To find the length of the bridge, we need to calculate the total distance covered by the car. We are given the speed of the car and the time it takes to cross the bridge. The key is to ensure all units are consistent (e.g., convert speed to meters per second and time to seconds) before applying the formula: Distance = Speed × Time.
Given speed \( = 99 \text{ km/hr}\).
To convert km/hr to m/s, multiply by \(\frac{5}{18}\).
\[ \text{Speed} = 99 \times \frac{5}{18} \text{ m/s} \]\[ \text{Speed} = \frac{11 \times 5}{2} \text{ m/s} \]\[ \text{Speed} = \frac{55}{2} \text{ m/s} \]\[ \text{Speed} = 27.5 \text{ m/s} \]Given time \( = 1.2 \text{ minutes}\).
Since \(1 \text{ minute} = 60 \text{ seconds}\),
\[ \text{Time} = 1.2 \times 60 \text{ seconds} \]\[ \text{Time} = 72 \text{ seconds} \]The distance covered by the car is equal to the length of the bridge.
Using the formula: Distance = Speed × Time
\[ \text{Distance} = 27.5 \text{ m/s} \times 72 \text{ s} \]\[ \text{Distance} = \frac{55}{2} \times 72 \text{ m} \]\[ \text{Distance} = 55 \times 36 \text{ m} \]\[ \text{Distance} = 1980 \text{ m} \]Therefore, the length of the bridge is 1980 meters.
B) 1980 is the correct answer. The calculation shows that the length of the bridge is 1980 meters after converting the speed and time to consistent units.
The problem involves simple interest calculations for two different investments. We are given the total principal amount, two different interest rates, and the condition that the interests earned from both investments are equal after a certain period. We need to find the amount invested at the lower interest rate.
Correct Option: B) 12,350
In a business partnership, profits are typically distributed among partners in proportion to their investment amounts. First, we need to find the ratio of the investments made by Rahul, Monika, and Yuvraj. Then, we will use this ratio to divide the total profit and find Yuvraj's share.
The ratio of investments (Rahul : Monika : Yuvraj) is:
\[ 1310 : 1430 : 1040 \]Divide each number by the greatest common divisor. In this case, each number is divisible by 10.
\[ \frac{1310}{10} : \frac{1430}{10} : \frac{1040}{10} \]\[ 131 : 143 : 104 \]Sum of ratio parts \( = 131 + 143 + 104 = 378 \)
The total profit is ₹1,890. Yuvraj's share will be his ratio part divided by the sum of ratio parts, multiplied by the total profit.
\[ \text{Yuvraj's share} = \frac{\text{Yuvraj's ratio part}}{\text{Sum of ratio parts}} \times \text{Total profit} \]\[ \text{Yuvraj's share} = \frac{104}{378} \times 1890 \]Notice that \(1890 = 5 \times 378\). So, \(1890 / 378 = 5\).
\[ \text{Yuvraj's share} = 104 \times 5 \]\[ \text{Yuvraj's share} = 520 \]C) ₹520 is the correct share of Yuvraj in the profit, calculated based on his investment proportion.
To find the area of a square, we first need to determine the length of its side. The perimeter of a square is given by the formula \(P = 4s\), where \(s\) is the side length. Once we find the side length, the area of the square can be calculated using the formula \(A = s^2\).
D) 7056
The calculated area of the square is 7056 m², which matches option D.
This problem involves setting up and solving linear equations based on the given age conditions at different points in time. We will define variables for the current ages of the father and son, then translate the given information into equations, and finally solve for the required future time.
Define current ages:
Let the current age of the father be \(F\) years.
Let the current age of the son be \(S\) years.
Formulate the first equation (3 years ago):
3 years ago, the father's age was \(F-3\) years.
3 years ago, the son's age was \(S-3\) years.
According to the problem, 3 years ago, the father's age was 21 years more than twice his son's age. So,
Formulate the second equation (future age condition):
Let \(x\) be the number of years from now when the father will be twice his son's age.
After \(x\) years, the father's age will be \(F+x\) years.
After \(x\) years, the son's age will be \(S+x\) years.
According to the problem, after \(x\) years, the father will be twice his son's age. So,
Solve for \(x\):
Now we have two expressions for \(F\). We can equate them:
Simplify and find \(x\):
Subtract \(2S\) from both sides:
A) 18. After 18 years from now, the father will be twice his son's age. This is derived directly from the equations formulated based on the problem statement.
This problem involves calculating the total profit percentage when a trader makes a profit claim and also uses a faulty weight. We need to combine the profit from the claimed selling price and the additional profit from the dishonest weighing. We can assume a base cost price and a standard weight to simplify calculations.
Let the actual cost price (CP) of 1 kg of vegetables be Rs. 1000.
The trader claims to sell at a profit of 57%. So, the claimed selling price (SP) for 1 kg would be:
\[ \text{Claimed SP} = \text{CP} \times (1 + \text{Profit Percentage}) \\ \text{Claimed SP} = 1000 \times (1 + 0.57) = 1000 \times 1.57 = \text{Rs. } 1570 \]The trader uses a weight which is 13% less than what is mentioned. This means for every 1 kg (1000 g) the customer pays for, the customer receives 13% less.
Actual weight delivered for 1 kg paid = \(1000 \text{ g} - (0.13 \times 1000 \text{ g})\)
Actual weight delivered = \(1000 \text{ g} - 130 \text{ g} = 870 \text{ g}\)
If 1000 g costs Rs. 1000, then 870 g will cost Rs. 870.
So, the actual cost price for the quantity sold is Rs. 870.
The trader sells 870 g of vegetables for Rs. 1570 (which is the claimed SP for 1 kg).
Actual Cost Price (CP) = Rs. 870
Actual Selling Price (SP) = Rs. 1570
Total Profit = Actual SP - Actual CP
\[ \text{Total Profit} = 1570 - 870 = \text{Rs. } 700 \]Total Profit Percentage = \[\left( \frac{\text{Total Profit}}{\text{Actual CP}} \right) \times 100 \%\]
\[ \text{Total Profit Percentage} = \left( \frac{700}{870} \right) \times 100 \% \]\[ \text{Total Profit Percentage} \approx 0.804597 \times 100 \% \approx 80.4597 \% \]Rounding off to the nearest whole number, the total profit percentage is 80%.
A) 80% is the correct answer after calculating the combined effect of the claimed profit and the dishonest weighing, and then rounding to the nearest whole number.
To determine which number divides 86,16,16,843, we need to perform division for each option or look for divisibility rules. Since the number is large, direct division is the most reliable method for competitive exams, especially when divisibility rules for the given options (like 43 or 47) are not commonly known or simple.
Divide 86,16,16,843 by 43.
\[ \frac{861616843}{43} \]\[ 861616843 \div 43 \]The division results in \(20037601\) with a remainder of \(0\). This means 86,16,16,843 is perfectly divisible by 43.
Since 43 divides the number, 47 cannot also divide it perfectly unless the number is a multiple of both 43 and 47 (which would make it a multiple of \(43 \times 47 = 2021\)). We already found a divisor, so we can assume this is the correct answer. If we were to check, \(861616843 \div 47\) would not yield a zero remainder.
A number is divisible by 50 if its last two digits are 00. The given number ends in 43, so it is not divisible by 50.
A number is divisible by 36 if it is divisible by both 4 and 9.
A) 43 — As shown in the step-by-step division, 86,16,16,843 divided by 43 gives a quotient of 20,037,601 with no remainder.
To find the greatest ratio among the given options, we need to convert each ratio into a decimal or a fraction with a common denominator and then compare them. Converting to decimals is often the most straightforward method for comparison.
Correct Option: D) 50 : 60
To determine which statement is correct, we need to understand the definitions and properties of squares, rectangles, and rhombuses, and how they relate to each other in terms of hierarchical classification of quadrilaterals.
Definition of a Rectangle: A rectangle is a quadrilateral with four right angles. Its opposite sides are equal in length and parallel.
Definition of a Rhombus: A rhombus is a quadrilateral with all four sides equal in length. Its opposite angles are equal, and its diagonals bisect each other at right angles.
Definition of a Square: A square is a quadrilateral with four equal sides and four right angles. It combines the properties of both a rectangle and a rhombus.
Relationship between Square and Rectangle: Since a square has four right angles and opposite sides are equal (all sides are equal, so opposite sides are certainly equal), it satisfies all the properties of a rectangle. Therefore, a square is a special type of rectangle where all sides are equal.
Relationship between Square and Rhombus: Since a square has all four sides equal in length, it satisfies all the properties of a rhombus. Therefore, a square is a special type of rhombus where all angles are right angles.
Conclusion: A square possesses all the characteristics of both a rectangle and a rhombus. Hence, it is a special case of both.
Correct Option: C) A square is a special case of both a rectangle and a rhombus.