To find the ratio of the marked price to the cost price, we need to establish relationships between Cost Price (CP), Selling Price (SP), and Marked Price (MP) using the given discount percentages and profit percentage. We will first calculate the Selling Price in terms of Cost Price using the profit percentage. Then, we will calculate the Selling Price in terms of Marked Price using the successive discounts. By equating these two expressions for Selling Price, we can find the ratio of MP to CP.
Calculate Selling Price (SP) in terms of Cost Price (CP):
Given that there is a 10% profit.
\[ \text{SP} = \text{CP} + (0.10 \times \text{CP}) = 1.10 \times \text{CP} \]Calculate Selling Price (SP) in terms of Marked Price (MP):
First discount is 15%.
Price after first discount \( = \text{MP} \times (1 - 0.15) = 0.85 \times \text{MP} \)
Second discount is 12% on the discounted price.
Selling Price (SP) \( = (0.85 \times \text{MP}) \times (1 - 0.12) \)
\( \text{SP} = (0.85 \times \text{MP}) \times 0.88 \)
\( \text{SP} = 0.748 \times \text{MP} \)
Equate the two expressions for SP:
From step 1: \( \text{SP} = 1.10 \times \text{CP} \)
From step 2: \( \text{SP} = 0.748 \times \text{MP} \)
Therefore,
\[ 1.10 \times \text{CP} = 0.748 \times \text{MP} \]Find the ratio of Marked Price (MP) to Cost Price (CP):
\[ \frac{\text{MP}}{\text{CP}} = \frac{1.10}{0.748} \]To simplify the fraction, multiply numerator and denominator by 1000 to remove decimals:
\[ \frac{\text{MP}}{\text{CP}} = \frac{1100}{748} \]Divide both by common factors. Both are divisible by 4:
\[ \frac{\text{MP}}{\text{CP}} = \frac{1100 \div 4}{748 \div 4} = \frac{275}{187} \]Both 275 and 187 are divisible by 11:
\[ \frac{\text{MP}}{\text{CP}} = \frac{275 \div 11}{187 \div 11} = \frac{25}{17} \]So, the ratio of marked price to cost price is 25 : 17.
Correct Option: B) 25 : 17
Which of the following is NOT correct with respect to amendments in the Constitution of India?
72nd amendment - to make provisions for setting up Rent Control Tribunals
102nd amendment - to give constitutional status to the National Commission for Backward Classes
61st amendment - to reduce the voting age from 21 to 18 years
101st amendment - to introduce goods and services tax
The question asks to identify the incorrect statement regarding amendments to the Constitution of India. We need to evaluate each given amendment and its associated purpose to determine which one is wrongly matched.
Based on the analysis, Option A incorrectly matches the 72nd Amendment with the provision for Rent Control Tribunals.
Correct Option: A) 72 nd amendment - to make provisions for setting up Rent Control Tribunals
The problem involves three positive integers \(a, b, c\) with a given average and a condition on their median. We need to find the least possible value of \(c\). We will use the definitions of average and median, along with the given inequality \(a \le b \le c\), to set up equations and inequalities. To find the least value of \(c\), we will maximize the values of \(a\) and \(b\) as much as possible while adhering to the given conditions.
Understand the given information:
Formulate equations from the given information:
From the average, we have:
\[ a+b+c = 3 \times 28 \] \[ a+b+c = 84 \quad \text{(Equation 1)} \]From the median, we have:
\[ b = a+16 \quad \text{(Equation 2)} \]Substitute Equation 2 into Equation 1:
Replace \(b\) with \((a+16)\) in Equation 1:
\[ a + (a+16) + c = 84 \] \[ 2a + 16 + c = 84 \] \[ 2a + c = 84 - 16 \] \[ 2a + c = 68 \quad \text{(Equation 3)} \]Use the ordering condition to find the minimum value of \(c\):
We are given \(a \le b \le c\).
Substitute \(b = a+16\):
\[ a \le a+16 \le c \]The inequality \(a \le a+16\) is always true for positive integers. The crucial inequality for \(c\) is \(b \le c\), which means:
\[ a+16 \le c \quad \text{(Inequality 4)} \]From Equation 3, we can express \(c\) in terms of \(a\):
\[ c = 68 - 2a \]Substitute this expression for \(c\) into Inequality 4:
\[ a+16 \le 68 - 2a \]Solve the inequality for \(a\):
\[ a + 2a \le 68 - 16 \] \[ 3a \le 52 \] \[ a \le \frac{52}{3} \] \[ a \le 17.33... \]Since \(a\) must be a positive integer, the maximum possible value for \(a\) is 17.
Find the least possible value of \(c\):
To find the least possible value of \(c\), we need to maximize \(a\) (because \(c = 68 - 2a\), so a larger \(a\) results in a smaller \(c\)).
Using the maximum integer value for \(a\), which is 17:
\[ a = 17 \]Now, calculate \(b\) using \(b = a+16\):
\[ b = 17+16 = 33 \]Now, calculate \(c\) using \(c = 68 - 2a\):
\[ c = 68 - 2(17) \] \[ c = 68 - 34 \] \[ c = 34 \]Verify the conditions:
The integers are \(a=17, b=33, c=34\).
All conditions are satisfied, and this value of \(c\) is obtained by maximizing \(a\), thus making it the least possible value for \(c\).
Correct Option: D) 34
To find the next number in the series, we need to identify the pattern or rule governing the sequence. This usually involves checking the differences between consecutive terms.
\[ 656 - 619 = 37 \]
\[ 693 - 656 = 37 \]
\[ 730 - 693 = 37 \]
The difference between consecutive terms is consistently \(37\). This indicates an arithmetic progression where each subsequent term is obtained by adding \(37\) to the previous term.
To find the next number in the series, add \(37\) to the last given term, which is \(730\).
\[ 730 + 37 = 767 \]
Therefore, the next number in the series is \(767\).
B) 767 is the correct answer because the series follows an arithmetic progression with a common difference of \(37\).
The question asks to identify the person who made the famous remark, "Nehru is a patriot, while Jinnah is a politician." This is a direct factual recall question related to prominent figures in Indian history and their views on the leaders of the Indian independence movement.
Correct Option: B) Mohammed Iqbal — Mohammed Iqbal is widely credited with making the statement, "Nehru is a patriot, while Jinnah is a politician." This remark highlighted the perceived differences in their motivations and approaches to politics during the freedom struggle.
The problem provides the ratio of three numbers and the value of the smallest number. We need to find the largest number. We can represent the numbers using a common multiplier based on their ratio and then use the given smallest number to find the value of this multiplier. Once the multiplier is known, we can calculate the largest number.
Correct Option: D) 112
If θ = 30°, then what will be the value of sinθcosθ?
\(\frac{3}{8}\)
\(\frac{\sqrt{3}}{6}\)
\(\frac{\sqrt{3}}{2}\)
\(\frac{\sqrt{3}}{4}\)
To find the value of \(\sin\theta\cos\theta\) when \(\theta = 30^\circ\), we need to recall the standard trigonometric values for \(\sin 30^\circ\) and \(\cos 30^\circ\) and then multiply them.
Identify the given value of \(\theta\):
Given \(\theta = 30^\circ\).
Recall the value of \(\sin 30^\circ\):
The standard trigonometric value for \(\sin 30^\circ\) is \(\frac{1}{2}\).
Recall the value of \(\cos 30^\circ\):
The standard trigonometric value for \(\cos 30^\circ\) is \(\frac{\sqrt{3}}{2}\).
Calculate \(\sin\theta\cos\theta\):
Substitute the values of \(\sin 30^\circ\) and \(\cos 30^\circ\) into the expression \(\sin\theta\cos\theta\):
\[ \sin\theta\cos\theta = \sin 30^\circ \cdot \cos 30^\circ \] \[ = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \] \[ = \frac{1 \cdot \sqrt{3}}{2 \cdot 2} \] \[ = \frac{\sqrt{3}}{4} \]Correct Option: D) \(\frac{\sqrt{3}}{4}\)
The problem asks us to find a certain percentage of a number, given another percentage of the same number. We can solve this by first finding the full value of the number, and then calculating the required percentage. Alternatively, we can find the value of 1% of the number and then scale it up to the desired percentage.
We are given that 28% of this number \(X\) is 20.
28% of \(X\) can be written as \(\frac{28}{100} \times X\).
So, we have the equation:
\[ \frac{28}{100} \times X = 20 \]To find \(X\), we can rearrange the equation:
\[ X = \frac{20 \times 100}{28} \]\[ X = \frac{2000}{28} \]\[ X = \frac{500}{7} \]Now we need to find 49% of \(\frac{500}{7}\).
\[ 49\% \text{ of } X = \frac{49}{100} \times X \]\[ 49\% \text{ of } X = \frac{49}{100} \times \frac{500}{7} \]We can cancel out common factors:
\[ 49\% \text{ of } X = \frac{49}{7} \times \frac{500}{100} \]\[ 49\% \text{ of } X = 7 \times 5 \]\[ 49\% \text{ of } X = 35 \]Alternatively, we can find the value of 1% of the number first:
Given that 28% of the number is 20, we can find 1% by dividing 20 by 28:
\[ 1\% \text{ of the number} = \frac{20}{28} \]\[ 1\% \text{ of the number} = \frac{5}{7} \]Since 1% of the number is \(\frac{5}{7}\), 49% of the number will be 49 times this value:
\[ 49\% \text{ of the number} = 49 \times \frac{5}{7} \]\[ 49\% \text{ of the number} = \frac{49}{7} \times 5 \]\[ 49\% \text{ of the number} = 7 \times 5 \]\[ 49\% \text{ of the number} = 35 \]Both methods yield the same result.
Correct Option: A) 35
If θ is an acute angle and tanθ = \( \frac{48}{55} \), then what is the value of cosecθ?
\( \frac{73}{48} \)
\( \frac{73}{55} \)
\( \frac{55}{48} \)
\( \frac{48}{73} \)
We are given the value of \(\tan\theta\) for an acute angle \(\theta\). We need to find the value of \(\text{cosec}\theta\). We can use the definition of trigonometric ratios in a right-angled triangle or trigonometric identities to solve this problem.
Understand the given information: We are given \(\tan\theta = \frac{48}{55}\). For a right-angled triangle, \(\tan\theta\) is defined as the ratio of the length of the side opposite to angle \(\theta\) to the length of the side adjacent to angle \(\theta\).
\[ \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{48}{55} \]So, we can consider the opposite side to be \(48k\) and the adjacent side to be \(55k\) for some positive constant \(k\). For simplicity, we can take \(k=1\), so Opposite \(= 48\) and Adjacent \(= 55\).
Calculate the hypotenuse: Using the Pythagorean theorem in a right-angled triangle:
\[ (\text{Hypotenuse})^2 = (\text{Opposite})^2 + (\text{Adjacent})^2 \] \[ (\text{Hypotenuse})^2 = (48)^2 + (55)^2 \] \[ (\text{Hypotenuse})^2 = 2304 + 3025 \] \[ (\text{Hypotenuse})^2 = 5329 \]To find the hypotenuse, we take the square root:
\[ \text{Hypotenuse} = \sqrt{5329} \]We can find the square root by estimation or calculation. \(70^2 = 4900\) and \(80^2 = 6400\). The number ends in 9, so the unit digit of its square root must be 3 or 7. Let's try \(73^2\):
\[ 73 \times 73 = (70+3)(70+3) = 4900 + 2 \times 70 \times 3 + 9 = 4900 + 420 + 9 = 5329 \]So, Hypotenuse \(= 73\).
Calculate \(\text{cosec}\theta\): The cosecant of an angle \(\theta\) is defined as the ratio of the hypotenuse to the opposite side:
\[ \text{cosec}\theta = \frac{\text{Hypotenuse}}{\text{Opposite}} \]Substitute the values we found:
\[ \text{cosec}\theta = \frac{73}{48} \]Correct Option: A) \( \frac{73}{48} \)
The question asks to identify the incorrectly matched pair of a dam and the river it is built on. We need to check the river associated with each dam listed in the options.
Based on this analysis, the pair "Nathpa Jhakri Dam - Jhelum river" is incorrectly matched.
Correct Option: B) Nathpa Jhakri Dam - Jhelum river