Question

Only tenth, eleventh and twelfth-grade students attend Washington High School. The ratio of tenth graders to the school’s total student population is 86: 255, and the ratio of eleventh graders to the school’s total student population is 18: 51. If a student is selected at random from the entire school, the grade in which the student is most likely to be is: -

Answer 1

Hint: Assume the total number of students as ‘x’. Find the number of students studying in \[{{10}^{th}}\] grade by multiplying x with the ratio 86: 255. Similarly, multiply x with the ratio 18 : 51 to find the total number of students in \[{{12}^{th}}\] grade, take the sum of number of students in \[{{10}^{th}}\] and \[{{11}^{th}}\] grade and subtract it from x. Now, check the grade in which there are maximum students to get the answer.

Complete step-by-step solution
Let us assume the total number of students attending Washington High School is ‘x’.
Now, it is given that the ratio of tenth graders to the school’s total student population is 86: 255. Therefore, we have,
\[\Rightarrow \] (Number of students in \[{{10}^{th}}\] grade / Total number of students) = \[\dfrac{86}{255}\]
\[\Rightarrow \] (Number of students in \[{{10}^{th}}\] grade / x) = \[\dfrac{86}{255}\]
\[\Rightarrow \] Number of students in \[{{10}^{th}}\] grade = \[\dfrac{86x}{255}\]
Now, it is given that the ratio of eleventh graders to the school’s total student population is 18: 51. Therefore, we have,
\[\Rightarrow \] (Number of students in \[{{11}^{th}}\] grade / Total number of students) = \[\dfrac{18}{51}\]
\[\Rightarrow \] (Number of students in \[{{11}^{th}}\] grade / x) = \[\dfrac{18}{51}\]
\[\Rightarrow \] Number of students in \[{{11}^{th}}\] grade = \[\dfrac{18x}{51}\]
Multiplying R.H.S with \[\dfrac{5}{5}\], we get,
\[\Rightarrow \] Number of students in \[{{11}^{th}}\] grade = \[\dfrac{18x}{51}\times \dfrac{5}{5}=\dfrac{90x}{255}\]
So, the total number of students in twelth grade will be the difference between the total number of students in the school and the sum of total students in grades eleventh and tenth.
\[\Rightarrow \] Number of students in \[{{12}^{th}}\] grade = \[x-\left( \dfrac{86x}{255}+\dfrac{90x}{255} \right)\]
\[\Rightarrow \] Number of students in \[{{12}^{th}}\] grade = \[x-\dfrac{176x}{255}\]
\[\Rightarrow \] Number of students in \[{{12}^{th}}\] grade = \[\dfrac{255x-176x}{255}\]
\[\Rightarrow \] Number of students in \[{{12}^{th}}\] grade = \[\dfrac{79x}{255}\]
Now, if a student is selected at random from the entire school then that student will be more likely to be from that grade in which there are a maximum number of students. Clearly, we can see that \[\dfrac{90x}{255}>\dfrac{86x}{255}>\dfrac{79x}{255}\]. Therefore, the selected student is more likely to be from eleventh grade.
Hence, option (b) is the correct answer.

Note: One may note that while calculating the number of students in \[{{11}^{th}}\] grade. We multiplied \[\dfrac{18}{51}\] with \[\dfrac{5}{5}\]. This was done to make the denominator equal to 255 so that the comparison of the number of students became easy. Actually, the above question uses the concept of probability. The given ratios are nothing but the probability of students studying in that particular grade.