Question

There are 816 students in enrolled at a certain high school. Each of these students is taking at least one of the subjects economics, geography, and biology. The sum of the number of students taking exactly one of these subjects and the number of students taking all 3 of these subjects is 5 times the number of students taking exactly 2 of these subjects. The ratio of the number of students taking only the two subjects economics and geography to the number of students taking only the two subjects economics and biology to the number of students taking only the two subjects geography and biology is \(3: 6: 8 .\) How many of the students enrolled at this high school are taking only the two subjects geography and biology? a. 35 b. 42 c. 64 d. 136 e. 240

Short answer

64 students are taking only geography and biology.

Step-by-step solution

Define Variables

Let E, G, B represent the subjects economics, geography, and biology. Define the variables: x - Number of students taking exactly two subjects y - Number of students taking exactly one subject z - Number of students taking all three subjectsThe given ratio for the students taking exactly 2 subjects is: let a = students taking only economics and geography b = students taking only economics and biology c = students taking only geography and biology The ratio of a:b:c = 3:6:8 means a = 3k, b = 6k, c = 8k

Use Total Students Condition

The total number of students is 816: y + 3k + 6k + 8k + z = 816. This simplifies to: y + 17k + z = 816

Use Given Ratio Relation

According to the problem, the sum of the number of students taking exactly one subject and the number of students taking all 3 subjects is 5 times the number taking exactly 2 subjects: y + z = 5x x = a + b + c = 3k + 6k + 8k = 17k Therefore, the relation becomes: y + z = 5 * 17k = 85k

Solve System of Equations

Solve the simultaneous equations found in steps 2 and 3: y + 17k + z = 816 y + z = 85k Subtract the second equation from the first: (y + 17k + z) - (y + z) = 816 - 85k 17k = 816 - 85k Combining like terms results in: 102k = 816 k = 816 / 102 = 8

Calculate Number of Students in Each Category

With k=8: a = 3k = 3 * 8 = 24 b = 6k = 6 * 8 = 48 c = 8k = 8 * 8 = 64 These are the students taking exactly two subjects. The problem asks for the number of students taking only geography and biology, which is c.

Conclusion

The number of students taking only the two subjects geography and biology is 64.

Key concepts

students and subjects

Understanding the relationship between students and subjects is crucial in solving GMAT problems involving multiple categories. In this exercise, we are given a high school with 816 students who are enrolled in at least one of three subjects: economics, geography, and biology. This setup forms the basis of our problem, where we need to categorize students based on their enrolment in any combination of these subjects. The first step is defining variables to represent the different groups of students, making it easier to apply algebraic methods to organize and solve the problem. By breaking down the students into those taking one, two, or all three subjects, we establish a more comprehensive understanding of the interplay between the students and their chosen subjects.

ratio problems

Ratio problems are a common type of question on the GMAT, requiring the use of proportional reasoning to find relationships between different groups. In this exercise, the ratio of students taking two specific subjects is given as 3:6:8. This ratio helps break down the problem into manageable parts. By assigning a variable 'k' to represent these ratios, we can translate the ratio into concrete numbers of students. This step simplifies the problem, as the future calculations become straightforward multiplications of 'k'. The critical part is recognizing that ratios can serve as multipliers, easing the resolution of more complex algebraic equations.

system of equations

Solving a system of equations is essential when dealing with interconnected variables. Here, we start with two main equations derived from the problem statement. The first equation, \(y + 17k + z = 816\), represents the total number of students. The second equation, \(y + z = 85k\), comes from the relationship given between students taking exactly one or all three subjects and those taking exactly two subjects. By solving these equations simultaneously, we isolate 'k', enabling us to find specific numbers for each student category. The process involves subtracting one equation from the other to eliminate variables and simplify the solution. Once 'k' is found, we use it to determine the exact number of students in each category.

algebraic expressions

Algebraic expressions are fundamental for representing and solving problems involving unknown quantities. In this exercise, expressions like \(3k, 6k, 8k\) denote the number of students taking specific pairs of subjects. Using algebraic forms such as \(y + z = 85k\) and \(y + 17k + z = 816\), we can organize information and formulate equations. These expressions compactly encapsulate relationships between different student groups, making it easier to apply mathematical operations. Mastery of algebraic expressions allows for neat and efficient problem-solving, which is essential for tackling complex GMAT questions.