Question

If a group of students having an average age of 16 years joined a class, the average age of all the students in the class reduces from 18 years to 17 years. What is the ratio of the number of students who joined the class to the number of students who were initially in the class?

Hint:

The method of alligation is a very fast technique for problems involving the mixing of two groups with different averages. Notice that the final average (17) is exactly in the middle of the two group averages (16 and 18). This implies that the two groups must have been of equal size for their average to balance perfectly in the middle.

Answer 1

Step 1: Understanding the Concept: 
This is a problem about averages, specifically the combination of two groups. The final average of the combined group is a weighted average of the individual group averages. We can set up an equation relating the sum of ages before and after the new students join. Alternatively, this can be solved efficiently using the method of alligation. 
Step 2: Key Formula or Approach: 
Method 1: Using the definition of average 
Let \(n_1\) be the initial number of students and \(n_2\) be the number of new students. 
Initial sum of ages = \(n_1 \times 18\). 
Sum of ages of new students = \(n_2 \times 16\). 
Final total number of students = \(n_1 + n_2\). 
Final sum of ages = \(18n_1 + 16n_2\). 
Final average age = \(\frac{18n_1 + 16n_2}{n_1 + n_2} = 17\). 
We need to solve this equation for the ratio \(n_2 : n_1\). 
Method 2: Alligation 
Alligation is a rule that enables us to find the ratio in which two or more ingredients at the given prices must be mixed to produce a mixture of a desired price. Here, we use 'age' instead of 'price'. 
We set up a diagram: 

Age of Group 1		Age of Group 2
18		16
	\(\searrow\) \(\swarrow\)
	Mean Age (17)
	\(\nearrow\) \(\nwarrow\)
(17 - 16)		(18 - 17)
1		1

The ratio of the quantities (number of students) is the inverse of the ratio of the differences. 
Ratio \(n_1 : n_2\) = \((17-16) : (18-17)\). 
Step 3: Detailed Explanation: 
Using Method 1 (Algebraic Equation): 
\(\frac{18n_1 + 16n_2}{n_1 + n_2} = 17\) 
Multiply both sides by \((n_1 + n_2)\): 
\(18n_1 + 16n_2 = 17(n_1 + n_2)\) 
\(18n_1 + 16n_2 = 17n_1 + 17n_2\) 
Now, group the terms with \(n_1\) on one side and the terms with \(n_2\) on the other. 
\(18n_1 - 17n_1 = 17n_2 - 16n_2\) 
\(n_1 = n_2\) 
This means the number of initial students is equal to the number of new students. 
The ratio of the number of students who joined (\(n_2\)) to the number of students who were initially in the class (\(n_1\)) is: 
\(\frac{n_2}{n_1} = \frac{n_1}{n_1} = \frac{1}{1}\) 
So, the ratio is 1:1. 
Using Method 2 (Alligation): 
The difference between the initial average age (18) and the final average age (17) is \(18-17=1\). 
The difference between the new students' average age (16) and the final average age (17) is \(17-16=1\). 
The rule of alligation states that the ratio of the number of initial students (\(n_1\)) to the number of new students (\(n_2\)) is the inverse of the ratio of these differences. 
\(\frac{n_1}{n_2} = \frac{17 - 16}{18 - 17} = \frac{1}{1}\) 
This gives \(n_1 : n_2 = 1:1\). 
The question asks for the ratio of the number of students who joined (\(n_2\)) to the number of students who were initially in the class (\(n_1\)), which is also \(n_2 : n_1 = 1:1\). 
Step 4: Final Answer: 
The required ratio is 1:1.