Simplify \(\frac{423\times423-325\times325}{423-325}\)
The problem asks us to simplify a given algebraic expression. The expression is in the form of \((a^2 - b^2) / (a - b)\). We can use the algebraic identity for the difference of squares to simplify this expression efficiently.
Identify the structure of the given expression:
\[ \frac{423 \times 423 - 325 \times 325}{423 - 325} \]Recognize that \(423 \times 423\) can be written as \(423^2\) and \(325 \times 325\) can be written as \(325^2\). So the expression becomes:
\[ \frac{423^2 - 325^2}{423 - 325} \]Recall the algebraic identity for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).
Apply this identity to the numerator of the expression. Let \(a = 423\) and \(b = 325\). Then the numerator \(423^2 - 325^2\) becomes \((423 - 325)(423 + 325)\).
Substitute this back into the expression:
\[ \frac{(423 - 325)(423 + 325)}{423 - 325} \]Notice that the term \((423 - 325)\) appears in both the numerator and the denominator. Since \(423 - 325 \neq 0\), we can cancel this term out:
\[ 423 + 325 \]Perform the addition:
\[ 423 + 325 = 748 \]A) 748 is the correct answer, as derived from the simplification using the difference of squares identity.
What is the rank of Bengali language in the world among the most spoken languages as per Forbes (as of 2022) ?
The question asks for the rank of the Bengali language among the most spoken languages globally, according to Forbes in 2022. This requires knowledge of global language statistics.
Correct Option: A) 6th
Bengali is indeed ranked as the 6th most spoken language in the world as per Forbes' 2022 data, with approximately 272.7 million speakers.
In a certain code language, MARKS is written as QEVOW and CLOSE is written as GPSWI.
How will DRIVE be written in the same language?
The problem is a coding-decoding question. We need to identify the pattern or rule used to transform the given words into their coded forms. Once the rule is found, we apply it to the word "DRIVE" to find its coded form.
D) HVMZI is the correct answer because applying the +4 shift to each letter of DRIVE results in HVMZI.
Using π = \( \frac{22}{7} \), find the perimeter of the semi-circle whose diameter is 21 cm long.
To find the perimeter of a semi-circle, we need to calculate the length of its curved part (half the circumference of a full circle) and add the length of its diameter. The formula for the circumference of a circle is \( \pi d \) or \( 2 \pi r \), where \( d \) is the diameter and \( r \) is the radius. For a semi-circle, the curved length is \( \frac{1}{2} \pi d \). The total perimeter will be \( \frac{1}{2} \pi d + d \).
Identify the given information:
Calculate the length of the curved part of the semi-circle:
The curved part is half the circumference of a full circle.
\[ \text{Curved length} = \frac{1}{2} \pi d \]\[ \text{Curved length} = \frac{1}{2} \times \frac{22}{7} \times 21 \]\[ \text{Curved length} = \frac{1}{2} \times 22 \times 3 \]\[ \text{Curved length} = 11 \times 3 \]\[ \text{Curved length} = 33 \text{ cm} \]Calculate the total perimeter of the semi-circle:
The perimeter of a semi-circle is the sum of its curved length and its diameter.
\[ \text{Perimeter} = \text{Curved length} + \text{Diameter} \]\[ \text{Perimeter} = 33 \text{ cm} + 21 \text{ cm} \]\[ \text{Perimeter} = 54 \text{ cm} \]Correct Option: C) 54.0 cm
The calculated perimeter of the semi-circle is 54 cm, which matches option C.
The problem involves men and women working together to complete the same task in different numbers of days. We need to determine the time taken by one man to complete the entire work alone. To solve this, we will assume the total work to be 1 unit and represent the daily work rates of one man and one woman using variables. By forming two equations from the given conditions and solving them simultaneously, we can calculate the daily work rate of one man and hence the number of days he would require to finish the work alone.
Identify the given information:
Represent the individual work rates:
Let \(m\) be the fraction of the total work completed by one man in one day and \(w\) be the fraction completed by one woman in one day.
Assume the total work is 1 unit.
Form the first equation:
Since 6 men and 8 women complete the work in 14 days, their combined daily work rate is:
\[ 6m+8w=\frac{1}{14} \]Multiplying both sides by 14:
\[ 84m+112w=1 \]This is Equation (1).
Form the second equation:
Since 8 men and 12 women complete the same work in 10 days:
\[ 8m+12w=\frac{1}{10} \]Multiplying both sides by 10:
\[ 80m+120w=1 \]This is Equation (2).
Eliminate one variable by subtracting the equations:
Equation (1):
\[ 84m+112w=1 \]Equation (2):
\[ 80m+120w=1 \]Subtract Equation (2) from Equation (1):
\[ (84m-80m)+(112w-120w)=1-1 \] \[ 4m-8w=0 \] \[ 4m=8w \] \[ m=2w \]Therefore, one man's daily work rate is twice that of one woman.
Substitute the relationship into Equation (1):
Since \(m=2w\), we have:
\[ 84(2w)+112w=1 \] \[ 168w+112w=1 \] \[ 280w=1 \] \[ w=\frac{1}{280} \]Therefore, one woman completes \(\frac{1}{280}\) of the total work in one day.
Calculate the daily work rate of one man:
Using \(m=2w\):
\[ m=2\times\frac{1}{280} \] \[ m=\frac{1}{140} \]Therefore, one man completes \(\frac{1}{140}\) of the total work in one day.
Calculate the time taken by one man working alone:
\[ \text{Time}=\frac{\text{Total work}}{\text{Daily work rate}} \] \[ T=\frac{1}{\frac{1}{140}} \] \[ T=140\text{ days} \]Verify the result using the given conditions:
For 6 men and 8 women:
\[ 6\left(\frac{1}{140}\right)+8\left(\frac{1}{280}\right) =\frac{12+8}{280} =\frac{1}{14} \]Thus, they complete the work in 14 days.
For 8 men and 12 women:
\[ 8\left(\frac{1}{140}\right)+12\left(\frac{1}{280}\right) =\frac{16+12}{280} =\frac{1}{10} \]Thus, they complete the work in 10 days. Both conditions are satisfied.
Correct Option: A) 140 days — By solving the simultaneous equations representing the combined work rates, we find that one man's daily work rate is \(\frac{1}{140}\) of the total work. Therefore, one man working alone would complete the entire work in 140 days.
B) 175 days: This is incorrect. If one man completed the work in 175 days, his daily work rate would be \(\frac{1}{175}\), which does not satisfy the simultaneous equations obtained from the given working conditions.
C) 210 days: This is incorrect. A time of 210 days would imply a daily work rate of \(\frac{1}{210}\) for one man, whereas the calculated daily work rate is \(\frac{1}{140}\).
D) 280 days: This is incorrect. The value of 280 days represents the time taken by one woman working alone, since her daily work rate is \(\frac{1}{280}\). One man works twice as fast as one woman and therefore requires only 140 days.
The question asks to identify which of the given options is NOT a part of pubertal development. Puberty is a period of rapid growth and development during which a child's body matures into an adult body, capable of reproduction. This process involves significant hormonal changes leading to various physical transformations.
Based on this evaluation, obesity is the only option that is not a direct or universal part of pubertal development.
A) Obesity — Obesity is a health condition related to excessive body fat, influenced by genetics, diet, and lifestyle. While hormonal changes during puberty can affect body composition, obesity itself is not a normal or expected developmental milestone of puberty. It is a potential health issue that can occur at any age, including during puberty, but it is not a defining characteristic of the pubertal process.
Chandrashekhar sold 12 identical articles for a total of Rs 420, incurring a loss equal to the cost price of 7 such articles. The cost price of 1 such article is:
We are given the selling price of 12 articles and the loss incurred, which is expressed in terms of the cost price of a certain number of articles. We need to find the cost price of one such article. We will set up equations based on the given information relating Cost Price (CP), Selling Price (SP), and Loss.
Let the Cost Price (CP) of one article be \(x\) rupees.
The Cost Price of 12 articles will be \(12x\) rupees.
The Selling Price (SP) of 12 articles is given as Rs 420.
The loss incurred is equal to the cost price of 7 such articles. So, Loss \(= 7x\) rupees.
We know the formula relating CP, SP, and Loss:
\[ \text{Loss} = \text{CP} - \text{SP} \]Substitute the values for 12 articles into the formula:
\[ 7x = 12x - 420 \]Now, solve the equation for \(x\):
\[ 420 = 12x - 7x \]\[ 420 = 5x \]Divide by 5 to find \(x\):
\[ x = \frac{420}{5} \]\[ x = 84 \]Therefore, the cost price of 1 such article is Rs 84.
A) Rs 84 is the correct answer. This is derived directly from the calculation where the cost price of one article is found to be Rs 84.
Simplify:
\(\frac{4.32\times4.32-2.64\times2.64}{1.68}\)
The problem asks us to simplify a given expression involving multiplication, subtraction, and division of decimal numbers. The expression is in the form of \((a^2 - b^2) / c\). We can use the algebraic identity \(a^2 - b^2 = (a - b)(a + b)\) to simplify the numerator, which will make the calculation easier.
D) 6.96 is the simplified value of the given expression.
This problem involves calculating the final displacement from the starting point based on a series of movements and turns. We can solve this by visualizing the path taken, treating movements as vectors, and then finding the resultant displacement. We can use a coordinate system or simply draw the path to determine the net change in position along the East-West and North-South axes.
Distance \( = |0 - (-40)|\)
Distance \( = |40|\)
Distance \( = 40 \text{ m}\)
Alternatively, we can analyze the net displacement:
Since the net displacement in the East-West direction is 0, she is directly South of her starting point. The distance from the original starting point is the magnitude of the net displacement in the South direction, which is 40 m.
C) 40 m — As calculated, B is 40 m South of her starting point, so the distance from the original starting point is 40 m.
Which of the following is the correct empirical formula?
\( \frac{mode-mean}{3} \) = median - mean
\(\mathrm{\frac{Mode-Mean}{3\times2}}\) = median - mean
3(mode - mean) = median - mean
\( \frac{2}{3} \) (mode - mean) = median - mean
The question asks for the correct empirical formula relating the mean, median, and mode of a moderately skewed distribution. This is a well-known statistical relationship.
Subtract 'Mean' from both sides:
\[ \text{Mode} - \text{Mean} = 3 \text{ Median} - 2 \text{ Mean} - \text{Mean} \]Simplify the right side:
\[ \text{Mode} - \text{Mean} = 3 \text{ Median} - 3 \text{ Mean} \]Factor out 3 from the right side:
\[ \text{Mode} - \text{Mean} = 3 (\text{Median} - \text{Mean}) \]Divide both sides by 3:
\[ \frac{\text{Mode} - \text{Mean}}{3} = \text{Median} - \text{Mean} \]This matches option A.
A) \( \frac{mode-mean}{3} \) = median - mean. This is the correct empirical formula derived from the relationship Mode = 3 Median - 2 Mean.