In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes

Question:

In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes

A. 12 : 7
B. 10 : 7
C. 8 : 7
D. 4 : 3

Core Concept

This question deals with the concept of Ratios and Proportions in the context of a real-life scenario (students in a school). A ratio represents the relationship between two quantities, and proportions are used when these quantities are compared relative to each other.

Key Concepts:

1. Ratio: The given ratio of boys to girls is 8 : 5. This means for every 8 boys, there are 5 girls.
2. Proportion: The ratio is adjusted after the addition of 22 more girls, so we need to use the principle of proportions to calculate the new ratio of boys to girls.
3. Total Students: The total number of students in the school is the sum of boys and girls, which is 286.

Formula:

Let the number of boys be $8x$ and the number of girls be $5x$, where $x$ is a common multiplier.

Then:

$$8x + 5x = 286$$

Solving for $x$ will help determine the number of boys and girls before the additional girls are admitted. After adding 22 more girls, the new ratio will need to be calculated.

Step-by-Step Solution

Step 1: Let the number of boys and girls be in terms of $x$

The total number of students is 286, and the ratio of boys to girls is 8 : 5. Therefore, we can express the number of boys and girls as:

$$\text{Number of boys} = 8x \quad \text{and} \quad \text{Number of girls} = 5x$$

The total number of students is:

$$8x + 5x = 286$$

Step 2: Solve for $x$

Simplify the equation:

$$13x = 286$$

Now, solve for $x$:

$$x = \frac{286}{13} = 22$$

Step 3: Calculate the number of boys and girls

Now that we know $x = 22$, we can calculate the number of boys and girls:

$$\text{Number of boys} = 8x = 8 \times 22 = 176$$ $$\text{Number of girls} = 5x = 5 \times 22 = 110$$

Step 4: Add 22 more girls

The number of girls increases by 22, so the new number of girls is:

$$\text{New number of girls} = 110 + 22 = 132$$

Step 5: Calculate the new ratio of boys to girls

The new number of boys is still 176. The new number of girls is 132. Therefore, the new ratio of boys to girls is:

$$\text{New ratio} = \frac{\text{Number of boys}}{\text{Number of girls}} = \frac{176}{132}$$

Simplify the ratio:

$$\frac{176}{132} = \frac{44}{33} = \frac{4}{3}$$

Thus, the new ratio of boys to girls is 4 : 3.

Correct Answer and Option

The correct answer is:
D. 4 : 3

Answer Reasoning

After adding 22 more girls, the ratio of boys to girls becomes 4 : 3. The calculation shows that the total number of boys remains 176, while the number of girls increases to 132, resulting in the ratio 4 : 3.

Explanation of Incorrect Options

• A. 12 : 7: This is incorrect because the ratio does not simplify to 12 : 7 after adding the 22 girls.
• B. 10 : 7: This is incorrect because the numbers do not fit the ratio after the adjustment.
• C. 8 : 7: This is incorrect because the original ratio of boys to girls (8 : 5) doesn’t lead to the final ratio being 8 : 7 after the increase in the number of girls.

Common Mistakes to Avoid

1. Forgetting to adjust the number of girls after the increase: Some students may miss adding the 22 girls to the original number of girls, which will result in an incorrect ratio.
2. Misinterpreting the ratio: Students may confuse the ratio of boys to girls before and after the addition, leading to errors in simplification.

Simplest Way to Solve (Shortcut or Insight)

A quicker method is to first calculate the initial number of boys and girls using the ratio and total number of students. Then, simply add the 22 girls and calculate the new ratio. You can directly use the LCM or GCD method to simplify the resulting ratio quickly.

Visual Aid Suggestion

Here’s a simple table that visualizes the breakdown of the calculation:

| Number of Students | Boys (8x) | Girls (5x) | Total Students | New Girls (22) | New Girls Total | New Ratio (Boys : Girls) |
|---------------------|-----------|------------|----------------|----------------|-----------------|-------------------------|
| Before              | 176       | 110        | 286            |                |                 |                         |
| After               | 176       | 132        | 286 + 22       | 22             | 132             | 4 : 3                   |

This table helps clarify the step-by-step adjustment of the number of girls and the new ratio calculation.

Application

The concept of ratios and adjusting proportions is widely used in real-life situations such as budgeting, resource distribution, and demographic studies. For example, if you have a group of students in a school, and the ratio of boys to girls changes over time, you can use this same principle to quickly determine how the group’s makeup changes. Understanding ratios can help in making decisions about resource allocation, like how to distribute materials in proportion to the number of students of each gender in a school.

Practice Questions

1. In a school, the ratio of boys to girls is 3 : 4. If there are 315 students in total, how many boys are there?
   • A. 135
   • B. 120
   • C. 105
   • D. 150
     (Answer: A)
2. In a class of 200 students, the ratio of boys to girls is 5 : 3. How many girls are there in the class?
   • A. 60
   • B. 80
   • C. 90
   • D. 100
     (Answer: B)
3. The ratio of boys to girls in a school is 7 : 5. If there are 420 students, how many girls are there?
   • A. 140
   • B. 175
   • C. 200
   • D. 250
     (Answer: B)
4. The ratio of boys to girls in a class is 2 : 3. If there are 150 students in total, how many boys are there?
   • A. 60
   • B. 75
   • C. 90
   • D. 100
     (Answer: A)
5. In a school, the ratio of boys to girls is 5 : 7. If the total number of students is 240, how many boys are there?
   • A. 100
   • B. 120
   • C. 140
   • D. 150
     (Answer: B)