This problem involves the concept of relative speed. When a train passes a person walking in the opposite direction, the relative speed is the sum of the train's speed and the person's speed. The distance covered by the train in this context is its own length. We will set up two equations based on the given information and solve for the speed of the train and its length.
Let the speed of the train be \(S_T\) (in m/s) and its length be \(L\) (in meters).
When the train passes a person walking in the opposite direction, the relative speed is the sum of their speeds. The distance covered is the length of the train.
First case: The first person is walking at 4 m/s. The train passes this person in 10 seconds.
Second case: The second person is walking at 10 m/s. The train passes this person in 8 seconds.
Now we have two equations for \(L\). We can equate them to solve for \(S_T\):
\[ (S_T + 4) \times 10 = (S_T + 10) \times 8 \]Expand both sides of the equation:
\[ 10 S_T + 40 = 8 S_T + 80 \]Rearrange the terms to solve for \(S_T\):
\[ 10 S_T - 8 S_T = 80 - 40 \]\[ 2 S_T = 40 \]\[ S_T = \frac{40}{2} \]\[ S_T = 20 \text{ m/s} \]The speed of the train is 20 m/s.
B) 20 m/s — By setting up two equations based on the relative speed and time taken to pass each person, and equating the length of the train, we find the speed of the train to be 20 m/s.
We are given the ratio of the length and breadth of a rectangle, and the ratio of its perimeter and area. We need to find the actual length of the rectangle. We will use the given ratios to set up equations and solve for the dimensions.
Define variables based on the first ratio:
Let the length of the rectangle be \(L\) and the breadth be \(B\).
Given that the ratio of length to breadth is 2 : 1, we can write:
This implies \(L = 2B\).
Express perimeter and area in terms of one variable:
The perimeter \(P\) of a rectangle is given by \(P = 2(L + B)\).
Substitute \(L = 2B\) into the perimeter formula:
The area \(A\) of a rectangle is given by \(A = L \times B\).
Substitute \(L = 2B\) into the area formula:
Use the second ratio to form an equation:
Given that the numerical values of the perimeter and the area are in the ratio 3 : 4, we have:
Substitute the expressions for \(P\) and \(A\) in terms of \(B\):
\[ \frac{6B}{2B^2} = \frac{3}{4} \]Solve the equation for \(B\):
Simplify the left side of the equation:
From this, we can directly see that \(B = 4\).
Calculate the length \(L\):
We know that \(L = 2B\). Substitute the value of \(B\):
So, the length of the rectangle is 8 units.
D) 8 units is the correct length of the rectangle, as calculated from the given ratios.
Solve the following: 4\( \frac{1}{3} \) × 2\( \frac{4}{5} \) ÷ 2\( \frac{1}{3} \)= ?
6 \( \frac{1}{5} \)
5 \( \frac{1}{5} \)
6 \( \frac{6}{5} \)
1 \( \frac{6}{5} \)
To solve the given expression, we first convert all mixed fractions into improper fractions. Then, we follow the order of operations (BODMAS/PEMDAS), which dictates that division should be performed before multiplication. Finally, we simplify the result and convert it back to a mixed fraction if necessary.
The given expression is \[4\frac{1}{3} \times 2\frac{4}{5} \div 2\frac{1}{3}\].
The expression becomes: \[\frac{13}{3} \times \frac{14}{5} \div \frac{7}{3}\]
Dividing by a fraction is equivalent to multiplying by its reciprocal.
\[ \frac{14}{5} \div \frac{7}{3} = \frac{14}{5} \times \frac{3}{7} \]Now, simplify before multiplying:
\[ \frac{14}{5} \times \frac{3}{7} = \frac{2 \times 7}{5} \times \frac{3}{7} = \frac{2 \times 3}{5} = \frac{6}{5} \]Now, multiply the result from step 2 with the first fraction:
\[ \frac{13}{3} \times \frac{6}{5} \]Simplify before multiplying:
\[ \frac{13}{3} \times \frac{6}{5} = \frac{13}{1} \times \frac{2}{5} = \frac{13 \times 2}{1 \times 5} = \frac{26}{5} \]Divide 26 by 5:
\[ 26 \div 5 = 5 \text{ with a remainder of } 1 \]So, the mixed fraction is \( 5\frac{1}{5} \).
Correct Option: B) \( 5\frac{1}{5} \)
This problem involves work and time. We are given the efficiency relationship between two workers and the time they take to complete a task together. We need to find the time one worker takes alone. The key is to relate efficiency to the amount of work done per unit of time.
A) 24 days is the correct answer. Worker A, being thrice as efficient as B, will take 24 days to complete the work alone.
To solve this problem, we first need to determine the initial number of girls and boys in the class using the given total number of students and their ratio. Then, we will adjust the number of boys based on the information that some boys leave. Finally, we will calculate the new ratio of boys to girls.
Calculate the initial number of girls and boys:
Total number of students = 60
Ratio of girls to boys = 5 : 7
The sum of the ratio parts is \(5 + 7 = 12\).
Number of girls \( = \frac{5}{12} \times 60 = 5 \times 5 = 25 \)
Number of boys \( = \frac{7}{12} \times 60 = 7 \times 5 = 35 \)
Alternatively, if there are 25 girls, then the number of boys is \(60 - 25 = 35\).
Adjust the number of boys:
5 boys leave the class.
New number of boys \( = 35 - 5 = 30 \)
The number of girls remains the same, which is 25.
Calculate the new ratio of boys to girls:
New number of boys = 30
Number of girls = 25
The ratio of boys to girls \( = 30 : 25 \)
To simplify the ratio, divide both numbers by their greatest common divisor, which is 5.
New ratio \[= \frac{30}{5} : \frac{25}{5} = 6 : 5\]
Correct Option: A) 6 : 5 is the new ratio of boys to girls after 5 boys leave the class.
To find the area of the rectangle, we first need to determine its length and breadth. We are given two pieces of information: the relationship between the length and breadth, and the perimeter of the rectangle. We can set up equations based on these facts and solve for the dimensions, then calculate the area.
B) 24 cm² is the correct area calculated from the derived length and breadth.
The area of a triangle can be calculated using its inradius and perimeter. The formula relating these quantities is given by \( \text{Area} = r \times s \), where \( r \) is the inradius and \( s \) is the semi-perimeter of the triangle. The semi-perimeter is half of the perimeter.
B) 256 cm² is the correct area of the triangle calculated using the given inradius and perimeter.
The question asks to identify the personality known as the 'Missile Man of India'. This is a direct factual recall question related to prominent Indian scientists and their contributions.
Correct Option: B) Dr. APJ Abdul Kalam is famously known as the 'Missile Man of India' for his extensive work on the development of ballistic missile and launch vehicle technology. He played a crucial role in the Pokhran-II nuclear tests in 1998 and served as the 11th President of India.
To find the square root of 4096, we can use the prime factorization method or estimation by checking the last digit and squaring the given options.
B) 64 is the correct answer because \(64 \times 64 = 4096\).
Read the given statements and conclusions carefully.
Statements:
I. No woman is a cow.
II. Elif is a woman.
Conclusions:
(i) Elif is not a cow.
(ii) Elif may or may not be a cow.
Only conclusion (i) follows.
Either conclusion (i) or (ii) follows.
Both conclusions (i) and (ii) follow.
Neither conclusions (i) nor (ii) follow.
This is a syllogism problem where we need to deduce conclusions based on given statements. We will use logical reasoning to determine which conclusion(s) necessarily follow from the statements.
A) Only conclusion (i) follows. Conclusion (i) is a direct and necessary deduction from the given statements. The statements establish a clear exclusion: Elif is a woman, and no woman is a cow. Therefore, Elif cannot be a cow.