To determine Rohit's monthly income, we need to take into account the given fractions of his income that he spends on various expenses, as well as the percentage of his income that he is left with.
Let's denote Rohit's total monthly income by [tex]\( x \)[/tex].
Given:
- Rohit donates [tex]\( \frac{1}{5} \)[/tex] of his income.
- He spends [tex]\( \frac{1}{4} \)[/tex] of his income on food.
- He spends [tex]\( \frac{1}{3} \)[/tex] of his income on rent.
- He spends [tex]\( \frac{1}{15} \)[/tex] of his income on other expenses.
- He is left with [tex]\( 9000\% \)[/tex] of his income.
First, convert the percentage of the income he is left with into a fraction. [tex]\( 9000\% \)[/tex] is equivalent to [tex]\( \frac{9000}{100} = 90 \)[/tex].
The sum of all these fractions should equal his total income. We can write:
[tex]\[
\frac{1}{5}x + \frac{1}{4}x + \frac{1}{3}x + \frac{1}{15}x + 90x = x
\][/tex]
Combine the fractions on the left-hand side:
[tex]\[
\left( \frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{15} + 90 \right)x = x
\][/tex]
First, we find a common denominator for the fractions:
[tex]\[
\frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{15} = \frac{12}{60} + \frac{15}{60} + \frac{20}{60} + \frac{4}{60} = \frac{51}{60}
\][/tex]
Adding 90 to the sum of these fractions:
[tex]\[
\frac{51}{60} + 90 = \frac{51}{60} + \frac{5400}{60} = \frac{5451}{60}
\][/tex]
Thus, the equation becomes:
[tex]\[
\left( \frac{5451}{60} \right) x = x
\][/tex]
To solve this, we set the coefficients equal to each other:
[tex]\[
\frac{5451}{60} = 1
\][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[
5451 = 60
\][/tex]
Given the numerical result, we observe:
[tex]\[
5451 = 60 \times \text{{Rohit's income}}
\][/tex]
From this setup, it converts [tex]\( x \)[/tex] altogether:
Hence, [tex]\( x = 0 \)[/tex]
Now, let's verify the solution with the individual amounts for each category. We found the following:
- Rohit's donation amount: [tex]\( 0 \)[/tex]
- Amount spent on food: [tex]\( 0 \)[/tex]
- Amount spent on rent: [tex]\( 0 \)[/tex]
- Amount spent on other expenses: [tex]\( 0 \)[/tex]
- Amount left over: [tex]\( 0 \)[/tex]
So, the total salary of [tex]\( Rohit \)[/tex] is [tex]\( 0 \)[/tex].
