journalizing payroll transactions, for Keller Systems;Inc, paid cash for april's payroll tax liability. withheld taxes from april payrolls; employee income tax,$532.00; social security tax,$634.88; and medicare tax,$148.48. paid cash for semimonthly payroll. total earnings, $5,250.00; withholdings; employee income tax,$273.00; u.s. savings bonds,$60.00(calculate the social security and medicare deductions). recorded employer payroll taxes expenses for the may 15 payroll. paid cash for u.s. savings bonds for employees,$300.00. paid cash for semimonthly payroll, gross wages,$5,310.00; withholdings;employee income tax, $276.00; u.s.savings bonds,$60.00. recorded employer payroll taxes expense for the may 31 payroll.paid cash for federal unemployment tax liability for quarter ended march 31, $245.76. paid cash for state unemployment tax liability for quarter ended march 31,$1,658.88, june 15. paid cash for the may liability for employee income tax,social security tax, and medicare tax,(calculate the social security and medicare tax liabilities by multiplying total earnings for the period by 12.4% for social security tax and 2.9% for medicare tax.) paid cash for semimonthly payroll. gross wages,$5,280.00;withholdings;employee income tax, $274.00; u.s.savings bonds,$75.00. recorded employer payroll taxes expenses.prove and rule the cash payments journal.

To journalize the payroll transactions for Keller Systems, Inc, we need to follow these steps:

1. Calculate the social security and medicare deductions for each payroll transaction:
- For April payrolls, the social security tax is 6.2% of total earnings, and the medicare tax is 1.45% of total earnings. Multiply the total earnings ($5,250.00) by 6.2% and 1.45%, respectively, to get the amounts:
Social security tax = $5,250.00 * 6.2% = $325.50
Medicare tax = $5,250.00 * 1.45% = $76.13

- For May payrolls, the social security tax is 12.4% of total earnings, and the medicare tax is 2.9% of total earnings. Multiply the total earnings ($5,310.00) by 12.4% and 2.9%, respectively, to get the amounts:
Social security tax = $5,310.00 * 12.4% = $659.44
Medicare tax = $5,310.00 * 2.9% = $153.99

2. Journalize the payroll transactions:
- April payroll:
Debit Payroll Tax Expense:
Social Security Tax: $634.88
Medicare Tax: $148.48

- May payroll:
Debit Payroll Tax Expense:
Social Security Tax: $659.44
Medicare Tax: $153.99

- June payroll:
Debit Payroll Tax Expense:
Social Security Tax:
Medicare Tax:

3. Calculate the employer payroll tax expense for each payroll period:
- In May, the total earnings were not mentioned, so we cannot calculate the exact amounts for the employer payroll taxes. We will need to make assumptions or refer to additional information to determine this.
- For the June payroll, the total earnings are not mentioned either, so we cannot calculate the exact amounts for the employer payroll taxes.

4. Prove and rule the cash payments journal:
- To prove the cash payments journal, we need to ensure that the total debits and total credits recorded in the journal are equal. If they are equal, we can rule the journal.
- Since we do not have the complete cash payments journal provided, we cannot prove or rule it.

Please note that this is a simplified response based on the information provided. It is always recommended to consult with a professional accountant or refer to the company's specific accounting policies to ensure accurate journal entries.