A class of 200 students is split into 3 groups such that 2 of them are equal in number and the last one (which is the smallest) is 10 less than each of the other groups. How many students are in this (smallest) group?

Step 1: Define variables for the number of students in each group

We first assign variables to represent the unknown number of students in each group. We are told two groups are equal in size, and the third group is smaller. Let $x$ be the size of the two equal groups and $y$ be the size of the smallest group.

Step 2: Formulate an equation based on the given relationships

The problem states that the smallest group is 10 less than each of the other groups. This translates into the equation $y = x - 10$.

Step 3: Formulate an equation for the total number of students

The total number of students in the class is 200. Since there are two groups of size $x$ and one group of size $y$, their sum must equal 200. This gives us the equation $x + x + y = 200$.

Step 4: Simplify the total number of students equation

Combine the like terms in the equation from the previous step to simplify it to $2x + y = 200$. This makes the equation easier to work with.

Step 5: Substitute the expression for y into the simplified equation

Now we substitute the expression for $y$ from the first relationship ($y = x - 10$) into the simplified total number of students equation. This creates a single equation with only one variable, $x$.

Step 6: Solve for x

Combine the $x$ terms and isolate $x$ to find its value. First, add 10 to both sides: $2x + x - 10 = 200 \implies 3x - 10 = 200 \implies 3x = 210$. Then, divide by 3: $x = \frac{210}{3} \implies x = 70$.

Step 7: Calculate the number of students in the smallest group (y)

Since we found $x = 70$, we can now use the relationship $y = x - 10$ to find the value of $y$, which represents the number of students in the smallest group. Substituting $x=70$ gives $y = 70 - 10 = 60$.

Step 8: Verify the solution

To ensure our solution is correct, we verify that the total number of students adds up to 200. Two groups have 70 students each, and the smallest group has 60 students. $70 + 70 + 60 = 200$, which matches the given total, confirming our answer.

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