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280 students appeared for an interview, and \[50\% \] of girls and \[6\dfrac{1}{4}\% \] of boys were declared successful. If such students are \[25\% \] on the whole, how many girls and boys appeared for the interview?
A.150 girls; 200 boys
B.120 girls; 180 boys
C.120 girls; 160 boys
D.100 girls; 120 boys

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Last updated date: 20th Jun 2024
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Answer
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Hint: Here, we will assume the number of girls to be \[x\]. We will then frame a linear equation based on the given information. By solving this equation, we will get the value of \[x\] which is the number of girls. We will then subtract the number of students from the total number of students to get the number of boys. Hence, we will get the required number of girls and boys who appeared for the interview.

Complete step-by-step answer:
Number of students who appeared for the interview \[ = 280\]
Let the number of Girls who appeared for the interview \[ = x\]
Hence, the number of boys who appeared for the interview \[ = \left( {280 - x} \right)\]
Now, according to the question,
 \[50\% \] of girls were declared successful
Hence, successful girls \[ = \dfrac{{50}}{{100}}x = \dfrac{x}{2}\]
Also, \[6\dfrac{1}{4}\% \] or \[\dfrac{{25}}{4}\% \] of boys were declared successful
Hence, successful boys \[ = \dfrac{{25}}{{4 \times 100}}\left( {280 - x} \right) = \dfrac{{\left( {280 - x} \right)}}{{16}}\]
Now, it is given that such students are \[25\% \] on the whole
Hence, when we will add these successful students we will get \[25\% \] of total students
 \[ \Rightarrow \dfrac{x}{2} + \dfrac{{\left( {280 - x} \right)}}{{16}} = \dfrac{{25}}{{100}} \times 280 = \dfrac{{280}}{4}\]
Taking LCM and solving further, we get
 \[ \Rightarrow \dfrac{{8x + 280 - x}}{4} = 280\]
Multiplying both sides by 4, we get
 \[ \Rightarrow 7x + 280 = 4 \times 280 = 1120\]
 \[ \Rightarrow 7x = 1120 - 280 = 840\]
Dividing both sides by 7, we get
 \[ \Rightarrow x = 120\]
Therefore, the number of Girls who appeared for the interview \[ = x = 120\]
And, the number of boys who appeared for the interview \[ = \left( {280 - x} \right) = 280 - 120 = 160\]
Hence, 120 girls and 160 boys appeared for the interview.
Therefore, option C is the correct answer.

Note: We can also solve this question with the help of mixtures and allegations.
By allegations, the required ratio is: \[h - d:d - l\] , where \[h\] is the higher strength, \[l\] is the lower strength and \[d\] is the desired strength.
Now, according to the question,
 \[50\% \] of girls were declared successful
And, \[6\dfrac{1}{4}\% \] or \[\dfrac{{25}}{4}\% \] of boys were declared successful
Also, it is given that such students are \[25\% \] on the whole
Hence, in the formula of allegations, we will substitute \[l = \dfrac{{25}}{4}\] , \[h = 50\] and \[d = 25\],
Therefore by alligation method,
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Hence, by the allegation method, we know that, \[h - d:d - l\]
 \[ \Rightarrow 50 - 25:25 - \dfrac{{25}}{4} = 25:\dfrac{{75}}{4} = 100:75 = 4:3\] is the ratio of Boys and girls
Therefore, the ratio of girls and boys who appeared for the interview is \[3:4\]
Therefore, number of girls who appeared \[ = \dfrac{3}{7} \times 280 = 3 \times 40 = 120\]
And number of boys who appeared \[ = \dfrac{4}{7} \times 280 = 4 \times 40 = 160\]
Hence, 120 girls and 160 boys appeared for the interview.
Therefore, option C is the correct answer.