1. Correct Answer: _Option A: z < x < y_
 2. Explanation:
     Let's break down the problem step by step:
     Step 1: Understanding the given conditions
     ● Condition 1: When 70% of a number \( x \) is added to another number \( y \), the sum becomes 165% of the value of \( y \).  
       Mathematically, this can be expressed as:
       \[
       0.7x + y = 1.65y
       \]
     ● Condition 2: When 60% of the number \( x \) is added to another number \( z \), the sum becomes 165% of the value of \( z \).  
       Mathematically, this can be expressed as:
       \[
       0.6x + z = 1.65z
       \]
     Step 2: Solving the equations
     ● From Condition 1:  
       \[
       0.7x + y = 1.65y
       \]
       Rearranging gives:
       \[
       0.7x = 1.65y - y
       \]
       \[
       0.7x = 0.65y
       \]
       \[
       x = \frac{0.65}{0.7}y
       \]
       \[
       x = \frac{65}{70}y
       \]
       \[
       x = \frac{13}{14}y
       \]
     ● From Condition 2:  
       \[
       0.6x + z = 1.65z
       \]
       Rearranging gives:
       \[
       0.6x = 1.65z - z
       \]
       \[
       0.6x = 0.65z
       \]
       \[
       x = \frac{0.65}{0.6}z
       \]
       \[
       x = \frac{65}{60}z
       \]
       \[
       x = \frac{13}{12}z
       \]
     Step 3: Comparing the values
         ○ From \( x = \frac{13}{14}y \), we know \( x < y \) because \(\frac{13}{14} < 1\).
         ○ From \( x = \frac{13}{12}z \), we know \( x > z \) because \(\frac{13}{12} > 1\).
     Combining these inequalities, we have:
     \[
     z < x < y
     \]
     Therefore, the correct answer is _Option A: z < x < y_.

•   Container X has 100 ml milk, and Container Y has 100 ml water.

•   When 20 ml milk goes from X to Y, X is left with 80 ml milk, and Y now has 100 ml water plus 20 ml milk.

•   So total water and milk ratio in container Y will be 5:1.

•   Now 20ml of the mixture of Y is transferred to X.

•   So, transferred amount will be 5/6 x 20 ml water and 1/6 x 20 ml milk.

•   Or 50/3 ml water, and 10/3 ml milk.

•   Thus in Y , water left = 100 - 50/3 and milk left = 20 - 10/3.

•   So ratio of water in Y = (100 - 50/3) / 100 = 5/6 = n (as per question)

•   Now container X got 20 ml of mixture in Y. So milk = 80 + 10/3 = 250/3 ml and water = 50/3

•   So ratio of milk in X (m) = (250/3) / 100 = 5/6
•   Thus, m = n.

•   Let x and y be the price of pen and pencil respectively.

•   Now, according to given condition. 3x + 5y = 2x + 7y

•   ∴ x = 2y

•   Thus, the price of a pen is two times the price of a pencil. Hence, option (c) is the correct answer.

•   From statement (1), P > (q + r) i.e., A's amount is more than individual amount received by B and C.

•   Thus, A gets the maximum share. Hence statement (1) is correct.

•   From statement (2), r < (p + q) i.e., C's amount is less than the sum of amount of A and B. But Individual amount received by A or B may be greater than the amount received by C.

•   Thus, C may not get minimum share. Hence statement (2) is not correct. Thus, option (a) is correct answer

Simplify the fraction ration (1/2 : 3/4 : 5/6 ) to numerator ratio. Multiplying by 12 :

(1/2 : 3/4 : 5/6 ) = (12*1/2 : 12*3/4 : 12*5 /6 ) = 6: 9 : 10

Now, according to the question,

6n + 9n + 10n = Rs.2500

Or 25n = 2500

Or n = Rs. 100 Now, the difference between maximum share and minimum share = 10n – 6n = 4n = 4 × 100 = Rs. 400

Final Ratio: 12.8 : 7.2

=> 128 : 72 

=> 16 : 9

•   Let the weight of mixture = 100 g. So amount of milk in mixture = 50 g, and amount of water in mixture = 50 g.

•   We then took one-third of the mixture i.e. 100 g mixture (= 100/3 g).

•   It has (100/3) / 2 water and (100/3) / 2 milk. (coz 50% both)

•   We added 100/3 pure milk to it.

•   So finally we have a mixture equal to:

100/3 milk + (100/6) water + (100/6) milk = 400 / 6 g mixture.

•   So amount of Water in this final mixture=

[ 100 / 6 ] / [ 400 / 6 ] = 1 / 4 = 25 %.

•   So final percent of water in the mixture= 25%.

•   Let P work with speed 3x, and let Q work with speed x.

•   Now work speed of both p and q taken together = 4x.

•   This is faster than R’s work by 4 times, so R’s work speed = 4x / 4 = x.

•   So now finally P’s speed = 3x, Q’s speed = x, and R’s speed = x. So they should share earnings in the ratio 3 : 1 : 1.

•   The monthly incomes of X and Y are in the ratio of 4 : 3. Let them be 4a and 3a.

•   Their monthly expenses are in the ratio of 3:2. Let them be 3b and 2b.

•   As each saves Rs 6,000 per month, we get two equations:

•   Saving = Income – Expenditure

•   So, 4a – 3b = 6000

and 3a – 2b = 6000

•   On solving we get: a = 6000 and b = 6000

•   Their total monthly income = 4a + 3a = 7a = 7 × 6000 = Rs. 42,000

•   Content of sugar in A = 30g/180 ml = 1/6 g/ml

•   Content of sugar in B = 40g/280 ml = 1/7 g/ml

•   Content of sugar in C = 20g/100 ml = 1/5 g/ml

•   Hence vessel C is the sweetest, followed by vessels A and B.

•   Total collection = Rs 443.

•   Total collection of the class - additional contribution = 443 - 2 = Rs 441

•   Every student contributed an amount equal to total number of students.

•   Let total number of students in the class be N. So, N² = 441

•   Hence, N = 21. Hence Ans (b).

1. Correct Answer: Option D: _12 : 11_
 2. Explanation:
     Let's denote the basic pay of the first person as \( B_1 \) and the basic pay of the second person as \( B_2 \).
         ○ The first person receives allowances equal to 65% of his basic pay. Therefore, his total emoluments are:
       \[
       E_1 = B_1 + 0.65B_1 = 1.65B_1
       \]
         ○ The second person receives allowances equal to 80% of his basic pay. Therefore, his total emoluments are:
       \[
       E_2 = B_2 + 0.80B_2 = 1.80B_2
       \]
     Since the total emoluments of both persons are the same, we have:
     \[
     1.65B_1 = 1.80B_2
     \]
     To find the ratio of the basic pay of the former to the latter, we solve for \( \frac{B_1}{B_2} \):
     \[
     \frac{B_1}{B_2} = \frac{1.80}{1.65}
     \]
     Simplifying the fraction:
     \[
     \frac{B_1}{B_2} = \frac{180}{165}
     \]
     Reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15:
     \[
     \frac{B_1}{B_2} = \frac{180 \div 15}{165 \div 15} = \frac{12}{11}
     \]
     Therefore, the ratio of the basic pay of the former to the basic pay of the latter is 12 : 11.

1. Correct Answer: Option A: *7 : 17*
 2. Explanation:
     To solve this problem, we need to determine the total amount of milk and water in the pot and then find their ratio.
     Step 1: Calculate the amount of milk in each glass.
         ○ Let the total volume of each glass be *V*.
         ○ The first glass is \( \frac{1}{3} \) full of milk, so the amount of milk in the first glass is:
       \[
       \frac{1}{3}V
       \]
         ○ The second glass is \( \frac{1}{4} \) full of milk, so the amount of milk in the second glass is:
       \[
       \frac{1}{4}V
       \]
     Step 2: Calculate the total amount of milk.
         ○ Total milk from both glasses:
       \[
       \frac{1}{3}V + \frac{1}{4}V = \frac{4}{12}V + \frac{3}{12}V = \frac{7}{12}V
       \]
     Step 3: Calculate the total amount of water.
         ○ Since each glass is filled up with water, the amount of water added to each glass is the remaining volume after the milk.
         ○ Water in the first glass:
       \[
       V - \frac{1}{3}V = \frac{2}{3}V
       \]
         ○ Water in the second glass:
       \[
       V - \frac{1}{4}V = \frac{3}{4}V
       \]
         ○ Total water from both glasses:
       \[
       \frac{2}{3}V + \frac{3}{4}V = \frac{8}{12}V + \frac{9}{12}V = \frac{17}{12}V
       \]
     Step 4: Calculate the ratio of milk to water.
         ○ The ratio of milk to water is:
       \[
       \frac{\frac{7}{12}V}{\frac{17}{12}V} = \frac{7}{17}
       \]
     Therefore, the ratio of milk to water in the pot is 7 : 17.

Given, monthly incomes of Peter and Paul are in the ratio of 4 : 3 and their expenses are in the ratio of 3 : 2.

Let their monthly incomes be ₹ 4x and ₹ 3x and their expenditures be ₹ 3y and ₹ 2y, respectively

Then,  4x - 3y = 6000…(i)

[⸪ Income - Expenditure = Savings]

and 3x - 2y = 6000

From Eqs. (i) and (ii),

4x- 3y = 3x -2 y

= 4x- 3y = 3y - 2y

= x= y

On putting x = y in Eq. (i), we get

4x - 3x = 6000

= x = 6000

Hence, their monthly incomes are ₹(4 x 6000) and ₹(3 x 6000), i.e. ₹24000 and ₹18000

Let the number of gold coins initially be x and the number c` non-gold coins•, be y According to the Quesbon. 3x = y Whelf1 1() friOrP gold coins total number of gold cons become x A 10 arid the number of non-gold coins remain the same at ■,-

Nnw we have 2(10  y

SolvinQ these two equations we get x = 20and y = 60

Iota] number of coins in the collection at the end is equal to 

2 • 1(i, y= 204 10+ 60=90

1. Correct Answer: c: 3 to 7
 2. Explanation:
     To solve this problem, we need to determine the minimum and maximum number of males who have a driver's license.
     Step 1: Understand the given data.
         ○ Total number of applications: 120
         ○ Number of males: 70
         ○ Number of applicants with a driver's license: 80
     Step 2: Determine the minimum number of males with a driver's license.
         ○ To find the minimum number of males with a driver's license, assume that the maximum number of females have a driver's license.
         ○ Total females = Total applications - Total males = 120 - 70 = 50
         ○ If all 50 females have a driver's license, then the minimum number of males with a driver's license = Total with driver's license - Females with driver's license = 80 - 50 = 30
     Step 3: Determine the maximum number of males with a driver's license.
         ○ To find the maximum number of males with a driver's license, assume that the minimum number of females have a driver's license.
         ○ If no females have a driver's license, then all 80 people with a driver's license are males.
         ○ Therefore, the maximum number of males with a driver's license = 80
     Step 4: Calculate the ratio.
         ○ Minimum number of males with a driver's license = 30
         ○ Maximum number of males with a driver's license = 70 (since there are only 70 males in total, the maximum cannot exceed this number)
         ○ Ratio of minimum to maximum = 30:70 = 3:7
     Therefore, the ratio between the minimum to maximum number of males having a driver's license is 3 to 7.

1. Correct Answer: Option D: 5 : 3
 2. Explanation:
     Let's assume the volume of each glass is V.
     ● Glass 1:  
           ○ Milk = \( \frac{1}{2}V \)
           ○ Water added = \( \frac{1}{2}V \) (to fill the glass to the brim)
     ● Glass 2:  
           ○ Milk = \( \frac{3}{4}V \)
           ○ Water added = \( \frac{1}{4}V \) (to fill the glass to the brim)
     Now, let's calculate the total amount of milk and water when the contents of both glasses are poured into another vessel:
     ● Total Milk:  
       \[
       \frac{1}{2}V + \frac{3}{4}V = \frac{2}{4}V + \frac{3}{4}V = \frac{5}{4}V
       \]
     ● Total Water:  
       \[
       \frac{1}{2}V + \frac{1}{4}V = \frac{2}{4}V + \frac{1}{4}V = \frac{3}{4}V
       \]
     Therefore, the ratio of milk to water in the vessel is:
     \[
     \frac{\frac{5}{4}V}{\frac{3}{4}V} = \frac{5}{4} \div \frac{3}{4} = \frac{5}{4} \times \frac{4}{3} = \frac{5}{3}
     \]
     Thus, the ratio of milk to water is 5 : 3.