The 2 equations that approximate the solution to this are:
y = -9734.7 * x + 218,761
y = -16474 * x + 266,478
If you graph both these equations, you will see when they cross.
The graph of both of these equations is shown below:
You can see from the graph that the approximate intersection point is somewhere around 7 years.
You can calculate that by making the equations equal to each other.
When you do that, you are saying that the value of y in the first equation is equal to the value of y in the second equation.
you get:
-9734.7x + 218761 = -16474 * x + 266,478
Solve this equation algebraically to get an answer of:
x = 7.080408945
y = 149835.343
The approximation given by the graph is pretty close.
x is the number of years when the balances will be approximately equal.
y i the outstanding balance in each account at that time.
The estimate will be somewhat inaccurate because the declining balance equation is not a straight line.
What the equations show are probability the best fit of a straight line to the actual balances as they drop in time.
The actual point where the balances are equal can be calculated through the use of Present Value of A Loan Amount Formulas, or through the use of a mechanized program that will do the calculations for each year of each loan.
I used Excel to do that for you, after having calculated what the monthly payments would be using a financial calculator, and the results are shown below:
Remaining Balance
year Loan1 Loan2
0 $200,000.00 $250,000.00
1 $199,616.07 $249,158.53
2 $199,229.90 $248,312.67
3 $198,841.47 $247,462.41
4 $198,450.79 $246,607.72
5 $198,057.82 $245,748.58
6 $197,662.56 $244,884.96
7 $197,264.99 $244,016.84
8 $196,865.10 $243,144.21
9 $196,462.89 $242,267.03
10 $196,058.32 $241,385.28
11 $195,651.40 $240,498.93
12 $195,242.10 $239,607.98
13 $194,830.41 $238,712.38
14 $194,416.33 $237,812.11
15 $193,999.82 $236,907.16
16 $193,580.89 $235,997.50
17 $193,159.52 $235,083.09
18 $192,735.68 $234,163.93
19 $192,309.38 $233,239.97
20 $191,880.58 $232,311.21
21 $191,449.29 $231,377.60
22 $191,015.48 $230,439.14
23 $190,579.14 $229,495.79
24 $190,140.25 $228,547.52
25 $189,698.80 $227,594.31
26 $189,254.78 $226,636.14
27 $188,808.17 $225,672.98
28 $188,358.95 $224,704.81
29 $187,907.12 $223,731.59
30 $187,452.64 $222,753.30
31 $186,995.52 $221,769.91
32 $186,535.73 $220,781.41
33 $186,073.26 $219,787.75
34 $185,608.09 $218,788.92
35 $185,140.20 $217,784.89
36 $184,669.59 $216,775.63
37 $184,196.23 $215,761.11
38 $183,720.11 $214,741.31
39 $183,241.21 $213,716.20
40 $182,759.52 $212,685.75
41 $182,275.02 $211,649.93
42 $181,787.69 $210,608.72
43 $181,297.52 $209,562.08
44 $180,804.50 $208,509.99
45 $180,308.59 $207,452.42
46 $179,809.79 $206,389.35
47 $179,308.09 $205,320.74
48 $178,803.45 $204,246.56
49 $178,295.87 $203,166.78
50 $177,785.34 $202,081.39
51 $177,271.82 $200,990.34
52 $176,755.31 $199,893.60
53 $176,235.78 $198,791.16
54 $175,713.23 $197,682.97
55 $175,187.62 $196,569.02
56 $174,658.95 $195,449.25
57 $174,127.20 $194,323.66
58 $173,592.34 $193,192.21
59 $173,054.37 $192,054.86
60 $172,513.25 $190,911.59
61 $171,968.98 $189,762.36
62 $171,421.53 $188,607.15
63 $170,870.90 $187,445.92
64 $170,317.05 $186,278.65
65 $169,759.96 $185,105.29
66 $169,199.63 $183,925.82
67 $168,636.03 $182,740.21
68 $168,069.14 $181,548.43
69 $167,498.95 $180,350.44
70 $166,925.43 $179,146.20
71 $166,348.56 $177,935.70
72 $165,768.33 $176,718.89
73 $165,184.72 $175,495.74
74 $164,597.70 $174,266.23
75 $164,007.25 $173,030.31
76 $163,413.36 $171,787.95
77 $162,816.01 $170,539.12
78 $162,215.17 $169,283.79
79 $161,610.83 $168,021.92
80 $161,002.96 $166,753.48
81 $160,391.55 $165,478.43
82 $159,776.57 $164,196.74
83 $159,158.00 $162,908.37
84 $158,535.82 $161,613.29
85 $157,910.02 $160,311.47
86 $157,280.56 $159,002.87
87 $156,647.43 $157,687.45
88 $156,010.61 $156,365.18 *
89 $155,370.07 $155,036.03 *
90 $154,725.80 $153,699.95
91 $154,077.77 $152,356.92
92 $153,425.96 $151,006.88
93 $152,770.35 $149,649.82
94 $152,110.91 $148,285.69
95 $151,447.63 $146,914.45
96 $150,780.47 $145,536.08
97 $150,109.43 $144,150.52
98 $149,434.47 $142,757.75
99 $148,755.57 $141,357.72
100 $148,072.71 $139,950.40
101 $147,385.87 $138,535.75
102 $146,695.03 $137,113.73
103 $146,000.15 $135,684.31
104 $145,301.22 $134,247.44
105 $144,598.21 $132,803.09
106 $143,891.10 $131,351.22
107 $143,179.87 $129,891.78
108 $142,464.49 $128,424.74
109 $141,744.93 $126,950.06
110 $141,021.18 $125,467.71
111 $140,293.21 $123,977.63
112 $139,560.99 $122,479.79
113 $138,824.49 $120,974.14
114 $138,083.71 $119,460.66
115 $137,338.60 $117,939.29
116 $136,589.14 $116,410.00
117 $135,835.31 $114,872.75
118 $135,077.09 $113,327.49
119 $134,314.44 $111,774.18
120 $133,547.34 $110,212.78
121 $132,775.77 $108,643.25
122 $131,999.70 $107,065.54
123 $131,219.10 $105,479.61
124 $130,433.94 $103,885.43
125 $129,644.21 $102,282.94
126 $128,849.87 $100,672.11
127 $128,050.90 $99,052.89
128 $127,247.26 $97,425.23
129 $126,438.94 $95,789.10
130 $125,625.90 $94,144.44
131 $124,808.12 $92,491.22
132 $123,985.57 $90,829.39
133 $123,158.22 $89,158.90
134 $122,326.05 $87,479.71
135 $121,489.02 $85,791.78
136 $120,647.11 $84,095.05
137 $119,800.29 $82,389.49
138 $118,948.52 $80,675.05
139 $118,091.79 $78,951.67
140 $117,230.06 $77,219.32
141 $116,363.31 $75,477.95
142 $115,491.49 $73,727.50
143 $114,614.60 $71,967.94
144 $113,732.58 $70,199.22
145 $112,845.43 $68,421.28
146 $111,953.09 $66,634.09
147 $111,055.56 $64,837.58
148 $110,152.78 $63,031.72
149 $109,244.74 $61,216.45
150 $108,331.40 $59,391.73
151 $107,412.74 $57,557.51
152 $106,488.72 $55,713.73
153 $105,559.30 $53,860.35
154 $104,624.47 $51,997.31
155 $103,684.18 $50,124.58
156 $102,738.41 $48,242.08
157 $101,787.11 $46,349.79
158 $100,830.27 $44,447.64
159 $99,867.85 $42,535.58
160 $98,899.82 $40,613.56
161 $97,926.14 $38,681.53
162 $96,946.77 $36,739.44
163 $95,961.70 $34,787.23
164 $94,970.88 $32,824.86
165 $93,974.28 $30,852.27
166 $92,971.86 $28,869.40
167 $91,963.60 $26,876.20
168 $90,949.46 $24,872.63
169 $89,929.40 $22,858.61
170 $88,903.39 $20,834.11
171 $87,871.39 $18,799.06
172 $86,833.38 $16,753.42
173 $85,789.31 $14,697.12
174 $84,739.15 $12,630.11
175 $83,682.86 $10,552.33
176 $82,620.41 $8,463.74
177 $81,551.77 $6,364.26
178 $80,476.89 $4,253.85
179 $79,395.74 $2,132.45
180 $78,308.28 ($0.00)
181 $77,214.48
182 $76,114.30
183 $75,007.71
184 $73,894.65
185 $72,775.11
186 $71,649.03
187 $70,516.39
188 $69,377.13
189 $68,231.24
190 $67,078.65
191 $65,919.35
192 $64,753.28
193 $63,580.41
194 $62,400.70
195 $61,214.10
196 $60,020.59
197 $58,820.11
198 $57,612.63
199 $56,398.10
200 $55,176.50
201 $53,947.76
202 $52,711.86
203 $51,468.75
204 $50,218.38
205 $48,960.73
206 $47,695.73
207 $46,423.36
208 $45,143.56
209 $43,856.30
210 $42,561.53
211 $41,259.21
212 $39,949.29
213 $38,631.73
214 $37,306.49
215 $35,973.51
216 $34,632.76
217 $33,284.18
218 $31,927.74
219 $30,563.39
220 $29,191.08
221 $27,810.76
222 $26,422.39
223 $25,025.93
224 $23,621.31
225 $22,208.51
226 $20,787.46
227 $19,358.12
228 $17,920.45
229 $16,474.38
230 $15,019.89
231 $13,556.90
232 $12,085.39
233 $10,605.29
234 $9,116.56
235 $7,619.14
236 $6,112.98
237 $4,598.05
238 $3,074.27
239 $1,541.61
240 $0.00
The asterisks (*) show you that the balances become equal somewhere between the 88th and 89th month.
This would be somewhere between 7.33333333 years and 7.41666666667 years.
The actual graph of the declining balance is not a straight line.
It drops slowly in the early years and drops a lot faster in the later years.
You can link to the following webiter and scroll down to the bottom and a graph of a remaining balance on a loan will show up.
You can see that it is not a straight line.
http://www.tvmcalcs.com/calculators/apps/excel_loan_amortization