Mala saves money every month in a till, starting with $500$ in the first month, $520$ in the second month, $540$ in the third month, and so on. Ramani also saves money in a different till, starting with $150$ in the first month, $190$ in the second month, $230$ in the third month, and so on. We need to find after how many months from the start will the total amount of money saved by both of them be equal.

1. Problem Description

Mala saves money every month in a till, starting with $500$ in the first month, $520$ in the second month, $540$ in the third month, and so on. Ramani also saves money in a different till, starting with $150$ in the first month, $190$ in the second month, $230$ in the third month, and so on. We need to find after how many months from the start will the total amount of money saved by both of them be equal.

2. Solution Steps

Let $n$ be the number of months.

Mala's savings form an arithmetic progression with first term $a_1 = 500$ and common difference $d_1 = 20$. The total savings of Mala after $n$ months is given by the sum of the arithmetic progression:

$S_{Mala} = \frac{n}{2} [2a_1 + (n-1)d_1]$

$S_{Mala} = \frac{n}{2} [2(500) + (n-1)(20)]$

$S_{Mala} = \frac{n}{2} [1000 + 20n - 20]$

$S_{Mala} = \frac{n}{2} [980 + 20n]$

$S_{Mala} = n(490 + 10n)$

$S_{Mala} = 10n^2 + 490n$

Ramani's savings form an arithmetic progression with first term $b_1 = 150$ and common difference $d_2 = 40$. The total savings of Ramani after $n$ months is given by the sum of the arithmetic progression:

$S_{Ramani} = \frac{n}{2} [2b_1 + (n-1)d_2]$

$S_{Ramani} = \frac{n}{2} [2(150) + (n-1)(40)]$

$S_{Ramani} = \frac{n}{2} [300 + 40n - 40]$

$S_{Ramani} = \frac{n}{2} [260 + 40n]$

$S_{Ramani} = n(130 + 20n)$

$S_{Ramani} = 20n^2 + 130n$

We want to find the number of months $n$ such that the total savings are equal:

$S_{Mala} = S_{Ramani}$

$10n^2 + 490n = 20n^2 + 130n$

$0 = 10n^2 - 360n$

$0 = 10n(n - 36)$

So, $n = 0$ or $n = 36$.

Since $n$ must be greater than 0, $n = 36$.

3. Final Answer

After 36 months, the total amount of money saved by both of them will be equal.