This problem involves calculating the individual weight of Akshay using the concept of averages. We are given the average weights of three people, then two pairs of people. By setting up equations for the sum of weights based on the given averages, we can solve for the unknown individual weight.
D) The weight of Akshay is 45 kg.
To find the smallest number that should be added to 74978 so that the sum is completely divisible by 24, we first need to find the remainder when 74978 is divided by 24. If the remainder is \(R\), then the number to be added will be \(24 - R\).
For the sum to be completely divisible by 24, the remainder must be 0. Since the current remainder is 2, we need to add a number to 74978 such that the new remainder is 0. This number will be the difference between the divisor (24) and the remainder (2).
\[ \text{Number to be added} = \text{Divisor} - \text{Remainder} \] \[ \text{Number to be added} = 24 - 2 \] \[ \text{Number to be added} = 22 \]If we add 22 to 74978, we get:
\[ 74978 + 22 = 75000 \]Now, let's check if 75000 is divisible by 24:
\[ 75000 \div 24 = 3125 \]Since 75000 is completely divisible by 24, the smallest number to be added is 22.
A) 22 is the correct answer because when 22 is added to 74978, the sum 75000 is perfectly divisible by 24.
The value of 8² + \(\sqrt{9^2}\) − 4\(\sqrt{25}\) − 14 is
To solve this problem, we need to evaluate the given expression by performing the operations in the correct order, following the rules of arithmetic (PEMDAS/BODMAS). This involves calculating squares, square roots, multiplications, additions, and subtractions.
Let's break down the calculation:
The given expression is: \(8^2 + \sqrt{9^2} - 4\sqrt{25} - 14\)
Calculate \(8^2\):
\[ 8^2 = 8 \times 8 = 64 \]Calculate \(\sqrt{9^2}\):
\[ \sqrt{9^2} = \sqrt{81} = 9 \]Calculate \(4\sqrt{25}\):
\[ 4\sqrt{25} = 4 \times 5 = 20 \]Substitute these values back into the expression:
\[ 64 + 9 - 20 - 14 \]Perform the additions and subtractions from left to right:
\[ 64 + 9 = 73 \]\[ 73 - 20 = 53 \]\[ 53 - 14 = 39 \]A) The final calculated value of the expression is 39.
To find the original number, we can set up an algebraic equation. If a number is increased by 80%, it means the new number is 100% (original) + 80% (increase) = 180% of the original number. We are given that this increased value is 2520.
Let the unknown number be \(x\).
When the number is increased by 80%, it means we add 80% of \(x\) to \(x\). Mathematically, this can be written as:
\[ x + 80\% \text{ of } x \]Or, as a decimal:
\[ x + 0.80x \]Combine the terms:
\[ x + 0.80x = 1.80x \]We are given that this increased value is 2520. So, we can set up the equation:
\[ 1.80x = 2520 \]To find \(x\), divide both sides by 1.80:
\[ x = \frac{2520}{1.80} \]To simplify the division, we can multiply the numerator and denominator by 100 to remove the decimal:
\[ x = \frac{2520 \times 100}{1.80 \times 100} = \frac{252000}{180} \]Now, perform the division:
\[ x = \frac{25200}{18} \] \[ x = 1400 \]Thus, the original number is 1400.
B) 1400
A) 4200 — This would be the result if 2520 was 60% of the number, or if there was an error in calculation (e.g., multiplying instead of dividing, or incorrect percentage interpretation).
C) 2800 — This would be the result if 2520 was 90% of the number, or if 2520 was increased by 80% (which would be \(2520 \times 1.8 = 4536\)), which is not what the question asks.
D) 700 — This would be the result if 2520 was 360% of the number, or if 2520 was decreased by 80% (which would be \(2520 \times 0.2 = 504\)), or if 2520 was divided by 3.6, which is incorrect.
To find the Least Common Multiple (LCM) of given numbers expressed in their prime factorized form, we first need to express all numbers in terms of their prime factors. Then, for each prime factor, we take the highest power present in any of the numbers. The product of these highest powers will be the LCM.
C) \(2^8 \times 3^5 \times 5^2 \times 19^2\) is the correct LCM, as derived from taking the highest power of each prime factor present in the given numbers.
To solve this problem, we need to use the fundamental properties of regular polygons, specifically the relationships between interior angles, exterior angles, and the number of sides. We know that the sum of an interior angle and its corresponding exterior angle is 180°, and the sum of all exterior angles of any polygon is 360°.
Let \(I\) be the measure of one interior angle and \(E\) be the measure of one exterior angle of the regular polygon. For any polygon, an interior angle and its corresponding exterior angle are supplementary.
\[ I + E = 180^\circ \quad \text{(Equation 1)} \]The problem states that the interior angle is greater than its exterior angle by 144°.
\[ I = E + 144^\circ \quad \text{(Equation 2)} \]Substitute Equation 2 into Equation 1:
\[ (E + 144^\circ) + E = 180^\circ \] \[ 2E + 144^\circ = 180^\circ \]Solve for \(E\):
\[ 2E = 180^\circ - 144^\circ \] \[ 2E = 36^\circ \] \[ E = \frac{36^\circ}{2} \] \[ E = 18^\circ \]For a regular polygon with \(n\) sides, the measure of each exterior angle is given by the formula:
\[ E = \frac{360^\circ}{n} \]Substitute the value of \(E\) we found into this formula:
\[ 18^\circ = \frac{360^\circ}{n} \]Solve for \(n\):
\[ n = \frac{360^\circ}{18^\circ} \] \[ n = 20 \]Thus, the polygon has 20 sides.
D) 20
Pipe P can fill \( \frac{5}{7} \) part of a tank in 30 hours, and pipe Q can fill \( \frac{4}{7} \) part of the same tank in 12 hours. Both P and Q were kept open for 7 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 9 hours. Pipes P, Q and R together can fill the empty tank in:
This problem involves calculating the work rates of pipes P, Q, and R, and then combining them to find the time taken to fill the tank. We will first find the individual rates of P and Q, then the amount of tank filled by P and Q together. After that, we'll determine the amount of water emptied by R and its rate. Finally, we'll calculate the combined rate of P, Q, and R to find the time to fill the tank.
D) 63 hours is the correct answer, as calculated above.
This problem involves calculating individual investments based on their profit shares. In a business, if the time period of investment is the same for all partners, the profit is distributed in the ratio of their investments. We will first find the ratio of profits, then use this ratio to determine the individual investments, and finally calculate the difference between them.
Calculate Anshuman's share of profit:
The total profit is ₹14,000, and Kunal's share is ₹6,000.
\[ \text{Anshuman's Profit Share} = \text{Total Profit} - \text{Kunal's Profit Share} \\ \text{Anshuman's Profit Share} = \text{₹14,000} - \text{₹6,000} \\ \text{Anshuman's Profit Share} = \text{₹8,000} \]Determine the ratio of their profit shares:
The ratio of Anshuman's profit to Kunal's profit is:
\[ \text{Anshuman's Profit} : \text{Kunal's Profit} = \text{₹8,000} : \text{₹6,000} \]\[ \text{Ratio} = 8 : 6 \]\[ \text{Ratio} = 4 : 3 \]Relate profit ratio to investment ratio:
Since the investment period is the same for both, the ratio of their investments will be the same as the ratio of their profits.
So, \(\text{Anshuman's Investment} : \text{Kunal's Investment} = 4 : 3\).
Calculate individual investments:
The total investment is ₹48,300. Let Anshuman's investment be \(4x\) and Kunal's investment be \(3x\).
\[ 4x + 3x = \text{₹48,300} \]\[ 7x = \text{₹48,300} \]\[ x = \frac{\text{₹48,300}}{7} \]\[ x = \text{₹6,900} \]Anshuman's Investment \(= 4x = 4 \times \text{₹6,900} = \text{₹27,600}\)
Kunal's Investment \(= 3x = 3 \times \text{₹6,900} = \text{₹20,700}\)
Calculate the difference between their investments:
\[ \text{Difference} = \text{Anshuman's Investment} - \text{Kunal's Investment} \\ \text{Difference} = \text{₹27,600} - \text{₹20,700} \\ \text{Difference} = \text{₹6,900} \]D) ₹6,900 is the difference between Anshuman's and Kunal's investments, as calculated in the steps above.
To simplify the given algebraic expression, we will use the distributive property to expand the products, then combine like terms (terms with the same variable and exponent) to arrive at the simplest form.
Correct Option: B) \(15x^2 - 7x - 21\)
To find the average speed for the entire journey, we need to calculate the total distance traveled and the total time taken. The average speed is then the total distance divided by the total time. The journey is divided into three segments, each with a different distance and speed. We will calculate the time taken for each segment and sum them up.
Distance (\(D_1\)) = 152 km
Speed (\(S_1\)) = 38 km/hr
Time (\(T_1\)) = \(\frac{D_1}{S_1}\)
\[ T_1 = \frac{152 \text{ km}}{38 \text{ km/hr}} = 4 \text{ hours} \]Distance (\(D_2\)) = 189 km
Speed (\(S_2\)) = 63 km/hr
Time (\(T_2\)) = \(\frac{D_2}{S_2}\)
\[ T_2 = \frac{189 \text{ km}}{63 \text{ km/hr}} = 3 \text{ hours} \]Distance (\(D_3\)) = 375 km
Speed (\(S_3\)) = 75 km/hr
Time (\(T_3\)) = \(\frac{D_3}{S_3}\)
\[ T_3 = \frac{375 \text{ km}}{75 \text{ km/hr}} = 5 \text{ hours} \]Total Distance (\(D_{\text{total}}\)) = \(D_1 + D_2 + D_3\)
\[ D_{\text{total}} = 152 \text{ km} + 189 \text{ km} + 375 \text{ km} = 716 \text{ km} \]Total Time (\(T_{\text{total}}\)) = \(T_1 + T_2 + T_3\)
\[ T_{\text{total}} = 4 \text{ hours} + 3 \text{ hours} + 5 \text{ hours} = 12 \text{ hours} \]Average Speed = \(\frac{D_{\text{total}}}{T_{\text{total}}}\)
\[ \text{Average Speed} = \frac{716 \text{ km}}{12 \text{ hours}} \approx 59.6666 \text{ km/hr} \]Average Speed \(\approx 59.67\) km/hr
Correct Option: A) 59.67