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If you had $100.00 invested in a savings account that is compounded annually and yields 3% interest, how much interest would you have accumulated in 30 years? A

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6 Answers

There is a specific formula for calculating the value of future interest payments in today's present value.
The base value of the initial investment is combined with the future values (add each individual year) to the current investment.
 
you take the current value over (one + the interest rate)^(nth power)
n is the amount of payments, and in this case it would be the amount of years.
 
for year 0 or initial investment you do not earn interest
for year 1 you earn 100 divided by 1.03 to the first power and so on
Months Interest Balance
1 0.25 100.25
2 0.49 100.49
3 0.74 100.74
4 0.99 100.99
5 1.24 101.24
6 1.49 101.49
7 1.74 101.74
8 1.99 101.99
9 2.24 102.24
10 2.49 102.49
11 2.75 102.75
12 3 103
13 3.25 103.25
14 3.51 103.51
15 3.76 103.76
16 4.02 104.02
17 4.28 104.28
18 4.53 104.53
19 4.79 104.79
20 5.05 105.05
21 5.31 105.31
22 5.57 105.57
23 5.83 105.83
24 6.09 106.09
25 6.35 106.35
26 6.61 106.61
27 6.88 106.88
28 7.14 107.14
29 7.4 107.4
30 7.67 107.67
31 7.94 107.94
32 8.2 108.2
33 8.47 108.47
34 8.74 108.74
35 9 109
36 9.27 109.27
37 9.54 109.54
38 9.81 109.81
39 10.08 110.08
40 10.35 110.35
41 10.63 110.63
42 10.9 110.9
43 11.17 111.17
44 11.45 111.45
45 11.72 111.72
46 12 112
47 12.27 112.27
48 12.55 112.55
49 12.83 112.83
50 13.11 113.11
51 13.39 113.39
52 13.67 113.67
53 13.95 113.95
54 14.23 114.23
55 14.51 114.51
56 14.79 114.79
57 15.07 115.07
58 15.36 115.36
59 15.64 115.64
60 15.93 115.93
61 16.21 116.21
62 16.5 116.5
63 16.79 116.79
64 17.08 117.08
65 17.36 117.36
66 17.65 117.65
67 17.94 117.94
68 18.23 118.23
69 18.53 118.53
70 18.82 118.82
71 19.11 119.11
72 19.41 119.41
73 19.7 119.7
74 19.99 119.99
75 20.29 120.29
76 20.59 120.59
77 20.88 120.88
78 21.18 121.18
79 21.48 121.48
80 21.78 121.78
81 22.08 122.08
82 22.38 122.38
83 22.68 122.68
84 22.99 122.99
85 23.29 123.29
86 23.59 123.59
87 23.9 123.9
88 24.21 124.21
89 24.51 124.51
90 24.82 124.82
91 25.13 125.13
92 25.43 125.43
93 25.74 125.74
94 26.05 126.05
95 26.37 126.37
96 26.68 126.68
97 26.99 126.99
98 27.3 127.3
99 27.62 127.62
100 27.93 127.93
101 28.25 128.25
102 28.56 128.56
103 28.88 128.88
104 29.2 129.2
105 29.52 129.52
106 29.84 129.84
107 30.16 130.16
108 30.48 130.48
109 30.8 130.8
110 31.12 131.12
111 31.45 131.45
112 31.77 131.77
113 32.09 132.09
114 32.42 132.42
115 32.75 132.75
116 33.07 133.07
117 33.4 133.4
118 33.73 133.73
119 34.06 134.06
120 34.39 134.39
121 34.72 134.72
122 35.06 135.06
123 35.39 135.39
124 35.72 135.72
125 36.06 136.06
126 36.39 136.39
127 36.73 136.73
128 37.07 137.07
129 37.4 137.4
130 37.74 137.74
131 38.08 138.08
132 38.42 138.42
133 38.76 138.76
134 39.11 139.11
135 39.45 139.45
136 39.79 139.79
137 40.14 140.14
138 40.48 140.48
139 40.83 140.83
140 41.18 141.18
141 41.53 141.53
142 41.88 141.88
143 42.23 142.23
144 42.58 142.58
145 42.93 142.93
146 43.28 143.28
147 43.63 143.63
148 43.99 143.99
149 44.34 144.34
150 44.7 144.7
151 45.06 145.06
152 45.41 145.41
153 45.77 145.77
154 46.13 146.13
155 46.49 146.49
156 46.85 146.85
157 47.22 147.22
158 47.58 147.58
159 47.94 147.94
160 48.31 148.31
161 48.67 148.67
162 49.04 149.04
163 49.41 149.41
164 49.78 149.78
165 50.15 150.15
166 50.52 150.52
167 50.89 150.89
168 51.26 151.26
169 51.63 151.63
170 52.01 152.01
171 52.38 152.38
172 52.76 152.76
173 53.13 153.13
174 53.51 153.51
175 53.89 153.89
176 54.27 154.27
177 54.65 154.65
178 55.03 155.03
179 55.41 155.41
180 55.8 155.8
181 56.18 156.18
182 56.57 156.57
183 56.95 156.95
184 57.34 157.34
185 57.73 157.73
186 58.12 158.12
187 58.51 158.51
188 58.9 158.9
189 59.29 159.29
190 59.68 159.68
191 60.08 160.08
192 60.47 160.47
193 60.87 160.87
194 61.26 161.26
195 61.66 161.66
196 62.06 162.06
197 62.46 162.46
198 62.86 162.86
199 63.26 163.26
200 63.66 163.66
201 64.07 164.07
202 64.47 164.47
203 64.88 164.88
204 65.28 165.28
205 65.69 165.69
206 66.1 166.1
207 66.51 166.51
208 66.92 166.92
209 67.33 167.33
210 67.75 167.75
211 68.16 168.16
212 68.57 168.57
213 68.99 168.99
214 69.41 169.41
215 69.82 169.82
216 70.24 170.24
217 70.66 170.66
218 71.08 171.08
219 71.51 171.51
220 71.93 171.93
221 72.35 172.35
222 72.78 172.78
223 73.2 173.2
224 73.63 173.63
225 74.06 174.06
226 74.49 174.49
227 74.92 174.92
228 75.35 175.35
229 75.78 175.78
230 76.22 176.22
231 76.65 176.65
232 77.09 177.09
233 77.52 177.52
234 77.96 177.96
235 78.4 178.4
236 78.84 178.84
237 79.28 179.28
238 79.72 179.72
239 80.17 180.17
240 80.61 180.61
241 81.06 181.06
242 81.5 181.5
243 81.95 181.95
244 82.4 182.4
245 82.85 182.85
246 83.3 183.3
247 83.75 183.75
248 84.21 184.21
249 84.66 184.66
250 85.12 185.12
251 85.57 185.57
252 86.03 186.03
253 86.49 186.49
254 86.95 186.95
255 87.41 187.41
256 87.87 187.87
257 88.33 188.33
258 88.8 188.8
259 89.26 189.26
260 89.73 189.73
261 90.2 190.2
262 90.67 190.67
263 91.14 191.14
264 91.61 191.61
265 92.08 192.08
266 92.56 192.56
267 93.03 193.03
268 93.51 193.51
269 93.98 193.98
270 94.46 194.46
271 94.94 194.94
272 95.42 195.42
273 95.91 195.91
274 96.39 196.39
275 96.87 196.87
276 97.36 197.36
277 97.85 197.85
278 98.33 198.33
279 98.82 198.82
280 99.31 199.31
281 99.8 199.8
282 100.3 200.3
283 100.79 200.79
284 101.29 201.29
285 101.78 201.78
286 102.28 202.28
287 102.78 202.78
288 103.28 203.28
289 103.78 203.78
290 104.28 204.28
291 104.79 204.79
292 105.29 205.29
293 105.8 205.8
294 106.31 206.31
295 106.81 206.81
296 107.32 207.32
297 107.84 207.84
298 108.35 208.35
299 108.86 208.86
300 109.38 209.38
301 109.89 209.89
302 110.41 210.41
303 110.93 210.93
304 111.45 211.45
305 111.97 211.97
306 112.5 212.5
307 113.02 213.02
308 113.54 213.54
309 114.07 214.07
310 114.6 214.6
311 115.13 215.13
312 115.66 215.66
313 116.19 216.19
314 116.72 216.72
315 117.26 217.26
316 117.79 217.79
317 118.33 218.33
318 118.87 218.87
319 119.41 219.41
320 119.95 219.95
321 120.49 220.49
322 121.04 221.04
323 121.58 221.58
324 122.13 222.13
325 122.68 222.68
326 123.23 223.23
327 123.78 223.78
328 124.33 224.33
329 124.88 224.88
330 125.44 225.44
331 125.99 225.99
332 126.55 226.55
333 127.11 227.11
334 127.67 227.67
335 128.23 228.23
336 128.79 228.79
337 129.36 229.36
338 129.92 229.92
339 130.49 230.49
340 131.06 231.06
341 131.63 231.63
342 132.2 232.2
343 132.77 232.77
344 133.35 233.35
345 133.92 233.92
346 134.5 234.5
347 135.08 235.08
348 135.66 235.66
349 136.24 236.24
350 136.82 236.82
351 137.4 237.4
352 137.99 237.99
353 138.58 238.58
354 139.17 239.17
355 139.76 239.76
356 140.35 240.35
357 140.94 240.94
358 141.53 241.53
359 142.13 242.13
360 142.73 242.73
 
the value after maturity is 242.73
 
A=P(1+r/n)nT
A = amount;  P = principal; r = interest rate; n = number of compounding periods per year;
T = number of years
 
A=100(1+.03/1)30×1=242.7262471
Interest = 242.7262471-100=142.7262471
3% interest is 1.03 mathematically
 
Each year yields 3% or 1.03 times the previous year's balance.
 
So it becomes Principal * 1.03 * 1.03 * 1.03 .... for 30 times.  Or the concise way to write it is:
 
Principal * (1.03^30) = Principal * 2.4273
 
For $100 that would be $242.73  total after 30 years and $142.73 of that would be accumulated interest. 
A = P (1 + r/n)^(nt) is the compound interest formula.
P is the original, or Principal, amount.
r is the interest rate.
n is the number of compounding periods per year.
t is the number of years.
A is the total amount in the account after t years.
 
A - P would be the interest accumulated after t years.
 
In this problem,
A = $100 (1 + 3%/1)^(1*30)
 = $100 (1.03)^30
 ≈ $242.73
 
and the Interest, I = A - P ≈ $142.73
Interest accumulated is the total sum on the account minus the principal.
 
Total sum accumulated is given by :
 
S=P(1+r)t, where P is the principal, t is the number of years passed, r is the annual interest rate (APR).
 
In your case,
S=$100*(1+0.03)30≈$242.73
 
Interest accumulated is S-P=$242.73-$100=$142.73
 
Answer: $142.73
This is a simple APR question.  Every year, your Present Value (F(0)=$100) will be compounded by 3% (1.03) annually.  Therefore, in 30 years, your Future Value (FV) will be:
 
FV(30)= F(0)*(1.03)^30 = $242.73  (rounded up to nearest cent)
Therefore, the interest accumulated is $142.73 on the initial $100
 
PS: If you can find a bank giving these rates....give me a call ;-)

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