To determine the total number of homework exercises the boy had, let's denote the total number of exercises by [tex]\( x \)[/tex].
According to the problem:
1. The boy does [tex]\( \frac{1}{4} \)[/tex] of his total exercises in the evening.
2. He does [tex]\( \frac{2}{3} \)[/tex] of his total exercises in the next morning.
3. After doing these exercises, he is still left with 2 exercises.
Let's express these pieces of information mathematically:
1. Exercises done in the evening: [tex]\( \frac{x}{4} \)[/tex]
2. Exercises done in the morning: [tex]\( \frac{2x}{3} \)[/tex]
3. Remaining exercises: 2
Since the total number of exercises [tex]\( x \)[/tex] is the sum of the exercises done in the evening, the exercises done in the morning, and the remaining exercises, we can set up the following equation:
[tex]\[
\frac{x}{4} + \frac{2x}{3} + 2 = x
\][/tex]
We'll solve this equation step-by-step:
1. Find a common denominator for the fractions [tex]\( \frac{x}{4} \)[/tex] and [tex]\( \frac{2x}{3} \)[/tex], which is 12.
[tex]\[
\frac{x}{4} = \frac{3x}{12} \quad \text{and} \quad \frac{2x}{3} = \frac{8x}{12}
\][/tex]
2. Substitute these into the equation:
[tex]\[
\frac{3x}{12} + \frac{8x}{12} + 2 = x
\][/tex]
3. Combine the fractions on the left side:
[tex]\[
\frac{3x + 8x}{12} + 2 = x \implies \frac{11x}{12} + 2 = x
\][/tex]
4. To clear the fraction, multiply every term by 12:
[tex]\[
11x + 24 = 12x
\][/tex]
5. Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side:
[tex]\[
12x - 11x = 24 \implies x = 24
\][/tex]
So, the total number of exercises he had for homework is [tex]\(\boxed{24}\)[/tex].
