To find the selling price after a discount, we first calculate the amount of the discount based on the marked price and the discount percentage. Then, we subtract this discount amount from the marked price to get the selling price.
Identify the given values:
Calculate the discount amount:
The discount amount is 13% of the Marked Price.
\[ \text{Discount Amount} = \frac{\text{Discount Percentage}}{100} \times \text{Marked Price} \] \[ \text{Discount Amount} = \frac{13}{100} \times 900 \] \[ \text{Discount Amount} = 13 \times 9 \] \[ \text{Discount Amount} = \text{₹}117 \]Calculate the Selling Price (SP):
The Selling Price is the Marked Price minus the Discount Amount.
\[ \text{Selling Price} = \text{Marked Price} - \text{Discount Amount} \] \[ \text{Selling Price} = 900 - 117 \] \[ \text{Selling Price} = \text{₹}783 \]Correct Option: C) ₹783
The problem asks us to find the third number given the average of three numbers and the values of two of those numbers. We will use the definition of average (arithmetic mean) to solve this problem.
C) 14 is the correct answer. The calculation shows that the third number is 14.
To solve this problem, we need to first understand how to divide an amount according to a given ratio. We will find the individual shares of each person and then calculate the difference between the largest and the smallest shares.
Identify the total amount and the ratio: Total amount to be divided = ₹638 Ratio among three persons = \(2 : 5 : 4\)
Calculate the sum of the ratio parts: Sum of ratio parts \( = 2 + 5 + 4 = 11\)
Determine the value of one ratio part: One ratio part \[= \frac{\text{Total amount}}{\text{Sum of ratio parts}} = \frac{638}{11}\]
\[\frac{638}{11} = 58\] So, one ratio part is ₹58.
Calculate the individual shares: First person's share \( = 2 \times 58 = 116\) Second person's share \( = 5 \times 58 = 290\) Third person's share \( = 4 \times 58 = 232\)
To verify, sum of shares \( = 116 + 290 + 232 = 638\), which matches the total amount.
Identify the largest and smallest shares: Largest share \( = 290\) (corresponding to ratio part 5) Smallest share \( = 116\) (corresponding to ratio part 2)
Calculate the difference between the largest and smallest shares: Difference \( = \text{Largest share} - \text{Smallest share} = 290 - 116 = 174\)
D) 174. The difference between the largest share (₹290) and the smallest share (₹116) is ₹174.
A) 76: This value is not obtained from the correct calculation of the difference between the largest and smallest shares.
B) 110: This value is not obtained from the correct calculation of the difference between the largest and smallest shares.
C) 157: This value is not obtained from the correct calculation of the difference between the largest and smallest shares.
This problem involves the concept of work and time. We need to determine the individual work rates of men and women and then combine them to find the time taken by a group of 6 women and 9 men to complete the work.
Given that 2 men can complete a piece of work in 18 days.
\[ \text{Total work units} = \text{Number of men} \times \text{Days} \times \text{Work rate of 1 man} \]Let the work rate of 1 man be \(m\) units/day. The total work is \(2 \times 18 \times m = 36m\) units. So, 1 unit of work is \(36m\).
Alternatively, if 2 men complete the work in 18 days, then 1 man would take \(2 \times 18 = 36\) days to complete the same work.
Therefore, the work done by 1 man in 1 day is:
\[ \text{Work by 1 man in 1 day} = \frac{1}{36} \text{ of the total work} \]Given that 4 women can complete the same work in 6 days.
Similarly, if 4 women complete the work in 6 days, then 1 woman would take \(4 \times 6 = 24\) days to complete the same work.
Therefore, the work done by 1 woman in 1 day is:
\[ \text{Work by 1 woman in 1 day} = \frac{1}{24} \text{ of the total work} \]Work done by 6 women in 1 day:
\[ 6 \times \frac{1}{24} = \frac{6}{24} = \frac{1}{4} \text{ of the total work} \]Work done by 9 men in 1 day:
\[ 9 \times \frac{1}{36} = \frac{9}{36} = \frac{1}{4} \text{ of the total work} \]Combined work done by 6 women and 9 men in 1 day:
\[ \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \text{ of the total work} \]If 6 women and 9 men can complete \(\frac{1}{2}\) of the work in 1 day, then they will complete the full work in:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Work done per day}} = \frac{1}{\frac{1}{2}} = 2 \text{ days} \]B) 2 days is the correct answer. The combined work rate of 6 women and 9 men is such that they can finish the entire work in 2 days.
We are given two statements relating the present ages of Mary and Diya. We can translate these statements into a system of two linear equations with two variables. By solving this system, we can find the individual ages of Mary and Diya, and then calculate the difference between their ages.
Let Mary's present age be \(M\) years and Diya's present age be \(D\) years.
Translate the first statement into an equation:
"2 times Mary's present age is 20 years more than 6 times Diya's present age"
\[ 2M = 6D + 20 \]Simplify the first equation by dividing by 2:
\[ M = 3D + 10 \quad \text{(Equation 1)} \]Translate the second statement into an equation:
"5 times Diya's present age is 6 years less than Mary's present age"
\[ 5D = M - 6 \]Rearrange the second equation to express \(M\) in terms of \(D\):
\[ M = 5D + 6 \quad \text{(Equation 2)} \]Now we have two expressions for \(M\). Set them equal to each other to solve for \(D\):
\[ 3D + 10 = 5D + 6 \]Subtract \(3D\) from both sides:
\[ 10 = 2D + 6 \]Subtract 6 from both sides:
\[ 4 = 2D \]Divide by 2 to find Diya's age:
\[ D = \frac{4}{2} = 2 \]So, Diya's present age is 2 years.
Substitute the value of \(D\) into either Equation 1 or Equation 2 to find Mary's age. Using Equation 1:
\[ M = 3D + 10 \]\[ M = 3(2) + 10 \]\[ M = 6 + 10 \]\[ M = 16 \]So, Mary's present age is 16 years.
Calculate the difference between their ages:
Difference \( = M - D \)
\[ \text{Difference} = 16 - 2 \]\[ \text{Difference} = 14 \]D) 14
A) 15 — This would be incorrect if, for example, Mary's age was 17 and Diya's was 2, or if there was an error in calculation leading to a difference of 15.
B) 18 — This would be incorrect if, for example, Mary's age was 20 and Diya's was 2, or if there was an error in setting up or solving the equations.
C) 16 — This is Mary's age, not the difference between their ages.
The problem requires calculating the remaining speed Anaya needs to maintain to reach Hyderabad on time. We need to calculate the total distance covered and time elapsed in the initial two phases of the journey. Then, we can find the remaining distance and remaining time to determine the required speed.
Calculate distance covered in the first phase:
\[ \text{Distance}_1 = \text{Speed}_1 \times \text{Time}_1 \] \[ \text{Distance}_1 = 36 \text{ km/hr} \times 11 \text{ hours} = 396 \text{ km} \]Calculate time taken for the second phase:
\[ \text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} \] \[ \text{Time}_2 = \frac{136 \text{ km}}{34 \text{ km/hr}} = 4 \text{ hours} \]Calculate total distance covered so far:
\[ \text{Total Distance Covered} = \text{Distance}_1 + \text{Distance}_2 \] \[ \text{Total Distance Covered} = 396 \text{ km} + 136 \text{ km} = 532 \text{ km} \]Calculate total time elapsed so far:
\[ \text{Total Time Elapsed} = \text{Time}_1 + \text{Time}_2 \] \[ \text{Total Time Elapsed} = 11 \text{ hours} + 4 \text{ hours} = 15 \text{ hours} \]Calculate remaining distance to cover:
\[ \text{Remaining Distance} = \text{Total Distance} - \text{Total Distance Covered} \] \[ \text{Remaining Distance} = 574 \text{ km} - 532 \text{ km} = 42 \text{ km} \]Calculate remaining time:
\[ \text{Remaining Time} = \text{Total Allowed Time} - \text{Total Time Elapsed} \] \[ \text{Remaining Time} = 18 \text{ hours} - 15 \text{ hours} = 3 \text{ hours} \]Calculate the required speed for the remaining journey:
\[ \text{Required Speed} = \frac{\text{Remaining Distance}}{\text{Remaining Time}} \] \[ \text{Required Speed} = \frac{42 \text{ km}}{3 \text{ hours}} = 14 \text{ km/hr} \]Correct Option: B) 14 km/hr
The simple interest on a certain sum for 7 \( \frac{1}{3} \) years at the rate of 4.5% per annum is ₹6,270. What will be the amount payable on the same sum at the rate of 8.6% per annum simple interest in 11 \( \frac{1}{2} \) years?
The problem involves two parts. First, we need to find the principal amount using the given simple interest, time, and rate. Second, we use this principal amount to calculate the future amount payable with a new simple interest rate and time period.
Given:
The formula for Simple Interest is:
\[ \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \]Rearranging to find P:
\[ \text{P} = \frac{\text{SI} \times 100}{\text{R} \times \text{T}} \]Substitute the given values:
\[ \text{P} = \frac{6270 \times 100}{4.5 \times \frac{22}{3}} \] \[ \text{P} = \frac{6270 \times 100 \times 3}{4.5 \times 22} \] \[ \text{P} = \frac{627000 \times 3}{99} \] \[ \text{P} = \frac{1881000}{99} \] \[ \text{P} = 19000 \]So, the principal amount is ₹19,000.
Now, we need to find the simple interest on the same principal (P = ₹19,000) with new parameters:
Using the simple interest formula:
\[ \text{SI}_2 = \frac{\text{P} \times \text{R} \times \text{T}}{100} \] \[ \text{SI}_2 = \frac{19000 \times 8.6 \times \frac{23}{2}}{100} \] \[ \text{SI}_2 = \frac{19000 \times 8.6 \times 23}{100 \times 2} \] \[ \text{SI}_2 = 190 \times 4.3 \times 23 \] \[ \text{SI}_2 = 817 \times 23 \] \[ \text{SI}_2 = 18791 \]The simple interest for the second scenario is ₹18,791.
The amount payable is the sum of the principal and the simple interest:
\[ \text{Amount} = \text{P} + \text{SI}_2 \] \[ \text{Amount} = 19000 + 18791 \] \[ \text{Amount} = 37791 \]The total amount payable will be ₹37,791.
B) ₹37,791 is the correct amount calculated.
To find the mean proportional between two numbers, we use the concept that if \(x\) is the mean proportional between \(a\) and \(b\), then the ratio \(a:x\) is equal to the ratio \(x:b\). This implies that \(x^2 = ab\), or \(x = \sqrt{ab}\).
Correct Option: A) 24 is the mean proportional between 18 and 32.
If x = a cosecA sinB, y = b cosecA cosB, and z = c cotA, then what is the value of \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}\)?
The problem provides three equations relating \(x, y, z\) to \(a, b, c\) and trigonometric functions. We need to find the value of the expression \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}\]. The strategy is to express \(\frac{x^2}{a^2}\), \(\frac{y^2}{b^2}\), and \(\frac{z^2}{c^2}\) in terms of trigonometric functions and then substitute these into the given expression. We will use the fundamental trigonometric identity \(\text{cosec}^2 A - \text{cot}^2 A = 1\).
Given the equations:
Express \(\frac{x^2}{a^2}\) in terms of trigonometric functions:
\[ \frac{x}{a} = \text{cosec} A \sin B \]\[ \frac{x^2}{a^2} = (\text{cosec} A \sin B)^2 = \text{cosec}^2 A \sin^2 B \]Express \(\frac{y^2}{b^2}\) in terms of trigonometric functions:
\[ \frac{y}{b} = \text{cosec} A \cos B \]\[ \frac{y^2}{b^2} = (\text{cosec} A \cos B)^2 = \text{cosec}^2 A \cos^2 B \]Express \(\frac{z^2}{c^2}\) in terms of trigonometric functions:
\[ \frac{z}{c} = \text{cot} A \]\[ \frac{z^2}{c^2} = (\text{cot} A)^2 = \text{cot}^2 A \]Substitute these expressions into the given expression \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}\]:
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = (\text{cosec}^2 A \sin^2 B) + (\text{cosec}^2 A \cos^2 B) - (\text{cot}^2 A) \]Factor out \(\text{cosec}^2 A\) from the first two terms:
\[ \text{cosec}^2 A (\sin^2 B + \cos^2 B) - \text{cot}^2 A \]Apply the trigonometric identity \(\sin^2 B + \cos^2 B = 1\):
\[ \text{cosec}^2 A (1) - \text{cot}^2 A \]\[ = \text{cosec}^2 A - \text{cot}^2 A \]Apply the fundamental trigonometric identity \(\text{cosec}^2 A - \text{cot}^2 A = 1\):
\[ = 1 \]D) The value of the expression is 1, as derived from the trigonometric identities.
The problem asks to calculate the simple interest. We will use the formula for simple interest, which is given by \(SI = \frac{P \times R \times T}{100}\), where \(P\) is the principal amount, \(R\) is the rate of interest per period, and \(T\) is the time period. It is crucial to ensure that the rate and time are consistent in their units (e.g., if the rate is per month, the time should be in months).
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]Since the rate is given per month and the time is in months, we can directly substitute the values:
\[ SI = \frac{1200 \times 4 \times 2}{100} \]First, multiply the numerator:
\[ 1200 \times 4 \times 2 = 4800 \times 2 = 9600 \]Then, divide by 100:\[ SI = \frac{9600}{100} = 96 \]The simple interest is ₹96.
Correct Option: A) ₹96 is the correct simple interest calculated based on the given principal, rate, and time.