To find the total number of digits used in numbering the pages of a book, we need to count the digits for single-digit page numbers, double-digit page numbers, and triple-digit page numbers separately, and then sum them up.
Count digits for single-digit page numbers (1 to 9):
There are 9 such pages (1, 2, ..., 9). Each page uses 1 digit.
\[ \text{Total digits for 1-digit numbers} = 9 \times 1 = 9 \]Count digits for double-digit page numbers (10 to 99):
The number of pages from 10 to 99 is \(99 - 10 + 1 = 90\).
Each of these pages uses 2 digits.
\[ \text{Total digits for 2-digit numbers} = 90 \times 2 = 180 \]Count digits for triple-digit page numbers (100 to 280):
The book has 280 pages. So, the triple-digit page numbers range from 100 to 280.
The number of pages from 100 to 280 is \(280 - 100 + 1 = 181\).
Each of these pages uses 3 digits.
\[ \text{Total digits for 3-digit numbers} = 181 \times 3 = 543 \]Calculate the total number of digits:
Sum the digits from all categories:
\[ \text{Total digits} = \text{Digits for 1-digit numbers} + \text{Digits for 2-digit numbers} + \text{Digits for 3-digit numbers} \] \[ \text{Total digits} = 9 + 180 + 543 = 732 \]Correct Option: A) 732
The total number of digits used in numbering the pages of a book having 280 pages is 732.
To find the average weight of all the boys, we need to calculate the total weight of all boys and divide it by the total number of boys. The total weight can be found by summing the total weight of the first group of boys and the total weight of the second group of boys.
Number of boys in the first group = 16
Average weight of the first group = 50.25 kgs
Total weight of the first group = Number of boys \(\times\) Average weight
\[ \text{Total weight}_1 = 16 \times 50.25 \text{ kgs} = 804 \text{ kgs} \]Number of boys in the second group = 8
Average weight of the second group = 45.15 kgs
Total weight of the second group = Number of boys \(\times\) Average weight
\[ \text{Total weight}_2 = 8 \times 45.15 \text{ kgs} = 361.2 \text{ kgs} \]Total weight of all boys = Total weight of the first group + Total weight of the second group
\[ \text{Total weight}_{\text{all}} = 804 \text{ kgs} + 361.2 \text{ kgs} = 1165.2 \text{ kgs} \]Total number of boys = Number of boys in the first group + Number of boys in the second group
\[ \text{Total number of boys} = 16 + 8 = 24 \]Average weight of all boys = \[\frac{\text{Total weight of all boys}}{\text{Total number of boys}}\]
\[ \text{Average weight}_{\text{all}} = \frac{1165.2 \text{ kgs}}{24} = 48.55 \text{ kgs} \]C) 48.55 kgs. This is the calculated average weight of all the boys in the class.
To find the average of a set of numbers, we sum all the numbers and then divide the sum by the count of the numbers. This is a fundamental concept in statistics and arithmetic.
Thus, the average of the given numbers is 2.5.
C) 2.50 — The calculation shows that the sum of the numbers (10.0) divided by the count of the numbers (4) results in 2.5. This matches option C.
This problem involves setting up a linear equation based on the given word problem. We need to find an unknown number. We are told that if we subtract 40% of this number from the number itself, the result is 30. We can represent the unknown number with a variable, form an equation, and then solve for the variable.
B) 50. As calculated, the number is 50.
To find the profit percentage, we first need to determine the profit amount. The problem states a relationship between the selling price (SP) and the cost price (CP). We can express SP in terms of CP, then calculate the profit (SP - CP), and finally use the profit percentage formula: \[\text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\].
Understand the given relationship:
The selling price (SP) of an article is \( \frac{7}{6} \) of its cost price (CP).
So, we can write this as:
Calculate the Profit:
Profit is the difference between the selling price and the cost price.
Calculate the Profit Percentage:
The formula for profit percentage is:
Substitute the value of Profit:
\[ \text{Profit Percentage} = \left( \frac{\frac{1}{6} \text{CP}}{\text{CP}} \right) \times 100\% \]\[ \text{Profit Percentage} = \left( \frac{1}{6} \right) \times 100\% \]\[ \text{Profit Percentage} = \frac{100}{6}\% \]\[ \text{Profit Percentage} = \frac{50}{3}\% \]Convert the fraction to a mixed number:
\[ \frac{50}{3} = 16 \frac{2}{3} \]So, the profit percentage is \( 16 \frac{2}{3}\% \).Correct Option: A) 16\( \frac{2}{3} \)%
The problem involves coins of different denominations (25 p, 10 p, 5 p) given in a specific ratio. We are given the total value of these coins and need to find the number of coins of a particular denomination. The approach is to represent the number of coins using the given ratio, calculate their total value in terms of a common variable, equate it to the total given amount, and then solve for the variable to find the number of 5 p coins.
Define the number of coins based on the ratio:
The ratio of 25 p, 10 p, and 5 p coins is 1 : 2 : 3. Let the common multiplier be \(x\).
Calculate the total value of the coins in terms of \(x\):
Convert all coin values to paise (p) for consistency. The total amount is Rs 30, which is \(30 \times 100 = 3000\) paise.
Total value of all coins:
\[ \text{Total Value} = 25x + 20x + 15x \]\[ \text{Total Value} = 60x \text{ p} \]Equate the total value to the given amount:
We know the total value is 3000 p.
Solve for \(x\):
\[ x = \frac{3000}{60} \]\[ x = 50 \]Find the number of 5 p coins:
The number of 5 p coins is \(3x\).
C) 150 — As calculated in the step-by-step solution, there are 150 five-paise coins.
This problem involves the concept of work and time, where the total work done remains constant. We can use the formula \(M_1 D_1 H_1 = M_2 D_2 H_2\), where \(M\) is the number of men, \(D\) is the number of days, and \(H\) is the number of hours per day.
Identify the given values for the first scenario:
Identify the given values for the second scenario:
Apply the work formula:
The total work done is constant in both scenarios. So, we can equate the product of men, days, and hours for both cases:
\[ M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2 \]Substitute the known values into the formula:
\[ 10 \times 18 \times 6 = 15 \times 12 \times H_2 \]Calculate the product for the first scenario:
\[ 10 \times 18 \times 6 = 180 \times 6 = 1080 \] So, the equation becomes: \[ 1080 = 15 \times 12 \times H_2 \]Calculate the product of men and days for the second scenario:
\[ 15 \times 12 = 180 \] So, the equation becomes: \[ 1080 = 180 \times H_2 \]Solve for \(H_2\):
\[ H_2 = \frac{1080}{180} \] \[ H_2 = 6 \] Therefore, 15 men must work 6 hours a day to complete the same work in 12 days.A) 6 hrs
This problem involves calculating the combined work rate of two pipes to fill a tank. We will determine the individual filling rates of each pipe and then sum them to find the combined rate. The reciprocal of the combined rate will give us the total time taken to fill the tank.
A) 12 min is the correct answer. This is the time calculated for both pipes working together.
This problem involves relative speed and distance. First, calculate the distance the thief covers before the policeman starts chasing. Then, determine the relative speed at which the policeman closes the distance. Finally, calculate the time taken to overtake and the total distance from the starting point.
Calculate the distance covered by the thief before the chase begins:
The thief runs for 20 seconds before the policeman starts chasing.
\[ \text{Thief's speed} = 8 \text{ m/s} \ \text{Time thief ran alone} = 20 \text{ s} \ \text{Distance covered by thief} = \text{Speed} \times \text{Time} = 8 \text{ m/s} \times 20 \text{ s} = 160 \text{ m} \]Calculate the relative speed of the policeman with respect to the thief:
Since both are moving in the same direction, the relative speed is the difference between their speeds.
\[ \text{Policeman's speed} = 10 \text{ m/s} \ \text{Thief's speed} = 8 \text{ m/s} \ \text{Relative speed} = \text{Policeman's speed} - \text{Thief's speed} = 10 \text{ m/s} - 8 \text{ m/s} = 2 \text{ m/s} \]Calculate the time taken for the policeman to overtake the thief:
The policeman needs to cover the initial distance of 160 m that the thief gained.
\[ \text{Time to overtake} = \frac{\text{Distance to cover}}{\text{Relative speed}} = \frac{160 \text{ m}}{2 \text{ m/s}} = 80 \text{ s} \]Calculate the total distance from the policeman's starting point where the overtaking occurs:
This is the distance the policeman covers in the time it takes to overtake the thief.
\[ \text{Distance covered by policeman} = \text{Policeman's speed} \times \text{Time to overtake} \ \text{Distance} = 10 \text{ m/s} \times 80 \text{ s} = 800 \text{ m} \]Therefore, the policeman will overtake the thief 800 m from his starting point.
Correct Option: B) 800 mtr
A train moves with a speed of 108 kmph. Its speed in metres per second is:
To convert a speed from kilometers per hour (kmph) to meters per second (m/s), we need to convert kilometers to meters and hours to seconds. We know that 1 kilometer equals 1000 meters and 1 hour equals 3600 seconds. This conversion can be simplified by multiplying the speed in kmph by a factor of \(\frac{5}{18}\).
To convert kmph to m/s, we multiply the speed by the ratio of meters per kilometer and seconds per hour:
\[ \text{Speed in m/s} = \text{Speed in kmph} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3600 \text{ s}} \]Now, multiply the given speed (108 kmph) by the simplified conversion factor \(\frac{5}{18}\):
\[ \text{Speed} = 108 \times \frac{5}{18} \text{ m/s} \]Correct Option: C) 30
The speed of the train in meters per second is 30 m/s.