In a college, the ratio of the number of boys to girls is 3 ∶ 8. If there are 180 boys, then how many girls are in the college?
480
Understanding Ratios: Boys and Girls in College
The question provides us with the ratio of the number of boys to girls in a college, which is given as 3 ∶ 8. We are also told that the actual number of boys in the college is 180. Our goal is to find out the total number of girls in the college based on this information.
A ratio compares the relative sizes of two or more values. In this case, the ratio 3 ∶ 8 means that for every 3 boys, there are 8 girls.
Calculating the Number of Girls using the Ratio
We can represent the given ratio and the actual numbers as a proportion. Let 'G' be the number of girls we want to find. The ratio of boys to girls can be written as a fraction:
\(\frac{\text{Number of Boys}}{\text{Number of Girls}} = \frac{3}{8}\)
We know the number of boys is 180, and we are looking for the number of girls (G). So, we can set up the proportion:
\(\frac{180}{G} = \frac{3}{8}\)
To solve for G, we can use cross-multiplication. Multiply the numerator of one fraction by the denominator of the other fraction and set them equal:
\(180 \times 8 = 3 \times G\)
\(1440 = 3G\)
Now, to find G, we need to divide both sides of the equation by 3:
\(G = \frac{1440}{3}\)
\(G = 480\)
So, there are 480 girls in the college.
Step-by-Step Solution
- Identify the given ratio of boys to girls: 3 ∶ 8.
- Identify the actual number of boys: 180.
- Let the unknown number of girls be G.
- Set up a proportion comparing the ratio and the actual numbers: \(\frac{180}{G} = \frac{3}{8}\).
- Cross-multiply: \(180 \times 8 = 3 \times G\).
- Simplify: \(1440 = 3G\).
- Solve for G by dividing both sides by 3: \(G = \frac{1440}{3}\).
- Calculate the result: \(G = 480\).
Therefore, the number of girls in the college is 480.
Revision Table: Ratio and Proportion
| Concept | Description | Example (Boys:Girls = 3:8) |
|---|---|---|
| Ratio | A comparison of two quantities. Can be written as a:b, a/b, or "a to b". | 3 ∶ 8 |
| Proportion | An equation stating that two ratios are equal. Used to find an unknown quantity when one ratio and part of the other are known. | \(\frac{3}{8} = \frac{180}{480}\) |
| Cross-Multiplication | A method used to solve proportions. If \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\). | From \(\frac{180}{G} = \frac{3}{8}\), we get \(180 \times 8 = 3 \times G\). |
Additional Information: Expanding on Ratio Problems
Understanding ratios is fundamental in many areas of mathematics and real life. Here are some related concepts:
- Finding Total Number: Once we know the number of boys (180) and girls (480), we can find the total number of students in the college by adding them: \(180 + 480 = 660\).
- Finding a Fraction of the Total: In the ratio 3:8, the total parts are \(3+8=11\). Boys make up \(\frac{3}{11}\) of the total students, and girls make up \(\frac{8}{11}\) of the total students. We can verify this with the total number of students (660):
- Number of boys = \(\frac{3}{11} \times 660 = 3 \times 60 = 180\) (Matches the given information).
- Number of girls = \(\frac{8}{11} \times 660 = 8 \times 60 = 480\) (Matches our calculated number of girls).
- Unit Rate Method: Another way to think about the ratio 3:8 is that for every 1 'part' in the ratio, there are a certain number of students. Since 3 parts represent 180 boys, each part is worth \(\frac{180}{3} = 60\) students. Since girls represent 8 parts, the number of girls is \(8 \times 60 = 480\).
These different approaches show how ratios can be used flexibly to solve problems involving proportional relationships.
Important Questions from Ratio and Proportion
If the ratio of three numbers A, B and C is 2 ∶ 3 ∶ 5, and the sum of the squares of these numbers is 3800, then the value of C is:
A bag contains ₹310 in the form of 5 rupee, 2 rupee and 1 rupee coins in the ratio 4 ∶ 3 ∶ 5. What is the number of 5 rupee coins?
The ratio of three numbers is 3 ∶ 5 ∶ 4 and the sum of their squares is 11250. Find the sum of the numbers.
When 'x' is subtracted from each of the numbers 22, 39, 56 and 107, then the resulting numbers, in this order, are in proportion. What is the mean proportional between (x + 3) and (3x - 7)?
The salaries of Ravi and Sumit are in the ratio 4 ∶ 5. If the salary of each is increased by Rs. 6,000 the new ratio becomes 35 ∶ 40. What will be Sumit's increased salary?
