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E
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Difficulty:
95%
(hard)
Question Stats:
41% (02:58) correct
59%
(03:03)
wrong
based on 34
sessions
History
Date
Time
Result
Not Attempted Yet
Leo and Agata each opened a savings account today, the last day of the month, but Agata’s account is empty. Beginning next month, they will each make a fixed deposit into their respective accounts on the last day of every month. One month from now, the balance of Leo’s account will be seven times the balance of Agata’s. Four months from now, the balance of Leo’s account will be three times that of Agata’s. How many months from now will the balance of Agata’s account first be at least one-half of that of Leo’s?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 16
(A) 6
(B) 8
(C) 10
(D) 12
(E) 16
25 Jun 2026, 20:36
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Let Leo's monthly deposit be L, Agata's monthly deposit be A, and Leo's initial balance be S. Agata starts with 0.
After the 1st deposit:
S + L = 7A
After the 4th deposit:
S + 4L = 12A
Subtract the first equation from the second:
3L = 5A
So,
L = 5A/3
Substitute into the first equation:
S + 5A/3 = 7A
S = 16A/3
After n deposits:
nA ≥ (1/2) × (16 + 5n)A/3
Cancel A:
n ≥ (16 + 5n)/6
6n ≥ 16 + 5n
n ≥ 16
So Agata's balance first becomes at least half of Leo's after the 16th deposit.
Since deposits begin next month, this is 16 months from now.
Answer: E. 16 months
After the 1st deposit:
- Leo's balance = S + L
- Agata's balance = A
S + L = 7A
After the 4th deposit:
- Leo's balance = S + 4L
- Agata's balance = 4A
S + 4L = 12A
Subtract the first equation from the second:
3L = 5A
So,
L = 5A/3
Substitute into the first equation:
S + 5A/3 = 7A
S = 16A/3
After n deposits:
- Leo's balance = 16A/3 + 5nA/3 = (16 + 5n)A/3
- Agata's balance = nA
nA ≥ (1/2) × (16 + 5n)A/3
Cancel A:
n ≥ (16 + 5n)/6
6n ≥ 16 + 5n
n ≥ 16
So Agata's balance first becomes at least half of Leo's after the 16th deposit.
Since deposits begin next month, this is 16 months from now.
Answer: E. 16 months
25 Jun 2026, 20:56
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