joel had $80 less than hazel.Hazel spent $128 on a vacuum cleaner.As a result,Joel had 4 times as much money as hazel.How much money did both of them have altogether at first

Answer 1

Let \( J \) be the amount of money Joel had and \( H \) be the amount of money Hazel had initially.

From the information given, we can set up two equations:

1. \( J = H - 80 \) (Joel had $80 less than Hazel)

2. After Hazel spent $128 on a vacuum cleaner, her remaining money is \( H - 128 \).

   According to the problem, after this purchase, Joel had 4 times as much money as Hazel: \[ J = 4(H - 128) \]

Now, we can substitute the expression for \( J \) from the first equation into the second equation:

\[ H - 80 = 4(H - 128) \]

Now, let's solve for \( H \):

1. Expand the right side: \[ H - 80 = 4H - 512 \]

2. Rearranging gives: \[ -80 + 512 = 4H - H \] \[ 432 = 3H \]

3. Solving for \( H \): \[ H = \frac{432}{3} = 144 \]

Now that we have \( H \), we can find \( J \):

\[ J = H - 80 = 144 - 80 = 64 \]

Now, let's find the total amount of money both had together initially:

\[ H + J = 144 + 64 = 208 \]

Thus, the total amount of money that both of them had altogether at first is \(\boxed{208}\).