JKSSB Aptitude MCQs

Explanation

To find the age of the youngest child, we can set up a simple algebraic equation based on the information provided.

1. Set up the variables:

Let the age of the youngest child be $x$.

Since the 5 children were born at intervals of 3 years, we can express their ages as follows:

• 1st child (youngest): $x$

• 2nd child: $x + 3$

• 3rd child: $x + 6$

• 4th child: $x + 9$

• 5th child (oldest): $x + 12$

2. Formulate the equation:

We are told that the sum of all their ages is 50 years.

$$x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50$$

3. Solve for $x$:

Combine all the $x$ terms and all the numbers:

$$5x + (3 + 6 + 9 + 12) = 50$$

$$5x + 30 = 50$$

Subtract 30 from both sides:

$$5x = 50 - 30$$

$$5x = 20$$

Divide by 5:

$$x = \frac{20}{5}$$

$$x = \mathbf{4}$$

The age of the youngest child is 4 years.

Explanation

1. Define the variables:

Let the time taken by the first pipe to fill the tank be $x$ hours.

• The second pipe fills the tank 5 hours faster than the first, so its time is $(x - 5)$ hours.

• The second pipe fills the tank 4 hours slower than the third pipe, which means the third pipe is 4 hours faster than the second. So, the third pipe's time is $(x - 5) - 4 =$ $(x - 9)$ hours.

2. Relate the times to their work rates:

The rate at which a pipe fills the tank is the reciprocal of the time it takes.

• Rate of first pipe = $\frac{1}{x}$

• Rate of second pipe = $\frac{1}{x - 5}$

• Rate of third pipe = $\frac{1}{x - 9}$

We are given that the first two pipes operating together fill the tank in the same amount of time as the third pipe alone. Therefore, the sum of their rates equals the rate of the third pipe:

$$\frac{1}{x} + \frac{1}{x - 5} = \frac{1}{x - 9}$$

3. Solve the equation:

First, find a common denominator for the left side of the equation:

$$\frac{(x - 5) + x}{x(x - 5)} = \frac{1}{x - 9}$$

$$\frac{2x - 5}{x^2 - 5x} = \frac{1}{x - 9}$$

Now, cross-multiply to solve for $x$:

$$(2x - 5)(x - 9) = 1(x^2 - 5x)$$

$$2x^2 - 18x - 5x + 45 = x^2 - 5x$$

$$2x^2 - 23x + 45 = x^2 - 5x$$

Subtract $(x^2 - 5x)$ from both sides to form a quadratic equation equal to zero:

$$x^2 - 18x + 45 = 0$$

Factor the quadratic equation by finding two numbers that multiply to 45 and add up to -18. Those numbers are -15 and -3:

$$(x - 15)(x - 3) = 0$$

So, $x = 15$ or $x = 3$.

4. Evaluate the solutions:

If we test $x = 3$, the time for the second pipe would be $3 - 5 = -2$ hours. Since time cannot be negative, $x = 3$ is not a valid solution.

Therefore, $x = 15$.

• First pipe: 15 hours

• Second pipe: $15 - 5 = 10$ hours

• Third pipe: $15 - 9 = 6$ hours

Explanation

To find the ratio between the speed of the boat and the speed of the water current, we can set up an equation based on the relationship between speed, distance, and time.

1. Define the variables:

• Let the speed of the boat in still water be $B$.

• Let the speed of the water current be $C$.

• The speed of the boat running downstream (with the current) is $B + C$.

• The speed of the boat running upstream (against the current) is $B - C$.

• Let the constant distance be $D$.

2. Convert time into a single unit (hours):

• Time taken downstream ($T_d$) = $4 \text{ hours}$

• Time taken upstream ($T_u$) = 8 hours and 48 minutes.

  To convert 48 minutes to hours, divide by 60: $\frac{48}{60} = \frac{4}{5} \text{ hours}$.

  So, $T_u = 8 + \frac{4}{5} = \mathbf{\frac{44}{5} \text{ hours}}$

3. Set up the distance equation:

Since the distance covered is the same in both directions, we can use the formula $\text{Distance} = \text{Speed} \times \text{Time}$ and set the two scenarios equal to each other:

$$\text{Downstream Distance} = \text{Upstream Distance}$$

$$(B + C) \times 4 = (B - C) \times \frac{44}{5}$$

4. Solve for the ratio of $B$ to $C$:

Divide both sides by 4 to simplify the equation:

$$B + C = (B - C) \times \frac{11}{5}$$

Multiply both sides by 5 to remove the fraction:

$$5(B + C) = 11(B - C)$$

$$5B + 5C = 11B - 11C$$

Now, rearrange the terms to group the $B$ variables on one side and the $C$ variables on the other:

$$5C + 11C = 11B - 5B$$

$$16C = 6B$$

Finally, find the ratio of $B$ (boat speed) to $C$ (current speed):

$$\frac{B}{C} = \frac{16}{6}$$

Simplify the fraction by dividing by 2:

$$\frac{B}{C} = \frac{8}{3}$$

The ratio between the speed of the boat and the speed of the water current is 8:3.

The correct option is 8:3.

Explanation

To determine which offer is better, we need to compare their values at the same point in time. We can do this by finding the Future Value of the Rs. 12,000 cash offer after 8 months and comparing it to the credit offer.

1. Calculate the Future Value of the Cash Offer:

If the man takes the Rs. 12,000 in cash today and invests it at the given rate of 18% per annum for 8 months, let's see how much it will be worth.

• Principal (P) = Rs. 12,000

• Rate (R) = 18% p.a.

• Time (T) = 8 months = $\frac{8}{12}$ years = $\frac{2}{3}$ years

First, calculate the Simple Interest (SI) he would earn:

$$SI = \frac{P \times R \times T}{100}$$

$$SI = \frac{12000 \times 18 \times \left(\frac{2}{3}\right)}{100}$$

$$SI = 120 \times 12 = \mathbf{Rs. 1440}$$

Now, add the interest to the principal to find the total Future Value:

$$\text{Future Value} = 12000 + 1440 = \mathbf{Rs. 13,440}$$

2. Compare the Two Offers:

• Value of Offer 1 after 8 months: Rs. 13,440

• Value of Offer 2 after 8 months: Rs. 12,880

Since Rs. 13,440 is greater than Rs. 12,880, taking the cash today and earning interest on it yields a higher return than accepting the future credit offer.

The correct answer is Rs. 12,000 in cash.

Explanation

To find the probability of drawing a white ball first and a black ball second without replacement, we need to calculate the probability of each event happening in sequence.

1. Probability of the first ball being white:

• Initial number of white balls = 12

• Initial number of black balls = 18

• Total number of balls = 12 + 18 = 30

• The probability of drawing a white ball first ($P(W_1)$) is:

  $$P(W_1) = \frac{12}{30} = \frac{2}{5}$$

2. Probability of the second ball being black:

• Since the first ball was drawn without replacement, there is now 1 less white ball in the bag.

• Remaining number of white balls = 11

• Remaining number of black balls = 18 (unchanged)

• New total number of balls = 11 + 18 = 29

• The probability of drawing a black ball second, given the first was white ($P(B_2|W_1)$) is:

  $$P(B_2|W_1) = \frac{18}{29}$$

3. Total Probability:

• To find the combined probability of both events happening in this exact order, multiply the two individual probabilities together:

  $$P(W_1 \text{ and } B_2) = P(W_1) \times P(B_2|W_1)$$

  $$P(W_1 \text{ and } B_2) = \frac{2}{5} \times \frac{18}{29} = \mathbf{\frac{36}{145}}$$

The correct answer is 36/145.

Explanation

1. Calculate the total possible outcomes:

• When two dice are thrown simultaneously, each die has 6 possible outcomes (1, 2, 3, 4, 5, 6).

• Total number of outcomes = $6 \times 6 = \mathbf{36}$

2. Determine the condition for an odd product:

• The product of two numbers is even if at least one of the numbers is even.

• The product is odd only if both numbers are odd.

• The odd numbers on a die are 1, 3, and 5 (there are 3 odd numbers).

3. Calculate the outcomes for an odd product:

• For both dice to show an odd number, we multiply the possibilities for each die:

• Odd outcomes = $3 \times 3 = \mathbf{9}$

• The probability of getting an odd product is $\frac{9}{36}$, which simplifies to $\mathbf{\frac{1}{4}}$.

4. Calculate the probability of an even product:

• Since the product must be either odd or even, we subtract the probability of an odd product from 1.

  $$P(\text{Even Product}) = 1 - P(\text{Odd Product})$$

  $$P(\text{Even Product}) = 1 - \frac{1}{4} = \mathbf{\frac{3}{4}}$$

Explanation

To find the length of the ladder, we can visualize the situation as a right-angled triangle where the ladder is the hypotenuse, the wall is the vertical side, and the ground is the base.

1. Identify the given information:

• Angle of elevation ($\theta$): 60°

• Distance from the foot to the wall (Adjacent side): 4.6 m

• Length of the ladder (Hypotenuse): Let's call this $L$.

2. Choose the correct trigonometric ratio:

We know the adjacent side and need to find the hypotenuse. The cosine function relates these two sides:

$$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$

3. Set up the equation and solve:

Substitute the known values into the formula:

$$\cos(60^\circ) = \frac{4.6}{L}$$

We know that the exact value of $\cos(60^\circ)$ is $\frac{1}{2}$:

$$\frac{1}{2} = \frac{4.6}{L}$$

Now, cross-multiply to solve for $L$:

$$L = 4.6 \times 2$$

$$L = 9.2$$

The correct answer is 9.2 m.

Explanation

To find the value of the expression, let's break it down and calculate each part step-by-step:

1. Calculate each percentage:

• 14% of 14 = $0.14 \times 14 = \mathbf{1.96}$

• 28% of 28 = $0.28 \times 28 = \mathbf{7.84}$

• 92% of 96 = $0.92 \times 96 = \mathbf{88.32}$

• 15% of 85 = $0.15 \times 85 = \mathbf{12.75}$

2. Put the values back into the equation:

$$1.96 + 7.84 + 88.32 - 12.75$$

3. Perform the addition and subtraction:

• Add the first two terms: $1.96 + 7.84 = \mathbf{9.80}$

• Add the third term: $9.80 + 88.32 = \mathbf{98.12}$

• Subtract the last term: $98.12 - 12.75 = \mathbf{85.37}$

The correct answer is 85.37.

Explanation

To find the speed of the train, we need to use the concept of relative speed since both the train and the man are moving.

1. Define the relative speed:

• Let the speed of the train be $S$ km/hr.

• The man is running at 5 km/hr in the same direction.

• When two objects move in the same direction, their relative speed is the difference between their speeds.

• $$\text{Relative Speed} = (S - 5) \text{ km/hr}$$

2. Convert relative speed to meters per second (m/s):

• To match the units of length (meters) and time (seconds) provided in the problem, convert km/hr to m/s by multiplying by $\frac{5}{18}$.

• $$\text{Relative Speed in m/s} = (S - 5) \times \frac{5}{18}$$

3. Relate distance, speed, and time:

• The distance covered by the train to pass the man is equal to its own length, which is 125 meters.

• The time taken is 10 seconds.

• Using the standard formula $\text{Distance} = \text{Speed} \times \text{Time}$:

• $$125 = \left((S - 5) \times \frac{5}{18}\right) \times 10$$

4. Solve for $S$ (the train's speed):

• Simplify the equation by multiplying the terms:

• $$125 = (S - 5) \times \frac{50}{18}$$

• Multiply both sides by 18 to remove the fraction:

• $$125 \times 18 = (S - 5) \times 50$$

• $$2250 = 50(S - 5)$$

• Divide both sides by 50:

• $$45 = S - 5$$

• Add 5 to both sides:

• $$S = 45 + 5 = \mathbf{50 \text{ km/hr}}$$

The correct answer is 50 km/hr.

Explanation

To find the time taken by the second train to cross the platform, we can break down the problem using the basic relationship between speed, distance, and time.

1. Analyze the first train

• Let the speed of the trains be $S$.

• Let the length of the first train be $L_1$.

• When a train crosses a stationary point (like a telegraphic post), the distance it travels is equal to its own length.

• It takes 5 minutes to cross the post, so we can establish the relationship:

  $$L_1 = S \times 5$$

2. Analyze the second train and the platform

• The length of the second train ($L_2$) is double the length of the first train:

  $$L_2 = 2 \times L_1$$

  $$L_2 = 2 \times (S \times 5) = \mathbf{10S}$$

• The train is crossing a platform of its own length. Let the length of the platform be $P$.

  $$P = L_2 = \mathbf{10S}$$

• When a train crosses a platform, the total distance it must cover is its own length plus the length of the platform:

  $$\text{Total Distance} = L_2 + P$$

  $$\text{Total Distance} = 10S + 10S = \mathbf{20S}$$

3. Calculate the time for the second train

• The second train is moving at the same speed ($S$).

• $$\text{Time} = \frac{\text{Total Distance}}{\text{Speed}}$$

• $$\text{Time} = \frac{20S}{S} = \mathbf{20 \text{ minutes}}$$

The correct answer is 20 minutes.

Explanation

To find the length of the platform, we need to determine the length of the train first, and then use that to find the platform's length.

Here is the step-by-step breakdown:

1. Convert the speed from km/hr to m/s:

• Speed of the train = 54 km/hr

• To convert to meters per second, multiply by $\frac{5}{18}$:

  $$\text{Speed} = 54 \times \frac{5}{18} = 15 \text{ m/s}$$

2. Calculate the length of the train:

• When a train passes a stationary object like a man, the distance it covers is equal to its own length.

• The train takes 20 seconds to pass the man.

• $$\text{Distance (Length of Train)} = \text{Speed} \times \text{Time}$$

• $$\text{Length of Train} = 15 \text{ m/s} \times 20 \text{ s} = \mathbf{300 \text{ meters}}$$

3. Calculate the total distance covered when passing the platform:

• When a train passes a platform, the total distance it covers is its own length plus the length of the platform.

• The train takes 36 seconds to pass the platform.

• $$\text{Total Distance} = \text{Speed} \times \text{Time}$$

• $$\text{Total Distance} = 15 \text{ m/s} \times 36 \text{ s} = \mathbf{540 \text{ meters}}$$

4. Find the length of the platform:

• Since the Total Distance (540 m) is the sum of the Train's Length and the Platform's Length:

  $$\text{Platform Length} = \text{Total Distance} - \text{Train Length}$$

  $$\text{Platform Length} = 540 - 300 = \mathbf{240 \text{ meters}}$$

The correct answer is 240 m.

Explanation

To find out how much B received from the profit, we need to determine the combined ratio of the investments made by A, B, and C.

1. Standardize the Ratios

We are given two separate ratios:

• Ratio of A to C ($A:C$) = 2:1

• Ratio of A to B ($A:B$) = 3:2

To combine these into a single ratio ($A:B:C$), we need to make the value for 'A' the same in both ratios. The lowest common multiple of 2 and 3 is 6.

• Multiply the $A:C$ ratio by 3:

  $A:C = (2 \times 3) : (1 \times 3) = \mathbf{6:3}$

• Multiply the $A:B$ ratio by 2:

  $A:B = (3 \times 2) : (2 \times 2) = \mathbf{6:4}$

Now that 'A' is 6 in both cases, we can combine them:

$A : B : C = 6 : 4 : 3$

2. Calculate the Total Parts of the Ratio

The profit is distributed based on the total parts of their investment ratio.

Total parts = $6 + 4 + 3 = \mathbf{13}$

3. Calculate B's Share of the Profit

The total profit is Rs. 157,300. B's share is 4 parts out of the total 13 parts.

$$B's Share = \left(\frac{4}{13}\right) \times 157300$$

$$B's Share = 4 \times 12100 = \mathbf{48400}$$

The correct answer is Rs. 48400.

Explanation

To find the amount borrowed at the 7% interest rate, we can set up an algebraic equation based on the formula for Simple Interest.

1. Set up the variables:

• Let the amount borrowed at 7% be $x$.

• Since the total amount borrowed is Rs. 2500, the amount borrowed at 5% is $2500 - x$.

• The total time ($T$) for both loans is 2 years.

2. Formulate the Simple Interest equations:

The formula for Simple Interest is:

$$SI = \frac{Principal \times Rate \times Time}{100}$$

• Interest from the 7% loan:

  $$SI_1 = \frac{x \times 7 \times 2}{100} = \frac{14x}{100} = 0.14x$$

• Interest from the 5% loan:

  $$SI_2 = \frac{(2500 - x) \times 5 \times 2}{100} = \frac{(2500 - x) \times 10}{100} = \frac{25000 - 10x}{100} = 250 - 0.10x$$

3. Set up the total interest equation:

We are given that the total interest paid for both loans is Rs. 276. Therefore, the sum of $SI_1$ and $SI_2$ must equal 276:

$$SI_1 + SI_2 = 276$$

$$0.14x + (250 - 0.10x) = 276$$

4. Solve for $x$:

Combine the $x$ terms:

$$0.04x + 250 = 276$$

Subtract 250 from both sides:

$$0.04x = 26$$

Divide by 0.04:

$$x = \frac{26}{0.04}$$

$$x = 650$$

The amount borrowed at the rate of 7% p.a. is Rs. 650.

Explanation

1. Analyze the Ratios

• Initial ratio of Milk to Water = 7:2

• Final ratio of Milk to Water = 7:3

Because only water is being added to the mixture, the proportion of milk (7 parts) conveniently remains constant in both ratios.

2. Calculate the Increase in Water Parts

• The water portion increases from 2 parts to 3 parts.

• Increase in water = 3 - 2 = 1 part

3. Find the Volume of One Part

• The total initial mixture is 729 liters, which represents the initial 9 parts (7 parts milk + 2 parts water).

• $$1 \text{ part} = \frac{729}{9} = \mathbf{81 \text{ liters}}$$

Since the water needs to increase by exactly 1 part to achieve the new ratio, you must add 81 liters of water.

The correct option is 81 litres.

Explanation

1. Determine the seating pattern:

We have a total of 7 people (4 boys and 3 girls) and 7 seats. For them to sit alternately, the arrangement must start and end with a boy, because there is one more boy than there are girls.

The only possible seating pattern is:

Boy - Girl - Boy - Girl - Boy - Girl - Boy

2. Calculate the arrangements for the boys:

There are 4 boys who need to be seated in the 4 designated "Boy" spots. The number of ways to arrange them is 4 factorial ($4!$):

$$4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}$$

3. Calculate the arrangements for the girls:

There are 3 girls who need to be seated in the 3 designated "Girl" spots. The number of ways to arrange them is 3 factorial ($3!$):

$$3! = 3 \times 2 \times 1 = 6 \text{ ways}$$

4. Calculate the total number of ways:

To find the total possible seating arrangements, multiply the number of ways to arrange the boys by the number of ways to arrange the girls:

$$\text{Total ways} = 4! \times 3!$$

$$\text{Total ways} = 24 \times 6 = \mathbf{144}$$

The correct answer is 144.

Explanation

To find out how many days it will take Amit to plough the field alone, we can calculate the amount of work each person does in a single day.

1. Calculate the combined 1-day work:

• Amit and Sumit together can plough the field in 4 days.

• Therefore, the amount of work they do together in 1 day is:

  $$\text{Amit's 1-day work} + \text{Sumit's 1-day work} = \frac{1}{4}$$

2. State Sumit's 1-day work:

• Sumit alone can plough the field in 6 days.

• Therefore, the amount of work Sumit does in 1 day is:

  $$\text{Sumit's 1-day work} = \frac{1}{6}$$

3. Calculate Amit's 1-day work:

• To find Amit's 1-day work, subtract Sumit's 1-day work from their combined 1-day work:

  $$\text{Amit's 1-day work} = \frac{1}{4} - \frac{1}{6}$$

• Find a common denominator to subtract the fractions (the lowest common multiple of 4 and 6 is 12):

  $$\text{Amit's 1-day work} = \frac{3}{12} - \frac{2}{12}$$

  $$\text{Amit's 1-day work} = \frac{1}{12}$$

4. Determine the total days for Amit:

• If Amit completes $\frac{1}{12}$ of the work in 1 day, it will take him 12 days to complete the entire field alone.

The correct answer is 12 days.

Explanation

1. Calculate the total work:

• If 10 men can do the work in 12 days, the total amount of work is:

• Total Work = 10 men × 12 days = 120 man-days

2. Calculate the time for 12 men:

• Now, we divide the total work by the new number of men to find the number of days needed.

• Time = Total Work / 12 men

• Time = 120 / 12 = 10 days

Alternatively, you can use the inverse proportion formula ($M_1 \times D_1 = M_2 \times D_2$):

• $10 \times 12 = 12 \times D_2$

• $120 = 12 \times D_2$

• $D_2 = 10$

The correct answer is 10 days.

Explanation

1. Determine the distance covered by each train:

• The total distance between the two stations is 200 km.

• The trains cross each other at a distance of 110 km from one of the stations.

• This means Train A traveled 110 km.

• Train B traveled the remaining distance: $200 - 110 = \mathbf{90 \text{ km}}$.

2. Relate distance to speed:

• Since both trains started at the same time and meet at the same moment, the time ($t$) they have been traveling is exactly the same.

• The formula for speed is $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$.

• Because time is constant for both, the ratio of their speeds is directly proportional to the ratio of the distances they covered.

3. Calculate the ratio:

• Let $S_1$ be the speed of Train A and $S_2$ be the speed of Train B.

• $$\frac{S_1}{S_2} = \frac{\text{Distance}_1}{\text{Distance}_2}$$

• $$\frac{S_1}{S_2} = \frac{110}{90}$$

• Simplify by canceling the zeros:

• $$\frac{S_1}{S_2} = \frac{11}{9}$$

The ratio of their speeds is 11:9.

Explanation

To make the calculations simple, let's assume the total distance is 100 km (since we are working with percentages).

1. Calculate the time for the first part of the journey:

• Distance: 30% of 100 km = 30 km

• Speed: 20 kmph

• Time ($T_1$): $\frac{\text{Distance}}{\text{Speed}} = \frac{30}{20} = \mathbf{1.5 \text{ hours}}$

2. Calculate the time for the second part of the journey:

• Distance: 60% of 100 km = 60 km

• Speed: 40 kmph

• Time ($T_2$): $\frac{\text{Distance}}{\text{Speed}} = \frac{60}{40} = \mathbf{1.5 \text{ hours}}$

3. Calculate the time for the remaining part of the journey:

• Remaining Distance: $100 \text{ km} - (30 \text{ km} + 60 \text{ km}) = \mathbf{10 \text{ km}}$

• Speed: 10 kmph

• Time ($T_3$): $\frac{\text{Distance}}{\text{Speed}} = \frac{10}{10} = \mathbf{1 \text{ hour}}$

4. Calculate the Total Average Speed:

• Total Distance: $30 + 60 + 10 = \mathbf{100 \text{ km}}$

• Total Time: $1.5 + 1.5 + 1 = \mathbf{4 \text{ hours}}$

$$\text{Average Speed} = \frac{100 \text{ km}}{4 \text{ hours}} = \mathbf{25 \text{ kmph}}$$

Explanation

1. Calculate the profit from the first batch

• Ranjan sold 30 tables at a profit of Rs. 12 per table.

• $\text{Profit}_1 = 30 \times 12 = \mathbf{Rs. 360}$

2. Calculate the profit from the second batch

• He sold 75 tables at a profit of Rs. 14 per table.

• $\text{Profit}_2 = 75 \times 14 = \mathbf{Rs. 1050}$

3. Calculate the loss from the remaining tables

• First, find the number of remaining tables: $120 - (30 + 75) = 120 - 105 = \mathbf{15 \text{ tables}}$.

• He sold these 15 tables at a loss of Rs. 7 per table.

• $\text{Loss} = 15 \times 7 = \mathbf{Rs. 105}$

4. Calculate the total net profit

• To find the overall profit, add the profits and subtract the loss.

• $\text{Net Profit} = 360 + 1050 - 105$

• $\text{Net Profit} = 1410 - 105 = \mathbf{Rs. 1305}$

5. Calculate the average profit per table

• Divide the net profit by the total number of tables (120).

• $$\text{Average Profit} = \frac{1305}{120} = \mathbf{10.875}$$

Explanation

The Shortcut Method

Whenever two items are sold at the same selling price, and one is sold at a profit of $x\%$ while the other is sold at a loss of $x\%$, there is always an overall loss.

The formula for the overall loss percentage is:

$$\text{Loss } \% = \frac{x^2}{100}$$

In this case, $x = 20$:

$$\text{Loss } \% = \frac{20^2}{100}$$

$$\text{Loss } \% = \frac{400}{100} = \mathbf{4\%}$$

Explanation

1. Set up the equations:

Let the shares of the four people be $A$, $B$, $C$, and $D$.

• Total Sum: $A + B + C + D = 1250$

• Condition 1: The total share of B and D is equal to 14/11 of the total share of A and C.

  $$B + D = \frac{14}{11}(A + C)$$

• Condition 2: The share of D is half of the share of A.

  $$D = \frac{A}{2}$$

• Condition 3: The share of C is 1.2 times the share of A.

  $$C = 1.2A$$

2. Find the combined share of A and C:

We know the sum of all shares is 1250. We can group them as $(A + C) + (B + D) = 1250$.

Substitute Condition 1 into this equation:

$$(A + C) + \frac{14}{11}(A + C) = 1250$$

Factor out $(A + C)$:

$$(A + C) \left(1 + \frac{14}{11}\right) = 1250$$

$$(A + C) \left(\frac{25}{11}\right) = 1250$$

Now, solve for $(A + C)$:

$$A + C = 1250 \times \frac{11}{25}$$

$$A + C = 50 \times 11 = \mathbf{550}$$

3. Find the individual shares of A and C:

Substitute Condition 3 ($C = 1.2A$) into the result from step 2:

$$A + 1.2A = 550$$

$$2.2A = 550$$

$$A = \frac{550}{2.2} = \mathbf{250}$$

Now, find C:

$$C = 1.2 \times 250 = \mathbf{300}$$

4. Find the shares of D and B:

Using Condition 2 to find D:

$$D = \frac{A}{2} = \frac{250}{2} = \mathbf{125}$$

Finally, we know that $(A + C) + (B + D) = 1250$. Since $(A + C) = 550$:

$$550 + (B + D) = 1250$$

$$B + D = 700$$

Substitute the value of D to find B:

$$B + 125 = 700$$

$$B = 700 - 125 = \mathbf{575}$$

Conclusion:

The shares are:

• A = Rs. 250

• B = Rs. 575

• C = Rs. 300

• D = Rs. 125

Explanation

To find the total number of students in the beginning, we can set up an equation using the given ratios.

1. Set up the initial variables:

Let the number of students in the three classes initially be $2x$, $3x$, and $4x$.

The total number of students in the beginning would be $2x + 3x + 4x = \mathbf{9x}$.

2. Formulate the equation after the increase:

If 12 students are added to each class, the new number of students in the classes becomes:

• $2x + 12$

• $3x + 12$

• $4x + 12$

We are told the new ratio is 8 : 11 : 14.

We can take the ratio of the first two classes and set it equal to 8/11:

$$\frac{2x + 12}{3x + 12} = \frac{8}{11}$$

3. Solve for $x$:

Cross-multiply to solve the equation:

$$11(2x + 12) = 8(3x + 12)$$

$$22x + 132 = 24x + 96$$

Subtract $22x$ from both sides:

$$132 = 2x + 96$$

Subtract 96 from both sides:

$$36 = 2x$$

$$x = 18$$

4. Calculate the total initial number of students:

Now substitute $x = 18$ back into the expression for the total number of students ($9x$):

$$\text{Total} = 9 \times 18 = 162$$

The total number of students in the three classes in the beginning was 162.

Explanation

To find the number of valid votes the other candidate received, we can break the problem down into a few simple steps:

1. Calculate the total number of valid votes:

• The total number of votes is 15,200.

• We know that 15% of the votes were invalid. This means that 85% of the total votes were valid (100% - 15%).

• Valid votes = 85% of 15,200

• Valid votes = 0.85 × 15,200 = 12,920

2. Calculate the percentage of valid votes for the other candidate:

• One candidate got 55% of the valid votes.

• Therefore, the other candidate must have received the remaining 45% of the valid votes (100% - 55%).

3. Calculate the number of votes the other candidate got:

• Votes for the other candidate = 45% of 12,920

• Votes = 0.45 × 12,920 = 5,814

The correct answer is 5814.

Explanation

The Correct Answer is 99

Here is the step-by-step mathematical breakdown of how to solve this:

1. Set up the initial equation

Let $N$ be the dividend, $D$ be the divisor, and $Q$ be the quotient.

Based on the first condition (a remainder of 71):

$$N = D \cdot Q + 71$$

(Note: A fundamental rule of division is that the divisor must always be strictly greater than the remainder. Therefore, $D > 71$).

2. Apply the second condition

The problem states the dividend is now twice as large. Let's multiply our initial equation by 2 to represent the new dividend, $2N$:

$$2N = 2(D \cdot Q + 71)$$

$$2N = 2DQ + 142$$

3. Determine the source of the new remainder

When we divide the new dividend ($2N$) by the same divisor ($D$), the first part of the expression ($2DQ$) is perfectly divisible by $D$ because $D$ is a factor.

This means the new remainder must come entirely from dividing $142$ by $D$.

The problem tells us this new remainder is $43$. We can express this as:

$$142 = k \cdot D + 43$$

(where $k$ is an unknown integer representing the quotient of this specific division)

4. Solve for the divisor ($D$)

Subtract $43$ from both sides to find the multiple of $D$:

$$k \cdot D = 142 - 43$$

$$k \cdot D = 99$$

This equation tells us that the divisor $D$ must be a factor of $99$. The factors of $99$ are: $1, 3, 9, 11, 33,$ and $99$.

Since we established in Step 1 that the divisor must be greater than $71$, the only valid factor is 99.

Explanation

The Correct Answer is: 324

Here is the step-by-step breakdown using a simple algebraic identity:

1. Identify the pattern in the expression:

The given expression is $287 \times 287 + 269 \times 269 - 2 \times 287 \times 269$.

Let's substitute the numbers with variables to make it easier to read:

• Let $a = 287$

• Let $b = 269$

Now, substitute these into the original expression:

$$a \times a + b \times b - 2 \times a \times b$$

$$a^2 + b^2 - 2ab$$

2. Apply the algebraic identity:

Recall the standard algebraic formula for the square of a difference:

$$(a - b)^2 = a^2 + b^2 - 2ab$$

3. Solve the simplified equation:

Since the expression is equal to $(a - b)^2$, we can simply plug our original numbers back into this much smaller formula:

$$(287 - 269)^2$$

First, subtract the numbers inside the parentheses:

$$287 - 269 = 18$$

Now, square the result:

$$18^2 = 18 \times 18 = 324$$

Explanation

The Correct Answer is 2

Here is the step-by-step breakdown of how to solve this:

1. Divide the number by the given divisor:

We need to find the remainder when 1056 is divided by 23.

Let's do the division:

• $1056 \div 23 = 45$ with some remainder.

• Multiply 45 by 23 to find the nearest multiple below 1056:

  $45 \times 23 = 1035$

2. Find the remainder:

Subtract that multiple from the original number:

• $1056 - 1035 = 21$

The remainder is 21. This means that 1056 is 21 greater than a perfect multiple of 23.

3. Calculate the number to be added:

To reach the next multiple of 23, we need to add the difference between the divisor (23) and our current remainder (21).

• Number to be added = Divisor - Remainder

• Number to be added = $23 - 21 = \mathbf{2}$

Verification:

If we add 2 to 1056, we get 1058.

$1058 \div 23 = 46$ (exactly divisible with no remainder).

Explanation

Scenario 1: The sum becomes 4 times itself.

The total amount is $A_1 = 4P$.

The simple interest earned is $SI_1 = A_1 - P = 3P$.

Given the rate $R_1 = 10\%$, we can use the standard simple interest formula to find the time $T$:

$$SI_1 = \frac{P \cdot R_1 \cdot T}{100}$$

$$3P = \frac{P \cdot 10 \cdot T}{100}$$

Dividing both sides by $P$ and simplifying:

$$3 = \frac{10 \cdot T}{100}$$

$$3 = \frac{T}{10}$$

$$T = 30 \text{ years}$$

Scenario 2: The sum becomes 7 times itself in the same duration.

The new total amount is $A_2 = 7P$.

The new simple interest earned is $SI_2 = A_2 - P = 6P$.

Using the calculated time $T = 30$, we solve for the new rate $R_2$:

$$SI_2 = \frac{P \cdot R_2 \cdot T}{100}$$

$$6P = \frac{P \cdot R_2 \cdot 30}{100}$$

Dividing by $P$ and simplifying:

$$6 = \frac{30 \cdot R_2}{100}$$

$$6 = \frac{3 \cdot R_2}{10}$$

$$60 = 3 \cdot R_2$$

$$R_2 = 20$$

The new rate of interest is 20%.

Explanation

The Correct Answer is: 50.7

Here is the step-by-step breakdown of how to solve this:

1. Find the initial (incorrect) total sum:

• We know the initial mean of 100 items was 46.

• Total Sum = Mean × Number of Items

• Incorrect Total Sum = 46 × 100 = 4600

2. Correct the misread items: To find the correct sum, we need to subtract the wrong values that were included and add the correct values back in.

• Wrong values to remove: 61 and 34 (Total = 95)

• Correct values to add: 16 and 43 (Total = 59)

• Correct Total Sum = (Incorrect Sum) - (Wrong Values) + (Correct Values)

• Correct Total Sum = 4600 - 95 + 59

• Correct Total Sum = 4564

3. Calculate the correct mean: We are given that the actual number of items was 90, not 100.

• Correct Mean = Correct Total Sum ÷ Correct Number of Items

• Correct Mean = 4564 ÷ 90

• Correct Mean = 50.711...

Rounding to one decimal place, the correct mean is 50.7.

Explanation

The Correct Answer is: 8 years 11 months

Here is the step-by-step breakdown of how to solve this:

1. Find the age gaps:

• Gap between A and B: A is 4 years and 7 months older than B.

• Gap between B and C: B is 3 years and 4 months older than C.

2. Determine everyone's current age: We are given that C is currently 5 years and 2 months old.

• B's Age: Add C's age to the gap between B and C.

  • 5 years 2 months + 3 years 4 months = 8 years and 6 months

• A's Age: Add B's age to the gap between A and B.

  • 8 years 6 months + 4 years 7 months = 12 years and 13 months

  • Since 12 months make a year, this simplifies to 13 years and 1 month

3. Calculate the total age: Add the ages of A, B, and C together:

• Years: 13 + 8 + 5 = 26 years

• Months: 1 + 6 + 2 = 9 months

• Total Age: 26 years and 9 months

4. Find the average: To easily divide by 3, convert the total age entirely into months:

• 26 years × 12 months = 312 months

• 312 + 9 months = 321 total months

Now, divide by 3 to find the average:

• 321 ÷ 3 = 107 months

Finally, convert 107 months back into years and months:

• 107 ÷ 12 = 8 years with a remainder of 11 months.

The average age is 8 years 11 months.