Answer :
To solve the given problems, let's break them down step by step:
Rice Eaten by Rats:
The total amount of rice in the bag is [tex]112\frac{1}{2}[/tex] kg. To find out how much rice was eaten by rats, we need to calculate two-fifth ([tex]\frac{2}{5}[/tex]) of the total rice.
First, convert the mixed fraction [tex]112\frac{1}{2}[/tex] into an improper fraction:
[tex]112\frac{1}{2} = \frac{225}{2}[/tex]
Now, calculate [tex]\frac{2}{5}[/tex] of [tex]\frac{225}{2}[/tex]:
[tex]\frac{2}{5} \times \frac{225}{2} = \frac{2 \times 225}{5 \times 2} = \frac{450}{10} = 45\ ext{kg}[/tex]
So, the amount of rice eaten by rats is 45 kg.
Man's Initial Income:
Let's denote the man's initial income as [tex]x[/tex]. According to the problem, he spends [tex]\frac{2}{5}[/tex] of his income and has ₹120 left.
This means that after spending, he has [tex]\frac{3}{5}[/tex] of his income left (since [tex]1 - \frac{2}{5} = \frac{3}{5}[/tex]).
Therefore, we can set up the equation:
[tex]\frac{3}{5}x = 120[/tex]
Solve for [tex]x[/tex] by multiplying both sides by 5:
[tex]3x = 120 \times 5[/tex]
[tex]3x = 600[/tex]
Now, divide both sides by 3:
[tex]x = \frac{600}{3} = 200[/tex]
The man's initial income was ₹200.
In summary, 45 kg of rice was eaten by rats, and the man's initial income was ₹200.
