To find the number of prime numbers between 476 and 491 (inclusive), we need to list all integers in this range and then test each integer for primality. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To test for primality, we check for divisibility by prime numbers up to the square root of the number.
Correct Option: D) 3
To solve this problem, we first need to calculate the total distance of the journey using the given speed and time. Once the distance is known, we can determine the new speed required to cover the same distance in a shorter time.
Calculate the total distance:
The formula relating distance, speed, and time is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]Given:
So, the distance (\(D\)) is:
\[ D = 75 \text{ km/hr} \times 17 \text{ hours} = 1275 \text{ km} \]Calculate the new speed:
We need to cover the same distance (\(D = 1275\) km) in a new time (\(T_2\)) of 5 hours. The formula for speed is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]Given:
So, the new speed (\(S_2\)) should be:
\[ S_2 = \frac{1275 \text{ km}}{5 \text{ hours}} = 255 \text{ km/hr} \]Correct Option: A) 255 km/hr
Find the value of 48-17 × 48-12 × 48-13.
The problem involves simplifying an expression with exponents. We will use the property of exponents that states: when multiplying terms with the same base, we add their exponents. That is, \(a^m \times a^n = a^{m+n}\).
Correct Option: D) \(48^{-42}\)
When partners invest different amounts in a business, the profit is distributed among them in the ratio of their investments. First, calculate the ratio of the investments made by Sneha, Yash, and Rohan. Then, use this ratio to determine Rohan's share of the total profit.
The ratio of their investments is:
\[ \text{Sneha} : \text{Yash} : \text{Rohan} = 1090 : 1170 : 1340 \]All numbers are divisible by 10. So, we can simplify the ratio by dividing each number by 10:
\[ 109 : 117 : 134 \]Sum of the ratio terms \( = 109 + 117 + 134 \)
\[ = 360 \]Rohan's share of the profit is his ratio term divided by the sum of the ratio terms, multiplied by the total profit.
Total profit = ₹1,620
Rohan's share \[= \frac{\text{Rohan's ratio term}}{\text{Sum of ratio terms}} \times \text{Total profit}\]
\[ = \frac{134}{360} \times 1620 \]We can simplify the fraction before multiplying:
\[ \frac{134}{360} \times 1620 = 134 \times \frac{1620}{360} \]\[ = 134 \times \frac{162}{36} \]\[ = 134 \times \frac{9 \times 18}{2 \times 18} \]\[ = 134 \times \frac{9}{2} \]\[ = 67 \times 9 \]\[ = 603 \]So, Rohan's share of the profit is ₹603.
A) ₹603 is the correct share of the profit for Rohan, calculated based on his investment proportion.
To determine which number divides 587407, we will perform division with each of the given options. A number is divisible by another if the remainder of their division is zero.
Divide 587407 by 34.
\[ 587407 \div 34 \]Performing the division:
\[ 587407 = 34 \times 17276 + 23 \]Since the remainder is 23 (not 0), 587407 is not divisible by 34.
Divide 587407 by 43.
\[ 587407 \div 43 \]Performing the division:
\[ 587407 = 43 \times 13659 + 10 \]Since the remainder is 10 (not 0), 587407 is not divisible by 43.
Divide 587407 by 41.
\[ 587407 \div 41 \]Performing the division:
\[ 587407 = 41 \times 14327 \]Since the remainder is 0, 587407 is divisible by 41.
Divide 587407 by 47.
\[ 587407 \div 47 \]Performing the division:
\[ 587407 = 47 \times 12498 + 1 \]Since the remainder is 1 (not 0), 587407 is not divisible by 47.
Correct Option: C) 41
We need to find the marked price given the difference between two selling prices calculated using different discount schemes. We will calculate the effective discount for each scenario and then set up an equation based on the given difference to find the marked price.
To find the volume of a cylinder, we use the formula \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. We are given the radius, height, and the value of \(\pi\).
Identify the given values:
Recall the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
Substitute the given values into the formula:
\[ V = (3.14) \times (3 \text{ cm})^2 \times (7 \text{ cm}) \]
Calculate the square of the radius:
\[ r^2 = 3^2 = 9 \text{ cm}^2 \]
Perform the multiplication:
\[ V = 3.14 \times 9 \times 7 \]
\[ V = 3.14 \times 63 \]
Calculate the final volume:
\[ V = 197.82 \text{ cm}^3 \]
Thus, the volume of the cylinder is 197.82 cm³.
D) 197.82 cm 3 is the correct answer, as calculated by substituting the given radius and height into the volume formula for a cylinder.
A) 193.58 cm 3 — This value might result from an arithmetic error during multiplication or using a slightly different value for \(\pi\) (e.g., if \(\pi\) was approximated as 22/7 and then rounded incorrectly, or if there was a calculation mistake).
B) 190.01 cm 3 — This value is significantly different and likely results from a major calculation error or incorrect formula application.
C) 195.68 cm 3 — This value is close but incorrect. It could be due to a minor rounding error or a small mistake in multiplication (e.g., \(3.14 \times 62.3\), or similar).
This problem involves calculating the selling price of articles to achieve a desired profit percentage, given an initial loss percentage. We will first determine the cost price (CP) of one article using the initial selling price (SP) and loss percentage. Then, we will calculate the new selling price required to achieve the target profit percentage, and finally, find out how many articles must be sold for a rupee at this new selling price.
Calculate the Selling Price (SP) of one article:
A man sells 6 articles for a rupee. So, the selling price of 1 article is:
\[ \text{SP of 1 article} = \frac{1}{6} \text{ Rupee} \]Calculate the Cost Price (CP) of one article:
He loses 30% by selling at this price. This means the SP is 70% of the CP.
\[ \text{SP} = \text{CP} \times \left(1 - \frac{\text{Loss percentage}}{100}\right) \]So,
\[ \frac{1}{6} = \text{CP} \times \left(1 - \frac{30}{100}\right) \]\[ \frac{1}{6} = \text{CP} \times \frac{70}{100} \]\[ \frac{1}{6} = \text{CP} \times \frac{7}{10} \]Now, solve for CP:
\[ \text{CP} = \frac{1}{6} \times \frac{10}{7} \]\[ \text{CP} = \frac{10}{42} = \frac{5}{21} \text{ Rupee} \]Calculate the new Selling Price (New SP) for a 40% gain:
To gain 40%, the new SP should be 140% of the CP.
\[ \text{New SP} = \text{CP} \times \left(1 + \frac{\text{Gain percentage}}{100}\right) \]\[ \text{New SP} = \frac{5}{21} \times \left(1 + \frac{40}{100}\right) \]\[ \text{New SP} = \frac{5}{21} \times \frac{140}{100} \]\[ \text{New SP} = \frac{5}{21} \times \frac{14}{10} \]\[ \text{New SP} = \frac{1}{21} \times \frac{14}{2} \]\[ \text{New SP} = \frac{1}{21} \times 7 \]\[ \text{New SP} = \frac{7}{21} = \frac{1}{3} \text{ Rupee} \]Determine how many articles must be sold for a rupee:
The new selling price of 1 article is \(\frac{1}{3}\) Rupee. This means for 1 Rupee, the man must sell 3 articles.
If 1 article costs \(\frac{1}{3}\) Rupee, then for 1 Rupee, the number of articles sold is:
\[ \text{Number of articles} = \frac{1 \text{ Rupee}}{\text{New SP of 1 article}} \]\[ \text{Number of articles} = \frac{1}{\frac{1}{3}} = 1 \times 3 = 3 \]Therefore, to gain 40%, he must sell 3 articles for a rupee.
D) 3 — As calculated, to achieve a 40% profit, the selling price of one article must be \(\frac{1}{3}\) Rupee, which means 3 articles must be sold for 1 Rupee.
The mode of a data set is the value that appears most frequently in the set. To find the mode, we need to count the occurrences of each distinct value in the given data.
Correct Option: C) 11 — The value 11 appears 3 times, which is the highest frequency among all the given marks. Therefore, 11 is the mode.
This problem involves calculating the original sales amount given successive percentage increases over two years and the final sales amount. We can work backward from the current sales or set up an equation to find the initial sales.
Correct Option: D) ₹1,08,000