To find the simple interest, we use the formula \[\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}\], where \(\text{P}\) is the principal amount, \(\text{R}\) is the rate of interest, and \(\text{T}\) is the time period. It's crucial to ensure that the rate of interest and the time period are in consistent units (e.g., both per month or both per year).
The formula for simple interest is:
\[ \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \]Correct Option: D) 180
To calculate the effective discount percentage, we need to determine the total value of goods received and the actual amount paid. The discount is the value of the free goods, and the original price for calculating the percentage should be the total value of all goods received (paid for + free).
D) 14.29% is the correct effective discount percentage.
This problem involves calculating the original sales amount before successive percentage increases. We will work backward from the current sales amount, reversing the percentage increases for each year.
Thus, Rupesh's sales two years ago were ₹1,04,000.
B) ₹1,04,000 — This is the correct calculation by reversing the percentage increases.
This problem involves calculating the total profit percentage for a dishonest trader. The trader makes a profit in two ways: first, by claiming a certain profit percentage on the selling price, and second, by using a faulty weight. We need to combine these two profit sources to find the overall profit percentage.
The value of \(\frac{\sin^3A-\cos^3A}{1+\sin A\cdot\cos A}\) + 2 sin²54° sec²36° is:
sin A + cos A − 2
2(1 + sin A · cos A)
sin A − cos A + 2
sin A · cos A + 2
The problem requires simplifying a trigonometric expression involving powers of sine and cosine, and specific angle values. We will use algebraic identities for cubes, fundamental trigonometric identities, and complementary angle properties to simplify the expression.
The first term is \(\frac{\sin^3A-\cos^3A}{1+\sin A\cdot\cos A}\).
Use the algebraic identity \(a^3-b^3 = (a-b)(a^2+ab+b^2)\). Here, \(a = \sin A\) and \(b = \cos A\).
\[ \sin^3A-\cos^3A = (\sin A-\cos A)(\sin^2A+\sin A\cos A+\cos^2A) \]\[ = (\sin A-\cos A)((\sin^2A+\cos^2A)+\sin A\cos A) \]\[ = (\sin A-\cos A)(1+\sin A\cos A) \quad (\text{since } \sin^2A+\cos^2A=1) \]Therefore, the first term becomes:\[ \frac{(\sin A-\cos A)(1+\sin A\cos A)}{1+\sin A\cos A} = \sin A-\cos A \]The second term is \(2 \sin^254^\circ \sec^236^\circ\).
Use the complementary angle identity: \(\sin \theta = \cos(90^\circ - \theta)\) and \(\sec \theta = \csc(90^\circ - \theta)\).
We know that \(54^\circ = 90^\circ - 36^\circ\). So, \(\sin 54^\circ = \cos(90^\circ - 54^\circ) = \cos 36^\circ\).
Substitute this into the second term:
\[ 2 \sin^254^\circ \sec^236^\circ = 2 (\cos 36^\circ)^2 \sec^236^\circ \]\[ = 2 \cos^236^\circ \sec^236^\circ \]Since \(\sec \theta = \frac{1}{\cos \theta}\), we have \(\sec^2\theta = \frac{1}{\cos^2\theta}\).\[ 2 \cos^236^\circ \left(\frac{1}{\cos^236^\circ}\right) = 2 \cdot 1 = 2 \]Add the simplified first and second terms:
\[ (\sin A-\cos A) + 2 \]C) \(\sin A - \cos A + 2\) is the result obtained by simplifying both parts of the expression.
Shifa can do a certain piece of work in 20 days. Shifa and Afiya can together do the same work in 12 days, and Shifa, Afiya and Mariyam can do the same work together in 9 days. In how many days can Shifa and Mariyam do the same work together?
\( \frac{83}{8} \)
\( \frac{85}{9} \)
\( \frac{90}{7} \)
\( \frac{80}{7} \)
This problem can be solved by using the concept of work rates. We will calculate the individual work rates of Shifa, Afiya, and Mariyam, and then combine the rates of Shifa and Mariyam to find the time they take to complete the work together.
Calculate Shifa's work rate:
Shifa can do the work in 20 days. So, Shifa's one day's work (work rate) is:
\[ \text{Shifa's work rate} = \frac{1}{20} \text{ part of the work per day} \]Calculate the combined work rate of Shifa and Afiya:
Shifa and Afiya together can do the work in 12 days. Their combined one day's work is:
\[ (\text{Shifa} + \text{Afiya})'s \text{ work rate} = \frac{1}{12} \text{ part of the work per day} \]Calculate Afiya's work rate:
We can find Afiya's work rate by subtracting Shifa's work rate from their combined work rate:
\[ \text{Afiya's work rate} = (\text{Shifa} + \text{Afiya})'s \text{ work rate} - \text{Shifa's work rate} \]\[ \text{Afiya's work rate} = \frac{1}{12} - \frac{1}{20} \]\[ \text{Afiya's work rate} = \frac{5 - 3}{60} = \frac{2}{60} = \frac{1}{30} \text{ part of the work per day} \]Calculate the combined work rate of Shifa, Afiya, and Mariyam:
Shifa, Afiya, and Mariyam together can do the work in 9 days. Their combined one day's work is:
\[ (\text{Shifa} + \text{Afiya} + \text{Mariyam})'s \text{ work rate} = \frac{1}{9} \text{ part of the work per day} \]Calculate Mariyam's work rate:
We can find Mariyam's work rate by subtracting the combined work rate of Shifa and Afiya from the combined work rate of all three:
\[ \text{Mariyam's work rate} = (\text{Shifa} + \text{Afiya} + \text{Mariyam})'s \text{ work rate} - (\text{Shifa} + \text{Afiya})'s \text{ work rate} \]\[ \text{Mariyam's work rate} = \frac{1}{9} - \frac{1}{12} \]\[ \text{Mariyam's work rate} = \frac{4 - 3}{36} = \frac{1}{36} \text{ part of the work per day} \]Calculate the combined work rate of Shifa and Mariyam:
Now, we add Shifa's work rate and Mariyam's work rate:
\[ (\text{Shifa} + \text{Mariyam})'s \text{ work rate} = \text{Shifa's work rate} + \text{Mariyam's work rate} \]\[ (\text{Shifa} + \text{Mariyam})'s \text{ work rate} = \frac{1}{20} + \frac{1}{36} \]To add these fractions, find a common denominator, which is the LCM of 20 and 36. LCM(20, 36) = 180.
\[ (\text{Shifa} + \text{Mariyam})'s \text{ work rate} = \frac{9}{180} + \frac{5}{180} = \frac{9 + 5}{180} = \frac{14}{180} \]\[ (\text{Shifa} + \text{Mariyam})'s \text{ work rate} = \frac{7}{90} \text{ part of the work per day} \]Calculate the number of days Shifa and Mariyam take together:
The number of days they take to complete the work together is the reciprocal of their combined work rate:
\[ \text{Days} = \frac{1}{(\text{Shifa} + \text{Mariyam})'s \text{ work rate}} \]\[ \text{Days} = \frac{1}{\frac{7}{90}} = \frac{90}{7} \text{ days} \]Correct Option: C) \( \frac{90}{7} \)
The problem involves calculating distance, speed, and time. Since both Shahid and Mira travel from Delhi to Kanpur, the distance covered by both is the same. We can first calculate the total distance using Shahid's speed and time. Then, using this distance and Mira's speed, we can find the time taken by Mira.
We are given Shahid's speed and the time he takes to reach Kanpur.
The formula for distance is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]So, the distance from Delhi to Kanpur (\(D\)) is:
\[ D = S_S \times T_S = 132 \text{ km/hr} \times 5 \text{ hours} = 660 \text{ km} \]Now we know the total distance and Mira's speed.
The formula for time is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]So, the time taken by Mira (\(T_M\)) is:
\[ T_M = \frac{D}{S_M} = \frac{660 \text{ km}}{60 \text{ km/hr}} = 11 \text{ hours} \]C) 11 hours is the correct answer. Mira takes 11 hours to reach Kanpur.
This problem involves setting up and solving linear equations based on the given age conditions. We will define variables for the current ages of the father and son, translate the given information into equations, and then solve for the unknown variable representing the number of years until the next condition is met.
D) 34 — The calculations show that after 34 years, the father's age will be twice the son's age.
To find the difference between the largest and smallest shares, we first need to determine the value of one share unit. We can do this by summing the ratio parts and dividing the total amount by this sum. Once we have the value of one unit, we can calculate the individual shares and then find the required difference.
Total amount \( = \text{₹}456 \)
Ratio of shares \( = 16 : 14 : 8 \)
Sum of ratio parts \( = 16 + 14 + 8 = 38 \)
If the total amount is divided into 38 equal parts, then the value of one part (or one ratio unit) is:
\[ \text{Value of one unit} = \frac{\text{Total amount}}{\text{Sum of ratio parts}} = \frac{456}{38} = 12 \]Difference \( = \text{Largest share} - \text{Smallest share} = 192 - 96 = \text{₹}96 \)
Correct Option: C) 96
The difference between the largest share (₹192) and the smallest share (₹96) is ₹96.
The problem states that a number, when increased by 100%, results in 2690. We need to find the original number. We can set up an algebraic equation to represent this relationship, where the original number is represented by a variable.
Thus, the original number is 1345.
C) 1345 is the correct answer because when 1345 is increased by 100%, it becomes \(1345 + (100\% \text{ of } 1345) = 1345 + 1345 = 2690\).