A student gets an aggregate of 60% marks in five subjects in the ratio $10:9:8:7:6$. If the passing marks are 50% of the maximum marks and each subject has the same maximum marks, in how many subjects did he pass the exam? (a) 2 (b) 3 (c) 4 (d) 5
ANSWER
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Hint: We will first form a relation between marks obtained in each subject and passing marks of each subject. Then we will use that relation to check if the student passed the subject of not.
Complete Step-by-Step solution: We are given that marks obtained by a student in five different subjects are in the ratio $10:9:8:7:6$. So, there will be a factor which will be multiplied by all the terms of the given ratio to obtain actual marks of the student in each subject. Let this factor be $x$. So, the marks obtained by student in five subjects will be $10x,9x,8x,7x,6x.$ Therefore, the total marks of the student = $10x+9x+8x+7x+6x=40x.$ Let, maximum marks in each subject be $y$. Therefor, maximum of total marks $=y+y+y+y+y=5y.$ Also, percentage of marks is given by the formula, \[\text{Percentage of marks= }\dfrac{\text{Total marks obtained}}{\text{Maximum total marks}}\times 100%\cdots \cdots \left( i \right)\]. Here, we have, Percentage of marks of the student = $60%.$ Total marks of the student = $40x.$ Maximum total marks of the student = $5y.$ Therefore, putting all this values in equation $\left( i \right)$, we get, $60%=\dfrac{40x}{5y}\times 100%$ $\Rightarrow 60=\dfrac{4000x}{5y}$ $\Rightarrow 60=\dfrac{800x}{y}$ Multiplying $y$ on both sides of the equation, we get, $\Rightarrow 60y=800x$ Dividing 60 on both sides of the equation, we get, $\Rightarrow y=\dfrac{800x}{60}$ $\Rightarrow y=\dfrac{40}{3}x\cdots \cdots \left( ii \right)$ Now, according to question, Passing marks of each subject = $50%\times $ maximum marks of the subject. \[=50%\times y\] $=\dfrac{50}{100}\times y$ $=\dfrac{1}{2}y.$ Substituting value of $y$ from equation $\left( ii \right)$, we get, Passing marks of each subject $=\dfrac{1}{2}\times \dfrac{40}{3}x=\dfrac{20}{3}x=6.67x$. Therefore, marks obtained in four of the subjects, which is, $10x,9x,8x,7x$ is more than passing marks. Hence, the student passed in 4 subjects. Therefore, the correct answer is option (c).
Note: In this question you may get struct in finding the value of $x$. Avoid doing that, as here we donâ€™t need an exact value of $x$ because the question is not to find the marks obtained, but just to find if a student passed the subject or not.