is= (8/9) × 18 = 16

To get 18 litres of juice with a water-to-juice ratio of 1:2, the vendor should mix 2 litres from the first can and 16 litres from the second can.

৬২২.

The perimeter of a rectangle is 72 cm. If the ratio of the lengths of two adjacent sides is 7 : 5, find the lengths of these sides.

ক
28 cm and 20 cm
খ
21 cm and 15 cm
গ
24 cm and 18 cm
ঘ
30 cm and 12 cm

ব্যাখ্যা

Question: The perimeter of a rectangle is 72 cm. If the ratio of the lengths of two adjacent sides is 7 : 5, find the lengths of these sides.

Solution:
Perimeter of a rectangle = 2(Length + Breadth)
Also Length : Breadth = 7 : 5
Let actual values are 7x and 5x.

Hence,
2(7x + 5x) = 72
⇒ 12x = 36
∴ x = 3

Now,
Length = 7x = 7 × 3 = 21 cm
Breadth = 5x = 5 × 3 = 15 cm

∴ sides will be of 21 cm and 15 cm.

৬২৩.

Car A travels at the speed of 65 km/hr and reaches its destination in 8 hours. Car B travels at a speed of 70 km/hr and reaches its destination in 4 hours. What is the ratio of the distance covered by car A and car B respectively?

ক
ক) 7:11
খ
খ) 13:7
গ
গ) 7:13
ঘ
ঘ) 11:7

ব্যাখ্যা

Required ratio:
= (65×8):(70×4)
= 520:280
= 13:7

৬২৪.

The ratio between the two numbers is 3 : 4. If each number is increased by 12, the ratio becomes 5 : 6. The difference between the numbers will be:

ক
ক) 5
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 11

ব্যাখ্যা

The ratio between the two numbers = 3 : 4
Each number is increased by 12

Let the numbers be 3x and 4x

According to the question:
(3x + 12)/(4x + 12) = 5/6
⇒ 6 × (3x + 12) = 5 × (4x + 12)
⇒ 18x + 72 = 20x + 60
⇒ 20x - 18x = 72 - 60
⇒ 2x = 12
⇒ x = 6

Required numbers = 3x = 3 × 6 = 18
4x = 4 × 6 = 24

The difference between the numbers = 24 - 18 = 6
∴ The difference between the numbers will be 6.

৬২৫.

The ratio of Solution “A” and Solution “B” in the container is 3 : 2 when 10 liters of the mixture is taken out and is replaced by the Solution “B”, the ratio become 2 : 3. The total quantity of the mixture in the container is:

ক
ক) 20
খ
খ) 25
গ
গ) 30
ঘ
ঘ) 35
ঙ
ঙ) 40

ব্যাখ্যা

Ration of A : B = 3 : 2
10 L taken out and replaced by B
So, A remain = 3x - 3×10/5 = 3x - 6 And
B remain = 2x - 2×10/5 +10
= 2x + 6
ATQ,
(3x - 6)/(2x + 6) = 2/3
Or, 9x - 18 = 4x + 12
Or, 5x = 30
or, x = 6
The total quantity of mixture = (3x + 2x) = 5x = 5 × 6 = 30 L [Answer.]

৬২৬.

The annual income of Mim, Zim and Jorna taken together is Tk. 46000. Mim spends 70 % of income, Zim spends 80 % of her income and Jorna spends 92 % of her income. If their annual savings are 15 : 11 : 10, find the annual saving of Mim?

ক
ক) Tk. 10000
খ
খ) Tk. 3000
গ
গ) Tk. 2200
ঘ
ঘ) Tk. 7000

ব্যাখ্যা

Question: The annual income of Mim, Zim and Jorna taken together is Tk. 46000. Mim spends 70 % of income, Zim spends 80 % of her income and Jorna spends 92 % of her income. If their annual savings are 15 : 11 : 10, find the annual saving of Mim?

Solution:
Let,
Income of Mim, Zim and Jorna are A, B and C.
Annual income given is Tk. 46000

If 70 % income is spent by Mim, then that means she saves 30% = 0.3A
Similarly, Zim saves 20% = 0.2B and Jorna saves 8% = 0.08C

Given ratio of their annual savings are 15 : 11 : 10
∴ (0.3A)/15 = (0.2B)/11 = (0.08C)/10
=A/50 = B/55 = C/125
= A/10 = B/11 = C/25 = (A + B + C)/(10 + 11 + 25) = 46000/46 = 1000

∴ A = 1000 × 10 = 10000
∴ B = 1000 x 11 = 11000
∴ C = 1000 x 25 = 25000

∴ The annual saving of Mim = 0.3 × 10000
= 3000

৬২৭.

The ratio of milk to water in three containers of equal capacity is 1 : 2, 7 : 8 and 11 : 4, respectively. The three containers are mixed together. What is the ratio of milk to water after mixing?

ক
ক) 27 : 13
খ
খ) 29 : 12
গ
গ) 21 : 27
ঘ
ঘ) 23 : 22

ব্যাখ্যা

ধরি,
প্রত্যেকটি পাত্রের ধারণ ক্ষমতা x

১ম পাত্রে
দুধের পরিমাণ = x/3
পানির পরিমাণ = 2x/3

২য় পাত্রে
দুধের পরিমাণ = 7x/15
পানির পরিমাণ = 8x/15

৩য় পাত্রে
দুধের পরিমাণ =11x/15
পানির পরিমাণ = 4x/15

নতুন মিশ্রণে দুধের পরিমাণ = (x/3) + (7x/15) + (11x/15)
= (5x + 7x + 11x)/15
= 23x/15

নতুন মিশ্রণে পানির পরিমাণ = (2x/3) + (8x/15) + (4x/15)
= (10x + 8x + 4x)/15
= 22x/15

নতুন মিশ্রণে দুধ ও পানির অনুপাত = (23x/15) : (22x/15)
= 23 : 22

৬২৮.

Find the ratio in which rice at Tk. 7.20 a kg be mixed with rice at Tk. 5.70 a kg to produce a mixture worth Tk. 6.30 a kg-

ক
ক) 6 : 8
খ
খ) 2 : 3
গ
গ) 3 : 2
ঘ
ঘ) 2 : 5

ব্যাখ্যা

Question: Find the ratio in which rice at Tk. 7.20 a kg be mixed with rice at Tk. 5.70 a kg to produce a mixture worth Tk. 6.30 a kg-

Solution:
Let, Tk. 7.20 per kg rice is X kg
and Tk. 5.70 per kg rice is Y kg

ATQ,
7.2X + 5.7Y = 6.3(X + Y)
7.2X + 5.7Y = 6.3X + 6.3Y
0.9X = 0.6Y

X : Y = 0.6 : 0.9 = 2 : 3

৬২৯.

How much should be added to each term of 4 : 7 so that it becomes 2 : 3?

ক
2
খ
5
গ
3
ঘ
6

ব্যাখ্যা

Question: How much should be added to each term of 4 : 7 so that it becomes 2 : 3?

Solution:
Given that,
Ratio of two numbers is 4 : 7
Let the number added to denominator and numerator be 'x'

Now according to the question,
(4 + x) : (7 + x) = 2 : 3
⇒ (4 + x)/(7 + x) = 2/3
⇒ 12 + 3x = 14 + 2x
∴ x = 2

∴ 2 will be added to make the term in the ratio of 2 : 3.

৬৩০.

The ratio of milk and water in a solution is 25 : 12 and after adding 9 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution-

ক
15 liters
খ
25 liters
গ
35 liters
ঘ
45 liters

ব্যাখ্যা

Question: The ratio of milk and water in a solution is 25 : 12 and after adding 9 liters of water in it the ratio of milk and water becomes 5 : 3, then find the final amount of water in the final solution.

Solution:
Let,
The initial amount of milk be 25x liters
and the amount of water 12x liters.

Ratio of milk and water after adding 5 liters
25x/(12x + 9) = 5/3
⇒ 75x = 60x + 45
⇒ 15x = 45
∴ x = 3

∴ Final amount of water in solution = 12x + 9 liters.
= 36 + 9 liters.
= 45 liters.

৬৩১.

The sides of a triangle are in the ratio (1/2) : (1/3) : (1/4) and its perimeter is 104 cm . The length of the longest side(in cm ) is-

ক
ক) 42 cm
খ
খ) 44 cm
গ
গ) 24 cm
ঘ
ঘ) 48 cm

ব্যাখ্যা

Ratio of sides =(1/2) : (1/3) : (1/4)
= 6 : 4 : 3

Let
The sides is 6x, 4x and 3x

Now
6x + 4x + 3x = 104
13x = 104
x = 104/13
x = 8
The length of the longest side = 48 cm

৬৩২.

A jar contains a total of 400 coins of 5 cents and 10 cents that add up to Tk 30. Find the number of 10-cent coins.

ক
124
খ
140
গ
160
ঘ
200

ব্যাখ্যা

Question: A jar contains a total of 400 coins of 5 cents and 10 cents that add up to Tk 30. Find the number of 10-cent coins.

Solution:
Let,
the number of 10-cent coins be = x
So, the number of 5-cent coins is = (400 - x)

ATQ,
10x + {5 × (400 - x)} = (30 × 100)
⇒ 10x + 2000 - 5x = 3000
⇒ 5x = 1000
∴ x = 200

Therefore, there are 200 coins of 10 cents.

৬৩৩.

Two containers contain milk and water in the ratios 5 : 2 and 9 : 5. What ratio should the mixtures be combined in to achieve a final ratio of 2 : 1 milk to water?

ক
2 : 1
খ
1 : 2
গ
1 : 3
ঘ
1 : 4
ঙ
None

ব্যাখ্যা

Question: Two containers contain milk and water in the ratios 5 : 2 and 9 : 5. What ratio should the mixtures be combined in to achieve a final ratio of 2 : 1 milk to water?

Solution:
Let,
P unit of the first mixture is added to Q unit of the second mixture.

So, in the P unit of the first mixture,
Amount of milk present = (5/7) × P = 5P/7
Amount of water present = (2/7) × P = 2P/7

In the Q unit of the second mixture,
Amount of milk present = (9/14) × Q = 9Q/14
Amount of water present = (5/14) × Q = 5Q/14

ATQ,
{(5P/7) + (9Q/14)}/{(2P/7) + (5Q/14)} = 2/1
⇒ {(10P + 9Q)/14}/{(4P + 5Q)/14} = 2
⇒ 10P + 9Q = 8P + 10Q
⇒ 2P = Q

∴ P : Q = 1 : 2

৬৩৪.

A, B and C share the profit in the ratio of 2:3:7. If the average gain is Tk. 8000, then B's share is?

ক
ক) 2000
খ
খ) 1000
গ
গ) 1500
ঘ
ঘ) 6000

ব্যাখ্যা

Here, A : B : C
Ratio of Profit → 2 : 3 : 7
Average gain= (2+3+7)/3 = 4 units
According to the question,
4 units= Tk. 8000
1 unit= Tk. 2000
3 units=3×2000= Tk. 6000
∴Share of B= Tk. 6000

৬৩৫.

A wheel that has 5 cogs is meshed with a larger wheel of 15 cogs. When the smaller wheel has made 21 revolutions, the number of revolutions made by the larger wheel will be-

ক
ক) 9
খ
খ) 8
গ
গ) 6
ঘ
ঘ) 7

ব্যাখ্যা

Question: A wheel that has 5 cogs is meshed with a larger wheel of 15 cogs. When the smaller wheel has made 21 revolutions, the number of revolutions made by the larger wheel will be-

Solution:

smaller wheel cross = 5 × 21 = 105 cogs by 21 revolutions.
so,
larger wheel will made = 105/15 = 7 revolutions.

Shortcut:
cogs of A : cogs of B = reolution of B : revolution of A
hense,
5 : 15 = X : 21
X = 7

৬৩৬.

If a : b : c = 7 : 3 : 5, then (a + b + c) : (2a + b - c) is equal to-

ক
2 : 3
খ
4 : 3
গ
5 : 4
ঘ
6 : 5

ব্যাখ্যা

Question: If a : b : c = 7 : 3 : 5, then (a + b + c) : (2a + b - c) is equal to-

Solution:
Given,
a : b : c = 7 : 3 : 5

Let,
(a/7) = (b/3) = (c/5) = k
a = 7k, b = 3k, c = 5k

Now, (a + b + c) : (2a + b - c)
= (7k + 3k + 5k) : {(2 × 7k) + 3k - 5k)
= 15 k : 12 k
= 5 : 4

৬৩৭.

If (3/2)X = (5/7)Y = (6/5)Z, then what is X : Y : Z?

ক
105 : 50 : 84
খ
24 : 25 : 32
গ
15 : 21 : 25
ঘ
20 : 42 : 25

ব্যাখ্যা

Question: If (3/2)X = (5/7)Y = (6/5)Z, then what is X : Y : Z?

Solution:
(3/2)X = (5/7)Y = (6/5)Z .................(1)

LCM of their numerators = LCM of (3, 5, 6) = 30

Divide the eq. (1) by 30.
3X/(2 × 30) =5Y/(7 × 30) = 6Z/(5 × 30)
⇒ X/20 = Y/42 = Z/25

∴ X : Y : Z = 20 : 42 : 25

৬৩৮.

If 25% of (A + B) = 50% of (A - B), then find B : A -

ক
3 : 1
খ
5 : 1
গ
1 : 3
ঘ
1 : 1

ব্যাখ্যা

Question: If 25% of (A + B) = 50% of (A - B), then find B : A -

Solution:
25% of (A + B) = 50% of (A - B)
⇒ (A + B) × 25/100 = (A - B) × 50/100
⇒ (A + B)/4 = (A - B)/2
⇒ 2 (A + B) = 4 (A - B)
⇒ 2A + 2B = 4A - 4B
⇒ 4A - 2A = 2B + 4B
⇒ 2A = 6B
⇒ A/B = 6/2 = 3/1
∴ B/A = 1/3
= 1 : 3

৬৩৯.

A bag contains an equal number of 20Tk, 10Tk. and 5Tk. note. If the total value is Tk.700, how many notes of each type are there?

ক
ক) 15
খ
খ) 20
গ
গ) 25
ঘ
ঘ) 30

ব্যাখ্যা

Question: A bag contains an equal number of 20Tk, 10Tk. and 5Tk. note. If the total value is Tk.700, how many notes of each type are there?

Solution:
Let the note of each type is X.
X notes of 20Tk. is equal to 20X Tk.
X notes of 10Tk. is equal to 10X Tk. and
X notes of 5Tk. is equal to 5X

ATQ,
20X + 10X + 5X = 700
35X = 700
X = 20

Hence, there were 20 notes of each value and 60 notes in total.

৬৪০.

If X is 2/5 of Y and Y is 5/7 of Z, what is the ratio of Z : X?

ক
ক) 5 : 7
খ
খ) 2 : 7
গ
গ) 7 : 5
ঘ
ঘ) 7 : 2

ব্যাখ্যা

প্রশ্ন: If X is 2/5 of Y and Y is 5/7 of Z, what is the ratio of Z : X?

সমাধান:
X = 2Y/5
Y = 5Z/7

X = 2Y/5
⇒ X = (2/5) × (5Z/7)
⇒ X = 2Z/7
⇒ Z/X = 7/2
∴ Z : X = 7 : 2

৬৪১.

A and B together have Tk. 1210. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have?

ক
Tk. 460
খ
Tk. 484
গ
Tk. 550
ঘ
Tk. 664

ব্যাখ্যা

Question: A and B together have Tk. 1210. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have?

Solution:
(4/15) A = (2/5) B
⇒ A = (2/5 × 15/4) B
⇒ A = (3/2) B
⇒ A/B = 3/2
∴ A : B = 3 : 2

B's share = 1210 × (2/5) = Tk. 484

৬৪২.

What is the difference between the third proportional of 12 and 18, and mean proportional of 9 and 25?

ক
ক) 8
খ
খ) 12
গ
গ) 10
ঘ
ঘ) 9

ব্যাখ্যা

Question: What is the difference between the third proportional of 12 and 18, and mean proportional of 9 and 25?

Solution:
Third proportional = (18 × 18)/12 = 27
Mean proportional = √(9 × 25) = √225 = 15

So, the difference = 27 - 15 = 12

৬৪৩.

A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk is respectively -

ক
ক) 5 : 4
খ
খ) 4 : 5
গ
গ) 1 : 5
ঘ
ঘ) 1 : 4

ব্যাখ্যা

Question: A milkman mixed some water with milk to gain 25% by selling the mixture at the cost price. The ratio of water and milk is respectively -

Solution:
Cost price of 1 litres of milk = Tk 100
∴ Mixture sold for Tk 125
= 125/100 = 5/4 litres
∴ Quantity of mixture = 5/4 litres
∴ Quantity of milk = 1 litre
∴ Quantity of water =(5/4) - 1 = 1/4 litre
∴ Required ratio = 1/4 : 1 = 1 : 4

৬৪৪.

In a school there are 286 students where boys and girls are in the ratio 8:5. If 22 new girls are admitted, then new ratio of boys and girls:

ক
4 : 3
খ
12 : 7
গ
10 : 7
ঘ
8 : 7

ব্যাখ্যা

Question: In a school there are 286 students where boys and girls are in the ratio 8:5. If 22 new girls are admitted, then new ratio of boys and girls:

Solution:
দেওয়া আছে,
একটি বিদ্যালয়ে মোট ছাত্র-ছাত্রীর সংখ্যা 286 জন
বিদ্যালয়ে বালক ও বালিকার সংখ্যার অনুপাত 8:5

মনে করি,
বিদ্যালয়ে বালকের সংখ্যা 8x জন
বিদ্যালয়ে বালিকার সংখ্যা 5x জন

প্রশ্নমতে,
8x + 5x = 286
⇒ 13x = 286
⇒ x = 286/13
∴ x = 22

∴ বিদ্যালয়ে বালকের সংখ্যা (8 × 22) = 176 জন
∴ বিদ্যালয়ে বালিকার সংখ্যা (5 × 22) = 110 জন

∴ যদি 22 জন বালিকা নতুন করে ভর্তি হয় তাহলে বালিকার সংখ্যা হবে = (110 + 22) = 132 জন

∴ বিদ্যালয়ে বালক ও বালিকার সংখ্যার নতুন অনুপাত = 176 : 132
= 4 : 3 [44 দ্বারা ভাগ করে]

৬৪৫.

35% of Rifat's income is equal to 25% of Reaz's income. The ratio of their income is-

ক
ক) 7 : 5
খ
খ) 4 : 3
গ
গ) 4 : 7
ঘ
ঘ) 5 : 7
ঙ
ঙ) 3 : 4

ব্যাখ্যা

Question: 35% of Rifat's income is equal to 25% of Reaz's income. The ratio of their income is-

Solution:
Let,
The income of Rifat = x
The income of Reza = y

ATQ,
x × 35% = y × 25%
⇒ x/y = 25/35 = 5/7

∴ x : y = 5 : 7

৬৪৬.

If 25% of a number is subtracted from a second number, the second number reduces to the five-sixth. The ratio of the first number to the second number is

ক
ক) 1 : 3
খ
খ) 3 : 2
গ
গ) 2 : 3
ঘ
ঘ) None

ব্যাখ্যা

Question: If 25% of a number is subtracted from a second number, the second number reduces to the five-sixth. The ratio of the first number to the second number is

Solution:
Let,
A = First Number
B = Second Number

ATQ,
B - 25A/100 = 5B/6
⇒ B - A/4 = 5B/6
⇒ B = A/4 + 5B/6
⇒ B - 5B/6 = A/4
⇒ B/6 = A/4
⇒ 4B = 6A
⇒ 2B = 3A
∴ A : B = 2 : 3

৬৪৭.

Three girls have candies in the ratio of 7 : 4 : 5. If the girl with the least number of candies has 12 candies, how many candies does the girl with the greatest number have?

ক
18
খ
21
গ
24
ঘ
28

ব্যাখ্যা

Question: Three girls have candies in the ratio of 7 : 4 : 5. If the girl with the least number of candies has 12 candies, how many candies does the girl with the greatest number have?

Solution:
Let their amounts of candies be 7x, 4x, and 5x.

ATQ,
4x = 12
⇒ x = 3

So, the girl with the greatest number of candies has = 7 × 3
= 21 candies

৬৪৮.

A 144 liters of mixture contains milk and water in the ratio of 5 : 7. How much milk needs to be added to this mixture so that the new ratio is 23 : 21 respectively?

ক
ক) 30
খ
খ) 32
গ
গ) 34
ঘ
ঘ) 36
ঙ
ঙ) 38

ব্যাখ্যা

Milk = 144 × 5/12 = 60L
Water = (144 - 60) = 84 L
ATQ,
(60 + x)/(144 + x) = 23/44
Or, 2640 + 44x = 3312 + 23x
Or, 21x = 672
Or, x = 672/21
or, x = 32L

Solution 2:
ATQ,
(60 + x)/84 = 23/21
Or, 60×21 + 21x = 23×84
Or, 21x = 672
or, x = 32 L

৬৪৯.

A container contains 64 litres of milk. From this container 16 litres of milk was taken out and replaced by water. This process was repeated overall three times. How much milk is now contained by the container?

ক
ক) 20
খ
খ) 37
গ
গ) 27
ঘ
ঘ) 34

ব্যাখ্যা

Question: A container contains 64 litres of milk. From this container 16 litres of milk was taken out and replaced by water. This process was repeated overall three times. How much milk is now contained by the container?

Solution:
After first replacement the ratio of milk and water is 48 : 16 = 3 : 1

After second replacement,
remaining milk = 48 - (3/4 of 16) = 48 - 12 = 36
water = 16 +12= 16 + 12 = 28
∴ ratio = 36 : 28 = 9 : 7

After third replacement,
remaining milk = 36 - (9/16 of 16) = 36 - 9 = 27
water = 28 + 9 = 37
∴ ratio = 27 : 37

shortcut,
after three replacement the reamining milk will be = 64 × (3/4)³
= 64 × 27/64
= 27

৬৫০.

The monthly incomes of two persons are in the ratio 5 : 4 and their monthly expenditures are in the ratio of 9 : 7. If each saves BDT 50 per month, what would be the total amount of their monthly expenditure?

ক
900 tk
খ
800 tk
গ
750 tk
ঘ
850 tk

ব্যাখ্যা

Question: The monthly incomes of two persons are in the ratio 5 : 4 and their monthly expenditures are in the ratio of 9 : 7. If each saves BDT 50 per month, what would be the total amount of their monthly expenditure?

Solution:
Let,
Their monthly income 5a and 4a
Their monthly expenses 9b and 7b

ATQ,
5a - 9b = 50 ..............(1)
4a - 7b = 50 ...............(2)

Multiply equation (1) and (2) by 4 and 5 respectively,
20a - 36b = 200 .............(3)
20a - 35b = 250 ..............(4)

(3) - (4) ⇒
20a - 36b- 20a + 35b = 200 - 250
or, - b = - 50
∴ b = 50

Hence, their total monthly expenditure = 9 × 50 + 7 × 50 = 450 + 350 = 800 tk

৬৫১.

20 litres of a mixture contains milk and water 4 : 1. Then the amount of water to be added to the mixture so as to have milk and water in ratio 2 : 1 is-

ক
7 litres
খ
4 litres
গ
5 litres
ঘ
6 litres

ব্যাখ্যা

Question: 20 litres of a mixture contains milk and water 4 : 1. Then the amount of water to be added to the mixture so as to have milk and water in ratio 2 : 1 is-

Solution:
In 20 litres of mixture,
quantity of mik = 20 × (4/5) = 16 litres
quantity of water = 20 × (1/5) = 4 litres
Let,
The quantity of water be added m litres
ATQ,
16 : (4 + m) = 2 : 1
or, 16/(4 + m) = 2/1
or, 2m + 8 = 16
or, 2m = 16 - 8
or, 2m = 8
∴ m = 8/2 = 4

∴ 4 litres water to be added to the mixture.

৬৫২.

Two equal glasses respectively contain one-fourth and two-fifth of milk. What is the ratio of milk and water if the rest of the glasses are filled with water and then mixed in a tumbler?

ক
3 : 7
খ
13 : 17
গ
13 : 27
ঘ
10 : 20

ব্যাখ্যা

Question: Two equal glasses respectively contain one-fourth and two-fifth of milk. What is the ratio of milk and water if the rest of the glasses are filled with water and then mixed in a tumbler?

Solution:
in the first glass,
milk = 1/4
water = 1 - (1/4)
= 3/4

in the second glass,
milk = 2/5
water = 1 - (2/5)
= 3/5

so,
milk : water
(1/4 + 2/5) : (3/4 + 3/5)
(5+8)/20 : (15+12)/20
13 : 27

৬৫৩.

P and Q are two alloys of gold and copper prepared by mixing metals in the ratio 7 : 2 and 7 : 11 respectively. If equal amounts of both alloys are melted to form a third alloy R, then the ratio of gold and copper in R will be -

ক
19 : 5
খ
11 : 6
গ
14 : 13
ঘ
7 : 3
ঙ
7 : 5

ব্যাখ্যা

Question: P and Q are two alloys of gold and copper prepared by mixing metals in the ratio 7 : 2 and 7 : 11 respectively. If equal amounts of both alloys are melted to form a third alloy R, then the ratio of gold and copper in R will be -

Solution:
Alloy P (Gold : Copper) = 7:2
Gold fraction in P = 7/9
Copper fraction in P = 2/9

Alloy Q (Gold : Copper) = 7:11
Gold fraction in Q = 7/18
Copper fraction in Q = 11/18

Let, 1 kg of each alloy is mixed;
Gold in Alloy R = 7/9 + 7/18 = 21/18

Copper in Alloy R = 2/9 + 11/18 = 15/18

∴ The ratio of Gold:Copper in R = (21/18) : (15/18)
= 21 : 15
= 7 : 5

৬৫৪.

In a college, the ratio of foreign to local students is 3 : 7. If three-fourths of the local students are female and one-quarter of the foreign students is female. What fraction of the combined students is female?

ক
50%
খ
60%
গ
64%
ঘ
65%

ব্যাখ্যা

Question: In a college, the ratio of foreign to local students is 3 : 7. If three-fourths of the local students are female and one-quarter of the foreign students is female. What fraction of the combined students is female?

Solution:
Since the ratio of foreign : local is 3 : 7,
let's use a total of 10 parts
So if we have a total of 100 students, that means
Foreign students = 30 students
Local students = 70 students

Foreign female students = (1/4) × 30 = 15/2 students
Local female students = (3/4) × 70 = 105/2 students

Total female students = (15/2) + (105/2)
= (15 + 105)/2
= 60 students or 60% of the combined students are female.

৬৫৫.

The ratio of water and salt in a 16 kg of salt - water solution is 3 : 1. How much water in kg must be added to make the ratio of water to salt 4 : 1?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 6

ব্যাখ্যা

salt = 16 × 1/4 = 4 Kg
water = 16 - 4
= 12 kg.
এখন water/salt = 4
∴ water = 4 × 4 = 16 Kg
∴ (16 - 12) = 4 Kg water add করতে হবে।

৬৫৬.

The present ratio of ages of A and B is 4 : 5. 18 years ago, this ratio was 11 : 16. Find the average of their present ages.

ক
ক) 90 years.
খ
খ) 40 years.
গ
গ) 45 years.
ঘ
ঘ) 80 years.

ব্যাখ্যা

Question: The present ratio of ages of A and B is 4 : 5. 18 years ago, this ratio was 11 : 16. Find the average of their present ages.

Solution:
Let present age of A and B be 4x and 5x
18 years ago their ages
(4x - 18)/(5x - 18) = 11/16
⇒ 64x - 288 = 55x - 198
⇒ 64x - 55x = -198 + 288
⇒ 9x = 90
⇒ x = 90/9
∴ x = 10
Sum of the present ages = 40 + 50 = 90 years.

∴ Sum of the present ages = 90/2 years = 45 years.

৬৫৭.

How many kgs of Basmati rice costing Tk. 42 per kg should a shopkeeper mix with 25 kgs of ordinary rice costing Tk. 24 per kg so that he makes a profit of 25% on selling the mixture at Tk. 40 per kg?

ক
20.0 kgs
খ
12.5 kgs
গ
16.0 kgs
ঘ
200.0 kgs

ব্যাখ্যা

Question: How many kgs of Basmati rice costing Tk. 42 per kg should a shopkeeper mix with 25 kgs of ordinary rice costing Tk. 24 per kg so that he makes a profit of 25% on selling the mixture at Tk. 40 per kg?

Solution:
Let the amount of Basmati rice being mixed be x kgs.
As the trader makes 25% profit by selling the mixture at Tk. 40 per kg
∴ His cost per kg of the mixture = (100 × 40)/125 = Tk. 32 per kg

∴ (x × 42) + (25 × 24) = 32(x + 25)
⇒ 42x + 600 = 32x + 800
⇒ 10x = 200
∴ x = 20

৬৫৮.

A 24 liters of milk and water mixture contains milk and water in the ratio 3: 5. What litres of the mixture should be taken out and replaced with pure milk so that the final mixture contains milk and water in equal proportions?

ক
28/5 L
খ
32/5 L
গ
20/3 L
ঘ
24/5 L

ব্যাখ্যা

Question: A 24 liters of milk and water mixture contains milk and water in the ratio 3: 5. What litres of the mixture should be taken out and replaced with pure milk so that the final mixture contains milk and water in equal proportions?

Solution:
In 24 l of mixture, milk = (3/8) × 24 = 9 L
So water = 24 - 9 = 15 L
Now since the mixture is to be replaced with pure milk, the amount of mixture will remain same after replacement too.
In 24 L mixture, to have 12 L water and 12 L milk, 3 L of water should be taken out, since we are only adding milk.
Let x L of mixture taken out.
So (5/8) × x = 3
⇒ 5x = 24
Solve, x = 24/5 L

৬৫৯.

Two numbers are in ratio of 21 : 26. If 8 is added in each, the new numbers are in ratio of 5 : 6. Find the ratio of numbers, if 6 subtracted from each number?

ক
ক) 18 : 23
খ
খ) 19 : 25
গ
গ) 6 : 7
ঘ
ঘ) 9 : 16

ব্যাখ্যা

ধরি,
একটি সংখ্যা = 21x
অপর সংখ্যা = 26x

প্রশ্নমতে
(21x + 8)/(26x + 8) = 5/6
130x + 40 = 126x + 48
130x - 126x = 48 - 40
4x = 8
x = 8/4
x = 2

নতুন অনুপাত = (21x - 6)/(26x - 6)
= (21 × 2 - 6)/(26 × 2 - 6)
= 36/46
= 18/23
= 18 : 23

৬৬০.

Karim weighs 65.7 kg. If he reduces his weight in the ratio 5 : 4, find his reduced weight in kg.

ক
55.25
খ
52.56
গ
57.20
ঘ
61.25

ব্যাখ্যা

Question: Karim weighs 65.7 kg. If he reduces his weight in the ratio 5 : 4, find his reduced weight in kg.

Solution:
মনেকরি,
করিমের পূর্বের ওজন = 5x
করিমের পরের ওজন = 4x

প্রশ্নমতে,
⇒ 5x = 65.7
⇒ x = 65.7/5
∴ x = 13.14

∴ ওজন কমে যাওয়ার পর হবে = 4 × 13.14
= 52.56 kg

৬৬১.

In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes

ক
12 : 7
খ
4 : 3
গ
10 : 7
ঘ
8 : 7

ব্যাখ্যা

Question: In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes

Solution:
Boys : girls = 8 : 5
Let, the boys = 8x, girl = 5x

According to the 1st condition,
8x + 5x = 286
⇒ 13x = 286
⇒ x = 286/13
∴ x = 22
Boys = 8 × 22 = 176 and girls = 5 × 2 = 110

22 more girls get admitted then the number of girls become
(5x + 22) = 110 + 22 = 132

Now, new ratio of boys and girls = 176 : 132 = 4 : 3

৬৬২.

A jar contains white, red and green marbles in the ratio of 2 : 3 : 5. Six more green marbles are added to the jars, and then the ratio becomes 2 : 3 : 7. How many white marbles are there in the jar?

ক
ক) 2
খ
খ) 3
গ
গ) 4
ঘ
ঘ) 5
ঙ
ঙ) 6

ব্যাখ্যা

W : R : G = 2 : 3 : 5 [Initial]
W : R : G = 2 : 3 : 7 [Final]
Let, Initial Green = 5x Final Green = 7x
ATQ,
(7x - 5x) = 6
=> 2x = 6
=> x = 3
So, white marbles in the jar = (2 × 3) = 6

৬৬৩.

The expense of 9 pens and 5 pencils is the same as the expense of 7 pens and 8 pencils. What is the ratio between the price of one pen and one pencil?

ক
5 : 2
খ
3 : 2
গ
1 : 2
ঘ
3 : 1

ব্যাখ্যা

Question: The expense of 9 pens and 5 pencils is the same as the expense of 7 pens and 8 pencils. What is the ratio between the price of one pen and one pencil?

Solution:
Let,
The price of one pen X Tk.
The price of one pencil Y Tk.

ATQ,
9X + 5Y = 7X + 8Y
⇒ 9X - 7X = 8Y - 5Y
⇒ 2X = 3Y
⇒ X/Y = 3/2
∴ X : Y = 3 : 2

৬৬৪.

Two brands of detergent are to be combined. Detergent A contains 30 percent bleach and 70 percent soap, while Detergent B contains 50 percent bleach and 50 percent soap. If the combined mixture is to be 40 percent bleach, what percent of the final mixture should be Detergent A?

ক
30%
খ
40%
গ
50%
ঘ
60%

ব্যাখ্যা

Question: Two brands of detergent are to be combined. Detergent A contains 30 percent bleach and 70 percent soap, while Detergent B contains 50 percent bleach and 50 percent soap. If the combined mixture is to be 40 percent bleach, what percent of the final mixture should be Detergent A?

Solution:
ধরি, মিশ্রণে Detergent A এর অংশ x%
∴ Detergent B এর অংশ হবে (100 - x%)

এখানে,
ডিটারজেন্ট A থেকে প্রাপ্ত ব্লিচের পরিমাণ হলো (x% এর 30%)।
ডিটারজেন্ট B থেকে প্রাপ্ত ব্লিচের পরিমাণ হলো (100 - x)% এর 50%।
চূড়ান্ত মিশ্রণে ব্লিচের মোট পরিমাণ হলো (100% এর 40%)।

প্রশ্নমতে,
0.30x + 0.50 × (100 - x) = 0.40 × 100
⇒ 0.30x + 50 - 0.50x = 40
⇒ - 0.20x = - 10
⇒ x = - 10/(- 0.20)
∴ x = 50

অর্থাৎ, মিশ্রণের 50% হবে Detergent A

৬৬৫.

If ratio of three numbers is 3 : 2 : 5 and their sum is 150, what is the largest number?

ক
ক) 30
খ
খ) 45
গ
গ) 60
ঘ
ঘ) 75

ব্যাখ্যা

Question: If ratio of three numbers is 3 : 2 : 5 and their sum is 150, what is the largest number?

Solution:
let the numbers be 3x, 2x, 5x

3x + 2x + 5x = 150
⇒ 10x = 150
∴ x = 15

largest number = 5 × 15
= 75

৬৬৬.

400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

ক
ক) 160 gm
খ
খ) 180 gm
গ
গ) 150 gm
ঘ
ঘ) 140 gm

ব্যাখ্যা

Question: 400 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
Sugar = 400 × 30% = 120 grams
Let,
We have to add x grams

ATQ,
120 + x = 50% (400 + x)
⇒ 120 + x = (50/100) × (400 + x)
⇒ 120 + x = (1/2) × (400 + x)
⇒ 240 + 2x = 400 + x
∴ x = 160

৬৬৭.

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

ক
400 kg
খ
600 kg
গ
500 kg
ঘ
630 kg

ব্যাখ্যা

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

Solution:
Let the sugar sell at 18% profit is = x
So, the sugar sell at of 8% profit = 1000-x

According to question,
18% of x + 8% of (1000 -x) = 14% of 1000
→ 18x/100 + (8000 - 8x)/100 = 14000/100
→ 18x + 8000 - 8x = 14000 (multiplying both sides by 100)
→ 10x = 6000
→ x = 600 kg

৬৬৮.

A, B, and C are partners in a business. Their shares are in the proposition of (1/3) : (1/4) : (1/5). A withdraws half of his capital after 15 months and after another 15 months, a profit of Tk. 4340 is divided. The share of C is-

ক
Tk. 1520
খ
Tk. 1435
গ
Tk. 1355
ঘ
Tk. 1240
ঙ
None

ব্যাখ্যা

Question: A, B, and C are partners in a business. Their shares are in the proposition of (1/3) : (1/4) : (1/5). A withdraws half of his capital after 15 months and after another 15 months the profit of Tk. 4340 is divided. The share of C is-

Solution:
Ratio of initial investments = 1/3 : 1/4 : 1/5
= 20 : 15 : 12

Let
their initial investments will be 20x, 15x and 12x respectively.

A : B : C = {(20x × 15) + (10x × 15)} : (15x × 30) : (12x × 30) [A's portion was calculated as half after the first 15 months.]
= 450x : 450x : 360x
= 5 : 5 : 4

Sum of the ratio = 5 + 5 + 4 = 14.
∴ C's share = 4340 × (4/14)
= Tk. 1240

৬৬৯.

The cost of Type 1 rice is Tk. 15 per kg and Type 2 rice is Tk. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is-

ক
Tk. 19.50
খ
Tk. 19
গ
Tk. 18
ঘ
Tk. 18.50

ব্যাখ্যা

Question: The cost of Type 1 rice is Tk. 15 per kg and Type 2 rice is Tk. 20 per kg. If both Type 1 and Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is-

Solution:
Let,
Quantity of type 1 rice is 2x kg.
Quantity of type 2 rice is 3x kg.
The price per kg of the mixed variety of rice is y taka

∴ Total price of type 1 rice is 15 × 2x = 30x Taka
∴ Total price of type 2 rice is 20 × 3x = 60x Taka

ATQ,
30x + 60x = y(2x + 3x)
⇒ 90x = y × 5x
⇒ y = (90x)/(5x)
∴ y = 18

৬৭০.

A chemist has two solutions, one containing 30% acid and the other containing 70% acid. How many liters of each solution should be mixed to get 10 liters of a solution containing 50% acid?

ক
ক) 2 liters
খ
খ) 3 liters
গ
গ) 4 liters
ঘ
ঘ) 5 liters

ব্যাখ্যা

Question: A chemist has two solutions, one containing 30% acid and the other containing 70% acid. How many liters of each solution should be mixed to get 10 liters of a solution containing 50% acid?

Solution:
Let x be the liters of the 30% acid solution.
Then, (10 - x) would be the liters of the 70% acid solution.

Amount of acid from 30% solution = 30% of x = 0.3x
Amount of acid from 70% solution = 70% of (10 - x) = 0.7(10 - x)

Total amount of acid in the mixture = Amount of acid from 30% solution + Amount of acid from 70% solution
0.3x + 0.7(10 - x) = 0.5(10)

Solve for x:
0.3x + 7 - 0.7x = 5
0.3x - 0.7x = 5 - 7
-0.4x = -2
x = -2 / -0.4
x = 5

So, 5 liters of the 30% acid solution should be mixed with (10 - 5) = 5 liters of the 70% acid solution to obtain 10 liters of the 50% acid solution.

৬৭১.

A 42 liter mixture contains milk and water in the ratio 3 : 4. How many liters of milk must be added to the mixture so that the ratio of milk to water becomes 1 : 1?

ক
3 liters
খ
6 liters
গ
5 liters
ঘ
7 liters

ব্যাখ্যা

Question: A 42 liter mixture contains milk and water in the ratio 3 : 4. How many liters of milk must be added to the mixture so that the ratio of milk to water becomes 1 : 1?

Solution:
The ratio of milk to water is 3 : 4
Total portion = 3 + 4 = 7

Quantity of milk = 42 × (3/7) = 18 liters.
Quantity of water = 42 × (4/7) = 24 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(18 + x) : 24 = 1 : 1
⇒ (18 + x)/24 = 1/1
⇒ 18 + x = 24
⇒ x = 24 - 18
⇒ x = 6

∴ Quantity of milk to be added = 6 liters

৬৭২.

A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallon?

ক
22
খ
24
গ
20
ঘ
16

ব্যাখ্যা

Question: A container is 1/2 full. When 8 gallons is removed the container is 1/10 full. What is the capacity of the container in gallon?

Solution:
let. the capacity of the container in gallon is x liter

ATQ,
(x/2) - 8 = x/10
⇒ (x - 16)/2 = x/10
⇒ 10(x - 16) = 2x
⇒ 10x - 160 = 2x
⇒ 10x - 2x = 160
⇒ 8x = 160
∴ x = 160/8 = 20 liter

৬৭৩.

The ratio of the number of boys and girls of a school with 504 students is 13 : 11. What will be the new ratio if 12 more girls are admitted?

ক
60 : 71
খ
10 : 9
গ
51 : 66
ঘ
91 : 81

ব্যাখ্যা

Question: The ratio of the number of boys and girls of a school with 504 students is 13 : 11. What will be the new ratio if 12 more girls are admitted?

Solution:
Total numbers of girls in the school = 504 × {11/(13 + 11)}
= 504 × (11/24) = 231

And, total numbers of boys in the school = 504 × {11/(13 + 11)}
= 504 × (11/24) = 231

Total no. of girls when 12 more girls are admitted = 231 + 12 = 243
∴ Required ratio = 273 : 243
= 91 : 81

৬৭৪.

The ratio of alcohol to water in a chemical solution is 3 : 2. If 4 liters of alcohol are added to the solution, the new ratio of alcohol to water becomes 7 : 4. Find the final amount of alcohol in the new solution.

ক
22 liters.
খ
25 liters.
গ
28 liters.
ঘ
32 liters.

ব্যাখ্যা

Question: The ratio of alcohol to water in a chemical solution is 3 : 2. If 4 liters of alcohol are added to the solution, the new ratio of alcohol to water becomes 7 : 4. Find the final amount of alcohol in the new solution.

Solution:
Let
the initial amount of alcohol be 3x liters and the amount of water 2x liters.

ATQ,
Ratio of alcohol and water after adding 4 liters of alcohol
(3x + 4)/2x = 7/4
⇒ 12x + 16 = 14x
⇒ 2x = 16
∴ x = 8

∴ Final amount of alcohol in solution = 3x + 4 = (3 × 8) + 4 = 28 liters.

৬৭৫.

Silver is 17 times as heavy as water and copper is 7 times as heavy as water. In what ratio should these be mixed to get alloy 13 times as heavy as water?

ক
1 : 1
খ
2 : 3
গ
1 : 2
ঘ
3 : 2

ব্যাখ্যা

Question: Silver is 17 times as heavy as water and copper is 7 times as heavy as water. In what ratio should these be mixed to get alloy 13 times as heavy as water?

Solution:
Given that,
Density of silver is 17 times as heavy as water
Density of copper is 7 times as heavy as water
Mixture should be 13 times as heavy as water

Let the weights of silver and copper be x and y respectively. Then we get,
⇒ (17x + 7y)/(x + y) = 13
⇒ 17x + 7y = 13(x + y)
⇒ 17x + 7y = 13x + 13y
⇒ 17x - 13x = 13y - 7y
⇒ 4x = 6y
⇒ x/y = 6/4 = 3/2
∴ x : y = 3 : 2

So Ratio of silver to copper = 3 : 2

৬৭৬.

There are two numbers. 1st number is 12 more than the 2nd number. The average of the two numbers is 19. If 2 is added in both numbers, find the ratio of the numbers.

ক
ক) 5 : 9
খ
খ) 11 : 9
গ
গ) 4 : 9
ঘ
ঘ) 9 : 5

ব্যাখ্যা

ATQ,
x - y = 12 ...... (i)
x + y = 38 ........ (ii)
(i) + (ii), 2x = 50
Or, x = 25
So, y = 13
If 2 is added in both the numbers, then their ratio is:
x+2 / y+2
= 25+2 / 13+2
= 27/15
= 9/5

৬৭৭.

94 is divided into two parts such that the fifth part of the first and the eighth part of the second are in the ratio 3 : 4. Find the first part.

ক
48
খ
36
গ
42
ঘ
30

ব্যাখ্যা

Question: 94 is divided into two parts such that the fifth part of the first and the eighth part of the second are in the ratio 3 : 4. Find the first part.

Solution:
Let the two parts be x and 94 - x.

According to the problem,
(x/5) : (94 - x)/8 = 3 : 4
⇒ (x/5)/{(94 - x)/8} = 3/4
⇒ 8x/5(94 - x) = 3/4
⇒ 32x = 15(94 - x)
⇒ 32x = 15 × 94 - 15x
⇒ 47x = 15 × 94
⇒ x = (15 × 94)/47
∴ x = 30

∴ First part is 30

৬৭৮.

A mixture of 150 liters of wine and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?

ক
10 liters
খ
20 liters
গ
30 liters
ঘ
40 liters

ব্যাখ্যা

Question: A mixture of 150 liters of wine and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?

Solution:
Number of liters of water in 125 liters of the mixture = 20% of 150 = 1/5 of 150 = 30 liters
Let us Assume that another 'P' liters of water are added to the mixture to make water 25% of the new mixture. So, the total amount of water becomes (30 + P) and the total volume of the mixture becomes (150 + P)
Thus, (30 + P) = 25% of (150 + P)
Solving, we get P = 10 liters

৬৭৯.

A mixture of 20 kg of spirit and water contains 10% water. How much water must be added to this mixture to raise the percentage of water to 25%?

ক
ক) 2
খ
খ) 4
গ
গ) 5
ঘ
ঘ) 6

ব্যাখ্যা

In 1st mixture, water = 10/100 × 20 = 2 kg
So, Spirit = 20-2 = 18 kg
In 2nd mixture where the water is 25%,
75 kg of spirit is contained in 100 kg mixture
So, 18 kg spirit is contained in = (100×18)/75 = 24 kg
So, water to be added = 24-20 = 4 kg

৬৮০.

Aman and Ajay started a business by investing Tk. 85000 and Tk. 15000 each. What be will the ratio of profit earned after 2 years between Aman and Ajay respectively?

ক
3 : 5
খ
17 : 23
গ
15 : 23
ঘ
None of the above

ব্যাখ্যা

Question: Aman and Ajay started a business by investing Tk. 85000 and Tk. 15000 each. What be will the ratio of profit earned after 2 years between Aman and Ajay respectively?

Solution:
the ratio of profit earned after 2 years between Aman and Ajay respectively = (85000 × 2) : (15000 × 2)
= 170000 : 30000
= 17 : 3

৬৮১.

800 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

ক
300 gm
খ
320 gm
গ
350 gm
ঘ
420 gm

ব্যাখ্যা

Question: 800 grams of sugar solution has 30% sugar in it. How much sugar should be added to make 50% in the solution?

Solution:
Amount of sugar = 800 × 30/100
= 240 grams

Let, x gm sugar to be added
ATQ,
(240 + x)/(800 + x) = 50%
⇒ (240 + x)/(800 + x) = = 1/2
⇒ 480 + 2x = 800 + x
⇒ 2x - x = 800 - 480
∴ x = 320 gm

৬৮২.

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is-

ক
4 : 7
খ
2 : 3
গ
2 : 5
ঘ
4 : 5
ঙ
None of the above

ব্যাখ্যা

Question: Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is-

Solution:
Let the third number be x

Then, first number = 120% of x = 120x/100 = 6x/5
Second number = 150% of x = 150x/100 = 3x/2

∴ Ratio of first two numbers = 6x/5 : 3x/2
= 12x : 15x
= 4 : 5

৬৮৩.

The first number is 5% greater and the second number is 25% greater than a third number. What is the ratio of first two numbers?

ক
21 : 25
খ
15 : 22
গ
7 : 20
ঘ
5 : 21

ব্যাখ্যা

Question: The first number is 5% greater and the second number is 25% greater than a third number. What is the ratio of first two numbers?

Solution:
Let
the third number be x

Then, first number = 105% of x
= 105x/100
= 21x/20

Second number = 125% of x
= 125x/100
= 5x/4

∴ Ratio of first two numbers = 21x/20 : 5x/4
= (21x × 4) : (5x × 20)
= 21x : 25x
= 21 : 25

৬৮৪.

In a class, the number of girls is 20% more than that of boys. The strength of the class is 66. If 4 more girls are admitted to the class, the ratio of the number of boys to that of the girls is -

ক
ক) 1 : 2
খ
খ) 1 : 4
গ
গ) 3 : 5
ঘ
ঘ) 3 : 4

ব্যাখ্যা

ধরি,
বালক আছে x জন
বালিকা আছে = x এর 120% = 120x/100
= 1.2x জন।
প্রশ্নমতে, x + 1.2x = 66
⇒ 2.2x = 66
⇒ x = 66/2.2
⇒ x = 30
অতএব বালিকা আছে = 1.2x = 1.2 × 30 = 36
4 জন বালিকা ভর্তি হলে = 36 + 4 = 40 জন
∴ বালকঃ বালিকা = 30 : 40
= 3 : 4

৬৮৫.

If x is 90% of y then what percent of x is y?

ক
1.11
খ
101.1
গ
11.1
ঘ
111.1

ব্যাখ্যা

Question: If x is 90% of y then what percent of x is y?

Solution:
x = 90% of y
⇒ x = 90y/100
⇒ x = 9y/10
⇒ y/x = 10/9
= (10/9) × 100%
= 111.1%

৬৮৬.

A, B and C rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing and the rent of the pasture is Tk. 175, then how much should C pay as his share of the rent?

ক
60
খ
35
গ
55
ঘ
45

ব্যাখ্যা

A : B : C
= 10 × 7 : 12 × 5 : 15 × 3
= (2 × 7) : (12 × 1) : (3 × 3)
= 14 : 12 : 9
Amount that C should pay
= 175 × (9/35)
= 5 × 9
= 45.

৬৮৭.

A piece of string 70 cm in length was cut into pieces, the ratio of whose lengths was 3: 7. Find the length of longest piece.

ক
21 cm
খ
70 cm
গ
49 cm
ঘ
7 cm

ব্যাখ্যা

Question: A piece of string 70 cm in length was cut into pieces, the ratio of whose lengths was 3: 7. Find the length of longest piece.

Solution:
Total length = 70 cm.
Ratio is 3 : 7
∴ Length of longest piece is (70 × 7)/10 = 49 cm

৬৮৮.

The ratio of green marbles to red marbles in a box 3:5. If there are 24 marbles in the box, how many additional green marbles will be required to make the ratio of green marbles to red marbles 1:1?

ক
12
খ
9
গ
6
ঘ
3
ঙ
2

ব্যাখ্যা

Number of green ball = 24 × (3/8) = 9
Number of red ball= 24 × (5/8) = 15
since the required ratio is 1:1
so, additional green ball = (15 - 9) = 6

৬৮৯.

A 40-meter long rope is cut into two unequal pieces. If one piece is 18 meter longer than the other, what is the length (in meter) of the shorter piece?

ক
ক) 10
খ
খ) 11
গ
গ) 18
ঘ
ঘ) None

ব্যাখ্যা

Question: A 40-meter long rope is cut into two unequal pieces. If one piece is 18 meter longer than the other, what is the length (in meter) of the shorter piece?

Solution:
ধরি, বড় টুকরোটির দৈর্ঘ্য x মিটার
ছোট টুকরোটির দৈর্ঘ্য x - 18 মিটার

x + x - 18 = 40
⇒ 2x = 58
∴ x = 29

ছোট টুকরোটির দৈর্ঘ্য = 29 - 18
= 11 মিটার

৬৯০.

In a mixture, the ratio of the milk and water is 6: 5. When 22 liter mixture is replaced by water, the ratio becomes 9: 13. What is the quantity of water after replacement?

ক
62 liter
খ
50 liter
গ
40 liter
ঘ
52 liter

ব্যাখ্যা

Question: In a mixture, the ratio of the milk and water is 6: 5. When 22 liter mixture is replaced by water, the ratio becomes 9 : 13. What is the quantity of water after replacement?

Solution:
Given that,
milk : water = 6 : 5
And 22 liter mixture are replaced by water

Now,
Let milk = 6x and water = 5x
In 22 liter mixture, milk removed = (6/11) × 22 = 12 liter
And water removed = (5/11) × 22 = 10 liter

According to question,
(6x - 12) : (5x - 10 + 22) = 9 : 13
⇒ 13(6x - 12) = 9(5x + 12)
⇒ 78x - 156 = 45x + 108
⇒ 78x - 45x = 108 + 156
⇒ 33x = 264
⇒ x = 8
∴ Initial water = 5x = 5 × 8 = 40 liters
Water removed in 22 L mixture = 10 liters
And water added back = 22 liters

∴ Water after replacement = Initial water - water removed + water added
= 40 - 10 + 22
= 52 liters

So the quantity of water after replacement is 52 liters.

৬৯১.

A mixture of 20 kg of spirit and water contains 10% water. How much water must be added to mixture to raise the percentage of water to 25%

ক
ক) 2 kg
খ
খ) 4 kg
গ
গ) 5 kg
ঘ
ঘ) 6 kg

ব্যাখ্যা

৬৯২.

Two numbers are in the ratio 2 : 3. If 4 is subtracted from the first number, the ratio becomes 1 : 2. What are the numbers?

ক
12, 18
খ
10, 15
গ
14, 21
ঘ
16, 24

ব্যাখ্যা

Question: Two numbers are in the ratio 2 : 3. If 4 is subtracted from the first number, the ratio becomes 1 : 2. What are the numbers?

Solution:
Let the two numbers be: 2x and 3x

According to the question,
(2x - 4)/3x = 1/2
⇒ 2(2x - 4) = 3x
⇒ 4x - 8 = 3x
⇒ x = 8

∴ First number = 2 × 8 = 16
∴ Second number = 3 × 8 = 24

৬৯৩.

A mixture of 150 liters of milk and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?

ক
18 liters
খ
15 liters
গ
10 liters
ঘ
9 liters

ব্যাখ্যা

Question: A mixture of 150 liters of milk and water contains 20% water. How much more water should be added so that water becomes 25% of the new mixture?

Solution:
Amount of water in the 150 liters mixture = 20% of 150
= 1/5 of 150
= 30 liters

Let,
P liters of water be added.
So, new amount of water = (30 + P)
and new total mixture = (150 + P)

ATQ,
(30 + P) = 25% of (150 + P)
⇒ 30 + P = (25/100) × (150 + P)
⇒ 30 + P = (1/4) × (150 + P)
⇒ 120 + 4P = 150 + P
⇒ 4P - P = 150 - 120
⇒ 3P = 30
∴ P = 10

∴ 10 liters more water should be added.

৬৯৪.

A 50 g gold-copper alloy contains 80% gold. How much additional gold is needed to raise the gold percentage to 90%?

ক
30 gm
খ
50 gm
গ
60 gm
ঘ
55 gm
ঙ
None of the above

ব্যাখ্যা

Question: A 50 g gold-copper alloy contains 80% gold. How much additional gold is needed to raise the gold percentage to 90%?

Solution:
Gold in alloy =50 × 80% = 40gm
Copper in alloy =50 × 20% =10gm

Now,
(40 + x)/10 = 90/10
⇒ 40 + x = 90
⇒ x = 90 - 40
∴ x = 50gm

৬৯৫.

A 60 litre mixture of sugar and water contains sugar and water in the ratio 2 : 3. How many litres of the mixture should be replaced by sugar so that the ratio of sugar and water becomes 1 : 1?

ক
6
খ
10
গ
15
ঘ
None

ব্যাখ্যা

Question: A 60 litre mixture of sugar and water contains sugar and water in the ratio 2 : 3. How many litres of the mixture should be replaced by sugar so that the ratio of sugar and water becomes 1 : 1?

Solution:
ধরি, mixture এর x পরিমাণকে sugar দ্বারা replace করতে হবে ।
অর্থাৎ , sugar/water = {2/5 × (60 - x) + x}/{(3/5) × (60 - x)} = 1/1
Or, (120 - 2x + 5x)/5 = (180 - 3x)/5
Or, 120 + 3x = 180 - 3x
Or, 6x = 60
Or, x = 10.

৬৯৬.

A shopkeeper blends two varieties of rice from two different regions, one variety costing tk 50 per kg and the other tk 30 per kg in the ratio 3 : 2. He sells the blended rice at tk 45 per kg. Find the profit or loss percent.

ক
7.14% loss
খ
9.21% loss
গ
7.14% profit
ঘ
9.21% profit

ব্যাখ্যা

Question: A shopkeeper blends two varieties of rice from two different regions, one variety costing tk 50 per kg and the other tk 30 per kg in the ratio 3 : 2. He sells the blended rice at tk 45 per kg. Find the profit or loss percent.

Solution:
Let 3 kg of the first variety be mixed with 2 kg of second variety.
Then total cost price of 5 kg of rice = (50 × 3) + (30 × 2) = tk 210
Selling price of 5 kg of rice = (45 × 5) = tk 225
∴ Profit = tk (225 - 210) = tk 15

∴ Percentage of profit = (15 × 100)/210 = 7.14%

৬৯৭.

The acid and water in two vessels A and B are in the ratio 4 : 3 and 2 : 3. In what ratio should the liquid in both the vessels be mixed to obtain a new mixture in vessel C containing half acid and half water?

ক
ক) 7 : 5
খ
খ) 5 : 7
গ
গ) 7 : 3
ঘ
ঘ) 5 : 3
ঙ
ঙ) 3 : 7

ব্যাখ্যা

According to the question,
Acid : Water -
Vessel A - 4 : 3
Vessel B - 2 : 3
Now using alligation,

৬৯৮.

If a² + b² + c² - ab - bc - ca = 0 then a : b : c is -

ক
ক) 1 : 1 : 2
খ
খ) 1 : 1 : 1
গ
গ) 1 : 2 : 1
ঘ
ঘ) 2 : 1 : 1

ব্যাখ্যা

a² + b² + c² - ab - bc - ca = 0 .....(i)
Multiple equation (i) by 2 we get
⇒ 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0
⇒ (a² + b² - 2ab) + (b² + c² - 2bc) + (c² + a² - 2ca) = 0 [ (a + b)² = a² + b² + 2ab]
⇒ (a - b)² + ((b - c)² + (c - a)² = 0 [if x² + y² + z² = 0 then x = 0, y = 0, z = 0]
∴ a - b = 0
⇒ a = b
b - c = 0
⇒ b = c
c - a = 0
⇒ c = a
∴ a : b : c = 1 : 1 : 1

৬৯৯.

A mixture of 200 liters of wine and water contains 30% water. How much more water should be added so that water becomes 40% of the new mixture?

ক
35.25 liters.
খ
30.50 liters.
গ
33.33 liters.
ঘ
25 liters.

ব্যাখ্যা

Question: A mixture of 200 liters of wine and water contains 30% water. How much more water should be added so that water becomes 40% of the new mixture?

Solution:
Number of liters of water in 200 liters of the mixture = 30% of 200 = 30/100 × 200 = 60 liters.

Let P liters of water added to the mixture to make water 25% of the new mixture.

Total amount of water becomes (60 + P) and total volume of mixture is (200 + P).
(60 + P) = 40/100 × (200 + P)
300 + 5P = 400 + 2P
=> P = 33.33 liters.

৭০০.

A mixture contains two liquids 'A' and 'B' in the ratio 5 : 3. If 8 litres of the mixture is withdrawn and replaced with 8 litres of 'A', the ratio becomes 2 : 1. What was the initial quantity of 'B'?

ক
45 litres
খ
24 litres
গ
27 litres
ঘ
32 litres

ব্যাখ্যা

Question: A mixture contains two liquids 'A' and 'B' in the ratio 5 : 3. If 8 litres of the mixture is withdrawn and replaced with 8 litres of 'A', the ratio becomes 2 : 1. What was the initial quantity of 'B'?

Solution:
ধরি, প্রাথমিক মিশ্রণের পরিমাণ ছিল 8x লিটার।
যেখানে A এর পরিমাণ = 5x লিটার এবং B এর পরিমাণ = 3x লিটার।

8 লিটার মিশ্রণ তুলে নেওয়ার পর,
মিশ্রণে A এর পরিমাণ = 5x - (5/8) × 8 = 5x - 5 লিটার।
মিশ্রণে B এর পরিমাণ = 3x - (3/8) × 8 = 3x - 3 লিটার।

নতুন 8 লিটার 'A' যোগ করার পর,
A এর নতুন পরিমাণ = (5x - 5) + 8 = 5x + 3 লিটার।

প্রশ্নানুযায়ী, নতুন অনুপাত,
⇒ (5x + 3) / (3x - 3) = 2/1
⇒ 1(5x + 3) = 2(3x - 3)
⇒ 5x + 3 = 6x - 6
⇒ 3 + 6 = 6x - 5x
⇒ 9 = x

সুতরাং, প্রাথমিকভাবে B এর পরিমাণ ছিল = 3x = 3 × 9 = 27 লিটার।