In a class of 72 students, the number of boys is twice the number of girls. Find the number of boys.
48
Solving the Ratio Problem in a Class
This question asks us to find the number of boys in a class given the total number of students and the relationship between the number of boys and girls. We are told the total number of students is 72, and the number of boys is twice the number of girls.
Understanding the Given Information
- Total number of students = 72
- Number of boys = 2 × Number of girls
Setting up Equations
Let's use variables to represent the unknown quantities:
- Let \(g\) be the number of girls.
- Let \(b\) be the number of boys.
Based on the problem statement, we can write two equations:
- The total number of students is the sum of boys and girls: \[b + g = 72\]
- The number of boys is twice the number of girls: \[b = 2g\]
Solving for the Number of Girls and Boys
We can use the substitution method to solve this system of equations. Since we know that \(b\) is equal to \(2g\), we can substitute \(2g\) for \(b\) in the first equation:
Substitute \(b = 2g\) into \(b + g = 72\):
\[(2g) + g = 72\]
Combine the terms involving \(g\):
\[3g = 72\]
Now, to find the number of girls (\(g\)), divide both sides by 3:
\[g = \frac{72}{3}\]
\[g = 24\]
So, there are 24 girls in the class.
Now that we know the number of girls, we can find the number of boys using the relationship \(b = 2g\):
\[b = 2 \times 24\]
\[b = 48\]
Thus, there are 48 boys in the class.
Verification
Let's check if our numbers add up correctly:
- Number of boys = 48
- Number of girls = 24
- Total students = Boys + Girls = \(48 + 24 = 72\)
The total number of students is indeed 72, and the number of boys (48) is twice the number of girls (24), since \(48 = 2 \times 24\). The solution is consistent with the problem statement.
Comparing with Options
The calculated number of boys is 48. Let's look at the given options:
- Option 1: 48
- Option 2: 36
- Option 3: 18
- Option 4: 24
Our calculated number of boys (48) matches Option 1.
Summary of Findings
| Item | Quantity |
|---|---|
| Total Students | 72 |
| Number of Girls (\(g\)) | 24 |
| Number of Boys (\(b\)) | 48 |
| Relationship (\(b = 2g\)) | \(48 = 2 \times 24\) (Verified) |
The number of boys in the class is 48.
Revision Table for Ratio Problems
When solving ratio or proportion problems involving totals, remember these steps:
| Step | Description | Example (from this problem) |
|---|---|---|
| 1 | Identify the total quantity. | Total students = 72 |
| 2 | Identify the parts of the total and their relationship (ratio or multiple). | Parts: Boys and Girls. Relationship: Boys = 2 × Girls. |
| 3 | Assign variables to the unknown quantities. | Let girls = \(g\), Boys = \(b\). |
| 4 | Write equations based on the given information. | \(b + g = 72\), \(b = 2g\). |
| 5 | Solve the equations (often using substitution). | Substitute \(2g\) for \(b\) in \(b + g = 72\) to get \(3g = 72\), solving for \(g\). Then use \(g\) to find \(b\). |
| 6 | Check your answer against the original problem statement. | Does \(48 + 24 = 72\)? Yes. Is \(48 = 2 \times 24\)? Yes. |
Additional Information on Ratios and Proportions
Problems like this involve ratios and proportions, which are fundamental concepts in mathematics. A ratio is a comparison of two quantities. In this problem, the ratio of boys to girls is \(b:g\), which is \(48:24\), simplifying to \(2:1\).
We could also solve this problem using ratios directly:
- The ratio of boys to girls is 2:1.
- This means for every 2 boys, there is 1 girl.
- The total ratio parts are \(2 + 1 = 3\).
- These 3 parts represent the total 72 students.
- Each ratio part is worth \(72 / 3 = 24\) students.
- Number of girls = 1 part = \(1 \times 24 = 24\).
- Number of boys = 2 parts = \(2 \times 24 = 48\).
Understanding ratios provides an alternative way to solve such problems efficiently.
Important Questions from Quant Based Puzzle
A number is subtracted from 4 times of it and then, the number obtained is added to its (the resultant’s) next number. If this gives the answer as 91, what was the original number?
When twice of a number added to 3 is multiplied by 5 and added to the number itself, it gives 158. What is the square of that number?
Two years ago, T was twice as old as P. P is thrice as old as R. In five years, P will be 29. What is the present age of T?
When a number is added to its multiple of 5 and its square, the sum of these three numbers is 91. Find the number.
If the diagonal of a square is increased by 4 cm, its area increases by 56 cm 2. Find the ratio of the new area of the square to the initial area of the square.
