Question

Beginning in January of last year, Carl made deposits of $120intohisaccountonthe15thofeachmonthforseveralconsecutivemonthsandthenmadewithdrawalsof$50 from the account on the 15th of each of the remaining months of last year. There were no other transactions in the account last year. If the closing balance of Carl's account for May of last year was $2,600,whatwastherangeofthemonthlyclosingbalancesofCar{l}^{\prime }saccountlastyear?\left(1\right)LastyeartheclosingbalanceofCar{l}^{\prime }saccountforAprilwaslessthan$2,625.
(2) Last year the closing balance of Carl's account for June was less than $2,675.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (C)

Both statements together are sufficient, but neither statement alone is sufficient.

• If Carl began making $50withdrawalsonorbeforeMay15,hisaccountbalanceonApril16wouldbeatleast$50 greater than it was on the last day of May. Thus, his account balance on April 16 would be at least $2600+$50 = $2650,whichiscontrarytotheinformationgivenin\left(1\right).Therefore,Carldidnotbeginmaking$50 withdrawals until June 15 or later. These observations can be used to give at least two possible ranges.
  

(i) Carl could have had an account balance of $2,000onJanuary1,made$120 deposits in each of the first 11 months of the year, and then made a $50 withdrawal on December 15, which gives a range of monthly closing balances of (120) * (10).
(ii) Also, Carl could have had an account balance of $2,000onJanuary1,made$120 deposits in each of the first 10 months of the year, and then made $50 withdrawals on November 15 and on December 15, which gives a range of monthly closing balances of (120) * (9); NOT sufficient.

• On June 1, Carl’s account balance was the same as its closing balance was for May, namely $2,600.DependingonwhetherCarlmadea$120 deposit or a $50withdrawalonJune15,Car{l}^{\prime }saccountbalanceonJune16waseither$2,720 or $2,550.Itfollowsfromtheinformationgivenin\left(2\right)thatCar{l}^{\prime }sbalanceonJune16was$2,550. Therefore, Carl began making $50 withdrawals on or before June 15. These observations can be used to give at least two possible ranges.
  

(i) Carl could have had an account balance of $2,680onJanuary1,madeone$120 deposit on January 15, and then made a $50withdrawalineachoftheremaining11monthsoftheyear(thisgivesaclosingbalanceof$2,600 for May), which gives a range of monthly closing balances of (50) * (11).
(ii) Also, Carl could have had an account balance of $2,510onJanuary1,made$120 deposits on January 15 and on February 15, and then made a $50withdrawalineachoftheremaining10monthsoftheyear(thisgivesaclosingbalanceof$2,600 for May), which gives a range of monthly closing balances of (50) * (10); NOT sufficient.

• Given both (1) and (2), it follows from the remarks above that Carl began making $50 withdrawals on June 15. Therefore, the changes to Carl’s account balance for each month of last year are known. Since the closing balance for May is given, it follows that the closing balances for each month of last year are known, and hence the range of these 12 known values can be determined. Therefore, (1) and (2) together are sufficient. The correct answer is C; both statements together are sufficient.