This problem involves setting up and solving a system of two linear equations based on the given information about the ages of P and Q. We will define variables for their present ages, translate the given statements into algebraic equations, and then solve these equations simultaneously to find the present age of Q.
Define variables: Let the present age of P be \(P\) years. Let the present age of Q be \(Q\) years.
Formulate the first equation: The first statement says, "Six times the present age of P exceeds the present age of Q by 11 years." This can be written as:
\[ 6P = Q + 11 \quad \text{(Equation 1)} \]Formulate the second equation: The second statement talks about ages after 10 years. Age of P after 10 years will be \(P + 10\) years. Age of Q after 10 years will be \(Q + 10\) years.
The statement is, "After 10 years, three times Q's age will be 27 years less than six times P's age at that time." This translates to:
\[ 3(Q + 10) = 6(P + 10) - 27 \]Simplify the second equation:
\[ 3Q + 30 = 6P + 60 - 27 \] \[ 3Q + 30 = 6P + 33 \] \[ 3Q = 6P + 3 \] \[ 3Q - 6P = 3 \quad \text{(Equation 2)} \]Solve the system of equations: We have two equations:
Equation 1: \(6P = Q + 11 \implies 6P - Q = 11\)
Equation 2: \(3Q - 6P = 3\)
We can rearrange Equation 1 to express \(6P\):
\[ 6P = Q + 11 \]Substitute this expression for \(6P\) into Equation 2:
\[ 3Q - (Q + 11) = 3 \] \[ 3Q - Q - 11 = 3 \] \[ 2Q - 11 = 3 \] \[ 2Q = 3 + 11 \] \[ 2Q = 14 \] \[ Q = \frac{14}{2} \] \[ Q = 7 \]Verify the answer (optional but good practice): If \(Q = 7\), from Equation 1:
\[ 6P = 7 + 11 \] \[ 6P = 18 \] \[ P = 3 \]So, P's present age is 3 years and Q's present age is 7 years.
Check with the second condition: After 10 years, P's age will be \(3 + 10 = 13\). After 10 years, Q's age will be \(7 + 10 = 17\).
Is \(3 \times 17 = 6 \times 13 - 27\)?
\[ 51 = 78 - 27 \] \[ 51 = 51 \]The conditions are satisfied. Thus, the present age of Q is 7 years.
Correct Option: C) 7
To find the ratio of the volumes of the original and inflated balloons, we need to use the formula for the volume of a sphere. We will calculate the volume for both states of the balloon and then find their ratio.
B) 1 : 8 is the correct ratio of the volume of the original balloon to the volume of the inflated balloon.
A can lay railway track between two given stations in 18 days and B can do the same job in 10 days. With the help of C, they did the job in 4 days only. Then, C alone can do the job in
11 \( \frac{10}{17} \) days
14 \( \frac{10}{17} \) days
10 \( \frac{10}{17} \) days
19 \( \frac{10}{17} \) days
This is a work and time problem. We need to calculate the individual work rates of A, B, and the combined work rate of A, B, and C. Then, by subtracting the combined work rate of A and B from the combined work rate of A, B, and C, we can find C's individual work rate. Finally, we convert C's work rate back into the number of days C would take to complete the job alone.
A can do the job in 18 days. So, A's one day's work is:
\[ \text{A's one day's work} = \frac{1}{18} \text{ of the job} \]B can do the job in 10 days. So, B's one day's work is:
\[ \text{B's one day's work} = \frac{1}{10} \text{ of the job} \]With the help of C, they did the job in 4 days. So, (A + B + C)'s one day's work is:
\[ \text{(A + B + C)'s one day's work} = \frac{1}{4} \text{ of the job} \]C's one day's work can be found by subtracting the combined one day's work of A and B from the combined one day's work of A, B, and C.
\[ \text{C's one day's work} = \text{(A + B + C)'s one day's work} - (\text{A's one day's work} + \text{B's one day's work}) \]\[ \text{C's one day's work} = \frac{1}{4} - \left( \frac{1}{18} + \frac{1}{10} \right) \]\[ \text{C's one day's work} = \frac{1}{4} - \left( \frac{5 + 9}{90} \right) \]\[ \text{C's one day's work} = \frac{1}{4} - \frac{14}{90} \]\[ \text{C's one day's work} = \frac{1}{4} - \frac{7}{45} \]To subtract these fractions, find a common denominator, which is \(4 \times 45 = 180\).\[ \text{C's one day's work} = \frac{45}{180} - \frac{28}{180} \]\[ \text{C's one day's work} = \frac{45 - 28}{180} \]\[ \text{C's one day's work} = \frac{17}{180} \text{ of the job} \]If C does \(\frac{17}{180}\) of the job in one day, then C can do the entire job in the reciprocal of this fraction.
\[ \text{Days for C alone} = \frac{180}{17} \text{ days} \]To convert this improper fraction to a mixed fraction:\[ \frac{180}{17} = 10 \frac{10}{17} \text{ days} \]C) 10 \( \frac{10}{17} \) days. This matches our calculated value.
This problem involves calculating distance and time based on given speeds. First, we need to find the total distance between Delhi and Kanpur using Ritesh's speed and time. Once the distance is known, we can calculate the time taken by Gauri using her speed and the same distance.
We know the formula: \(\text{Distance} = \text{Speed} \times \text{Time}\).
For Ritesh:
So, the distance is:
\[ \text{Distance} = 66 \text{ km/hr} \times 9 \text{ hours} = 594 \text{ km} \]
Now we know the total distance is \(594 \text{ km}\).
For Gauri:
Using the formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
\[ \text{Time taken by Gauri} = \frac{594 \text{ km}}{99 \text{ km/hr}} \]
To simplify the division:
\[ \frac{594}{99} = \frac{6 \times 99}{99} = 6 \text{ hours} \]
Therefore, Gauri takes 6 hours to reach Kanpur.
Correct Option: C) 6 hours
The problem asks us to find the greater of two consecutive positive natural numbers whose product is 182. We can solve this by setting up an algebraic equation or by testing the options provided.
B) 14 — As calculated, the two consecutive positive natural numbers are 13 and 14, and the greater of these is 14.
To determine which number divides 630605, we will use divisibility rules for each of the given options: 5, 7, 14, and 3.
The last digit of 630605 is 5. According to the divisibility rule for 5, since the last digit is 5, the number 630605 is divisible by 5.
Number: 630605
Step 1: Remove the last digit (5) and subtract twice its value from the remaining number (63060).
\[ 63060 - (2 \times 5) = 63060 - 10 = 63050 \]Step 2: Repeat the process for 63050.
\[ 6305 - (2 \times 0) = 6305 - 0 = 6305 \]Step 3: Repeat the process for 6305.
\[ 630 - (2 \times 5) = 630 - 10 = 620 \]Step 4: Repeat the process for 620.
\[ 62 - (2 \times 0) = 62 - 0 = 62 \]Since 62 is not divisible by 7, the number 630605 is not divisible by 7.
For a number to be divisible by 14, it must be divisible by both 2 and 7.
The last digit of 630605 is 5, which is an odd number. Therefore, 630605 is not divisible by 2. Since it's not divisible by 2, it cannot be divisible by 14.
(Also, we already found it's not divisible by 7).
Sum of the digits of 630605:
\[ 6 + 3 + 0 + 6 + 0 + 5 = 20 \]Since 20 is not divisible by 3, the number 630605 is not divisible by 3.
A) 5. The number 630605 ends with the digit 5, which makes it divisible by 5.
To find the surface area of a sphere, we need its radius. The problem provides the diameter, so we will first calculate the radius from the given diameter. Then, we will use the formula for the surface area of a sphere.
Thus, the surface area of the sphere is \(484 \pi \text{ cm}^2\).
Correct Option: A) \(484\pi \text{ cm}^2\)
This problem involves relative speed concepts for trains. First, we need to find the speed of the train using the information about it overtaking a man. Then, we will use this train's speed to calculate the time it takes to cross another train moving in the opposite direction.
Man's speed: \(9 \text{ km/hr}\)
\[ 9 \text{ km/hr} = 9 \times \frac{5}{18} \text{ m/s} = \frac{5}{2} \text{ m/s} = 2.5 \text{ m/s} \]Let the speed of the first train be \(S_T \text{ m/s}\).
When the train overtakes a man moving in the same direction, the relative speed is the difference between their speeds.
Relative speed \( = (S_T - 2.5) \text{ m/s}\)
The distance covered is the length of the train, \(490 \text{ m}\).
Time taken \( = 12 \text{ seconds}\)
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]\[ 490 = (S_T - 2.5) \times 12 \]\[ S_T - 2.5 = \frac{490}{12} \]\[ S_T - 2.5 = \frac{245}{6} \approx 40.833 \text{ m/s} \]\[ S_T = \frac{245}{6} + 2.5 = \frac{245}{6} + \frac{15}{6} = \frac{260}{6} = \frac{130}{3} \text{ m/s} \]Length of the first train \(L_1 = 490 \text{ m}\)
Length of the second train \(L_2 = 500 \text{ m}\)
Speed of the first train \(S_1 = \frac{130}{3} \text{ m/s}\)
Speed of the second train \(S_2 = 42 \text{ km/hr}\)
Convert \(S_2\) to m/s:
\[ 42 \text{ km/hr} = 42 \times \frac{5}{18} \text{ m/s} = \frac{7 \times 5}{3} \text{ m/s} = \frac{35}{3} \text{ m/s} \]Since the trains are moving in opposite directions, their relative speed is the sum of their speeds.
\[ \text{Relative Speed} = S_1 + S_2 = \frac{130}{3} + \frac{35}{3} = \frac{165}{3} = 55 \text{ m/s} \]When two trains cross each other, the total distance covered is the sum of their lengths.
\[ \text{Total Distance} = L_1 + L_2 = 490 \text{ m} + 500 \text{ m} = 990 \text{ m} \]C) 18
We need to calculate the price of the computer after the initial discount. Then, we will compare this price with the amount paid by the student to find the additional discount amount. Finally, we will express this additional discount as a percentage of the price after the initial discount.
Calculate the price after the first discount:
The marked price of the computer is ₹70,000.
The first discount offered is 16%.
\[ \text{Discount amount} = 16\% \text{ of } \text{₹}70,000 \]\[ \text{Discount amount} = \frac{16}{100} \times 70000 = 16 \times 700 = \text{₹}11,200 \]Price after first discount:
\[ \text{Price after first discount} = \text{Marked Price} - \text{Discount amount} \]\[ \text{Price after first discount} = 70000 - 11200 = \text{₹}58,800 \]Calculate the additional discount amount:
A student pays ₹57,036 for the computer.
The price after the first discount was ₹58,800.
Additional discount amount received by the student:
\[ \text{Additional discount amount} = \text{Price after first discount} - \text{Amount paid by student} \]\[ \text{Additional discount amount} = 58800 - 57036 = \text{₹}1,764 \]Calculate the additional discount percentage:
The additional discount is calculated on the price after the first discount (₹58,800).
\[ \text{Additional discount percentage} = \frac{\text{Additional discount amount}}{\text{Price after first discount}} \times 100\% \]\[ \text{Additional discount percentage} = \frac{1764}{58800} \times 100\% \]\[ \text{Additional discount percentage} = \frac{1764}{588} \% \]\[ \text{Additional discount percentage} = 3\% \]D) 3% is the correct additional discount percentage received by the student.