What should be added to \( \frac{10}{11} \) so that the sum becomes \( \frac{11}{10} \)?
\( \frac{1}{55} \)
\( \frac{2}{11} \)
\( \frac{1}{110} \)
\( \frac{21}{110} \)
To find the number that should be added to a given fraction to reach a target sum, we can set up an algebraic equation. Let the unknown number be \(x\). Then, we can write the equation: (given fraction) + \(x\) = (target sum). We then solve for \(x\) by subtracting the given fraction from the target sum.
Let the number to be added be \(x\). According to the problem statement:
\[ \frac{10}{11} + x = \frac{11}{10} \]To find \(x\), subtract \( \frac{10}{11} \) from both sides of the equation:
\[ x = \frac{11}{10} - \frac{10}{11} \]The least common multiple (LCM) of 10 and 11 is \(10 \times 11 = 110\). Convert both fractions to have this common denominator:
\[ \frac{11}{10} = \frac{11 \times 11}{10 \times 11} = \frac{121}{110} \] \[ \frac{10}{11} = \frac{10 \times 10}{11 \times 10} = \frac{100}{110} \]Now substitute these equivalent fractions back into the equation for \(x\):
\[ x = \frac{121}{110} - \frac{100}{110} \] \[ x = \frac{121 - 100}{110} \] \[ x = \frac{21}{110} \]Correct Option: D) \( \frac{21}{110} \)
To find the total time taken for the journey, we need to calculate the time taken for each half of the journey separately and then add them up. The formula for time is given by \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
Calculate the total distance:
The total distance covered by the train is 250 km.
Calculate the distance for the first half:
Half of the total distance is \( \frac{250 \text{ km}}{2} = 125 \text{ km} \).
Calculate the time taken for the first half:
The speed for the first half is 50 km/h.
\[ \text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{125 \text{ km}}{50 \text{ km/h}} = 2.5 \text{ hours} \]Calculate the distance for the second half:
The remaining half of the distance is also 125 km.
Calculate the time taken for the second half:
The speed for the second half is 75 km/h.
\[ \text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{125 \text{ km}}{75 \text{ km/h}} \]\[ \text{Time}_2 = \frac{5}{3} \text{ hours} \approx 1.666... \text{ hours} \]Calculate the total time:
Add the time taken for both halves:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = 2.5 \text{ hours} + \frac{5}{3} \text{ hours} \]To add these, convert 2.5 hours to a fraction:
\[ 2.5 = \frac{5}{2} \]\[ \text{Total Time} = \frac{5}{2} + \frac{5}{3} \]Find a common denominator (6):
\[ \text{Total Time} = \frac{5 \times 3}{2 \times 3} + \frac{5 \times 2}{3 \times 2} = \frac{15}{6} + \frac{10}{6} = \frac{25}{6} \text{ hours} \]Convert total time to hours and minutes:
Divide 25 by 6:
\[ \frac{25}{6} = 4 \text{ with a remainder of } 1 \]So, it is 4 hours and \( \frac{1}{6} \) of an hour.
Convert the fraction of an hour to minutes:
\[ \frac{1}{6} \text{ hours} \times 60 \text{ minutes/hour} = 10 \text{ minutes} \]Therefore, the total time taken is 4 hours and 10 minutes.
Correct Option: D) 4 hours 10 minutes
This is a blood relation question. We need to establish the relationships between the individuals mentioned in the problem statement to determine how the daughter-in-law would refer to the man's wife.
A) Mother-in-law — As established in the step-by-step analysis, the man's wife is the mother of the daughter-in-law's husband, making her the daughter-in-law's mother-in-law.
The question asks to identify the term for the region of high particle density during sound propagation. This relates to the fundamental nature of sound waves as longitudinal waves.
Correct Option: D) Compression is the region in a sound wave where the particles of the medium are crowded together, resulting in high density and high pressure.
To calculate the total time duration, we need to consider the number of full days and the difference in hours and minutes between the start and end times. A crucial step is to identify if the period includes a leap year, as this adds an extra day.
C) 367 days 23 hours 51 minutes
The question asks to identify a mineral acid that is NOT produced in the human stomach. We need to recall the primary acid produced in the stomach and distinguish between mineral acids and organic acids, then evaluate each option.
D) Sulphuric acid — Sulphuric acid (H\(_2\)SO\(_4\)) is a strong mineral acid. It is a highly corrosive industrial chemical and is not naturally produced in the human body, including the stomach. The stomach produces hydrochloric acid.
The problem presents a sequence consisting of an alphabet and a number. We need to identify the pattern for both the alphabetical and numerical parts separately to determine the next term in the sequence.
The given alphabetical sequence is E, I, M.
The pattern for the alphabetical part is an increment of 4 letters each time.
Applying this pattern to M:
The next letter in the sequence is Q.
The given numerical sequence is 2, 12, 72.
The pattern for the numerical part is multiplication by 6 each time.
Applying this pattern to 72:
The next number in the sequence is 432.
Combining the next letter (Q) and the next number (432), the next term in the sequence is Q432.
A) Q432 is the correct option because it follows both the alphabetical pattern (adding 4 to the letter) and the numerical pattern (multiplying by 6).
To find the Greatest Common Divisor (GCD) of three numbers, we can use the prime factorization method. This involves finding the prime factors of each number and then multiplying the common prime factors raised to their lowest powers.
Divide 306 by the smallest prime numbers:
\[ 306 \div 2 = 153 \n153 \div 3 = 51 \n51 \div 3 = 17 \n17 \div 17 = 1 \]So, the prime factorization of 306 is \(2 \times 3^2 \times 17\).
Divide 204 by the smallest prime numbers:
\[ 204 \div 2 = 102 \n102 \div 2 = 51 \n51 \div 3 = 17 \n17 \div 17 = 1 \]So, the prime factorization of 204 is \(2^2 \times 3 \times 17\).
Divide 136 by the smallest prime numbers:
\[ 136 \div 2 = 68 \n68 \div 2 = 34 \n34 \div 2 = 17 \n17 \div 17 = 1 \]So, the prime factorization of 136 is \(2^3 \times 17\).
The prime factors for each number are:
The common prime factors are 2 and 17.
Therefore, the GCD is \(2^1 \times 17^1 = 2 \times 17 = 34\).
Correct Option: B) 34
The question asks for the year Qutub-ud-din Aibak became the Sultan of Delhi. This is a direct historical fact related to the establishment of the Delhi Sultanate and the Mamluk dynasty.
Correct Option: C) 1206
Qutub-ud-din Aibak became the Sultan of Delhi in 1206 CE, marking the beginning of the Delhi Sultanate and the Mamluk dynasty.
The question asks to identify the base for an altitude drawn from vertex A in a right-angled triangle ABC, where the right angle is at A. An altitude from a vertex is a perpendicular line segment from that vertex to the opposite side. The side opposite to the vertex from which the altitude is drawn is considered the base.
D) BC
In any triangle, the altitude from a specific vertex is drawn to the side opposite that vertex. For vertex A, the opposite side is BC. When \(\angle A\) is a right angle, the altitude from A to BC is a segment within the triangle (or one of its legs if it's an obtuse angle, but here it's a right angle). The side BC is the hypotenuse in this right-angled triangle.