The number of students in 3 classes is in the ratio 2 : 3 : 4. If 12 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was

Question:

The number of students in 3 classes is in the ratio 2 : 3 : 4. If 12 students are increased in each class this ratio changes to 8 : 11 : 14. The total number of students in the three classes in the beginning was

A. 162
B. 108
C. 96
D. 54

Core Concept

This problem is based on Ratios and Proportions involving changes in quantities. We are asked to determine the initial total number of students in the classes before the increase, given that the ratio of students in the classes changes after adding 12 students to each class.

Key Concepts:

1. Ratios: Ratios represent the relationship between two or more quantities. Initially, the ratio of students in the three classes is 2 : 3 : 4. After adding 12 students to each class, the ratio changes to 8 : 11 : 14.
2. Proportions: We can set up proportional relationships based on the initial and final ratios to find the number of students in each class.
3. Solving Systems of Equations: By using the ratios, we can set up equations and solve them to find the initial number of students in each class.

Step-by-Step Solution

Step 1: Let the initial number of students in each class be represented by a variable

Let the initial number of students in the three classes be:

• $2x$ for the first class,
• $3x$ for the second class,
• $4x$ for the third class.

Here, $x$ is a common factor.

Step 2: After adding 12 students to each class

After adding 12 students to each class, the number of students becomes:

• First class: $2x + 12$,
• Second class: $3x + 12$,
• Third class: $4x + 12$.

Step 3: Set up the proportion based on the new ratio

We are given that the new ratio of students in the classes is 8 : 11 : 14. Therefore, we can set up the following proportion:

$$\frac{2x + 12}{8} = \frac{3x + 12}{11} = \frac{4x + 12}{14}$$

Step 4: Solve the proportion

To solve, we can compare the first two terms of the proportion:

$$\frac{2x + 12}{8} = \frac{3x + 12}{11}$$

Cross-multiply to solve for $x$:

$$11(2x + 12) = 8(3x + 12)$$ $$22x + 132 = 24x + 96$$

Simplify and solve for $x$:

$$22x – 24x = 96 – 132$$ $$-2x = -36$$ $$x = 18$$

Step 5: Calculate the initial number of students

Now that we know $x = 18$, we can calculate the initial number of students in each class:

• First class: $2x = 2 \times 18 = 36$,
• Second class: $3x = 3 \times 18 = 54$,
• Third class: $4x = 4 \times 18 = 72$.

Step 6: Total number of students

The total number of students in the three classes in the beginning is:

$$36 + 54 + 72 = 162$$

Correct Answer and Option

The correct answer is:
A. 162

Answer Reasoning

By setting up the initial ratios and applying the changes to the number of students, we derived a system of equations based on the proportions. Solving these equations gave us $x = 18$, which we then used to calculate the total number of students in the beginning as 162.

Explanation of Incorrect Options

• B. 108: This is incorrect because it does not match the result obtained by solving the proportion equation. The correct total is 162, not 108.
• C. 96: This is incorrect because the values derived from the given proportions lead to a total of 162 students, not 96.
• D. 54: This is incorrect as well, as it underestimates the total number of students in all three classes.

Common Mistakes to Avoid

1. Incorrectly applying the proportion: Some students may forget to set up the correct proportional relationship between the classes and the new ratio, leading to incorrect equations.
2. Not simplifying the equation correctly: Simplifying the equation after cross-multiplying is a critical step. Mistakes in simplifying terms can lead to an incorrect value for $x$.

Simplest Way to Solve (Shortcut or Insight)

The simplest way to solve this is by directly setting up the proportion based on the initial and final ratios, solving for $x$, and then calculating the total number of students by using the formula $2x + 3x + 4x$. This eliminates unnecessary steps and gives the result in a straightforward manner.

Visual Aid Suggestion

Here’s a simple table to visualize the setup and calculations:

| Class           | Initial Number of Students | Students after Increase | New Ratio |
|-----------------|----------------------------|-------------------------|-----------|
| First Class     | 2x = 36                    | 2x + 12 = 48            | 8         |
| Second Class    | 3x = 54                    | 3x + 12 = 66            | 11        |
| Third Class     | 4x = 72                    | 4x + 12 = 84            | 14        |
| Total           | 162                        | 198                     |           |

Explanation:

• Initial Number of Students: This shows the initial number of students in each class based on the variable $x$.
• Students after Increase: This column shows the new number of students after adding 12 students to each class.
• New Ratio: This shows the final ratio after the increase, which is used to verify the calculation.

This table helps visualize the relationship between the classes, their student counts, and how the total is derived after the increase.

Application

The concept of ratios and proportions is used in various real-world scenarios, such as resource allocation, project management, and financial distribution. For example, in budgeting, if a company allocates resources based on certain ratios, understanding how these ratios change with an increase or decrease in resources helps in making efficient decisions. Similarly, this method is useful in time management for dividing tasks or team distribution in a project.

Practice Questions

1. The number of students in 4 classes is in the ratio 3 : 5 : 7 : 9. If 5 students are added to each class, the ratio changes to 4 : 6 : 8 : 10. Find the total number of students initially.
   • A. 170
   • B. 180
   • C. 190
   • D. 200
     (Answer: B)
2. The ratio of the ages of A, B, and C is 4 : 5 : 6. If 5 years are added to each of their ages, the ratio becomes 6 : 7 : 8. Find their ages initially.
   • A. 20, 25, 30
   • B. 16, 20, 24
   • C. 12, 15, 18
   • D. 14, 18, 22
     (Answer: A)
3. The number of students in 3 classes is in the ratio 5 : 6 : 7. If 10 students are added to each class, the ratio becomes 6 : 7 : 8. Find the total number of students initially.
   • A. 120
   • B. 150
   • C. 180
   • D. 210
     (Answer: A)
4. In a box, the number of red balls, green balls, and blue balls is in the ratio 4 : 5 : 6. If 3 red balls, 4 green balls, and 5 blue balls are added, the ratio becomes 5 : 6 : 7. Find the initial number of balls in the box.
   • A. 40
   • B. 50
   • C. 60
   • D. 70
     (Answer: C)
5. If the ratio of the number of students in three groups is 2 : 3 : 4, and 5 students are added to each group, the ratio changes to 3 : 4 : 5. Find the total number of students in the beginning.
   • A. 150
   • B. 160
   • C. 170
   • D. 180
     (Answer: A)