Tom had $144 more than Ali at first. After Tom spent 25% of his money and Ali spent 1/3 of his money, Tom had $122 more than Ali. How much did Tom have at first?

Question

Tom had $144 more than Ali at first. After Tom spent 25% of his money and Ali spent 1/3 of his money, Tom had $122 more than Ali. How much did Tom have at first?

Answer 1

Let T = amount of money Tom had originally

A = amount of money Ali had originally

T = A + 144 (initially Tom had 144 dollars more than Ali).

If Tom spent 25% of his money, that means he has 75% (or 3/4) remaining.

(3/4)T = amount of money Tom has after spending 25% of his original holding.

Similarly Ali spend 1/3 if his money, so he has 2/3 remaining.

(2/3) A = amount of money Ali has remaining.

After the spending spree, Tom now has $122 more than Ali:

(3/4)T = (2/3)A + 122

We can solve the very first equation for A in terms of T and substitute into the last equation to have one equation containing variable T.

A = T - 144

So:

(3/4)T = (2/3)(T - 144) + 122

Multiply every term by 12 to clear the fractions:

9T = 8(T - 144) + 12(122)

9T = 8T - 8(144) + 12(122)

T = 12(122) - 8(144) = 312

T = 312 (money Tom had starting out)

Confirm:

A = 312 - 144 = 168 (amount Ali had initially)

After spending:

(3/4) 312 = 234

(2/3) 168 = 112

And 234 is 122 more than 112 (check)