Question

For dollar amounts, give your answer to the nearest cent. For interest rates, give our answer as a percentage rounded to 2 decimal places.

If any parts of the question use values from earlier parts, use the EXACT values from earlier parts.

QUESTION START

You are buying a car at a cost of $42,000 by taking a loan. The nominal interest rate is 9% per annum compounded monthly. The bank offers 2 options for the structure of the repayments.

• Option 1: The loan will be repaid over 10 years by equal month-end-instalments.

a) Calulate the monthly installment. (1 mark)

b) Calculate the interest component for the 20^th repayment.

c) Calculate the loan outstanding immediately after 8 years (immediately after the payment on that date).

d) Hence or otherwise, calculate the cumulative principal repayments during the 9^th year.

• Option 2: Month-end-instalments of $X will be made for the first 3.5 years. Then the bank offers you a payment free period (i.e., no repayments required) of 1 year. After that, the remaining balance will be repaid over 3.5 years by month-end-instalments of $2X.

e) Calculate X.

Answer:

Answer 1

Option 1:Interest rate per annum = 9% per annum, Compounding monthly

Interest rate per month = 9% /12 = 0.75% per month

Let the Monthly installment be X

We know that the Present value of future cash outflows is equal to the loan amount.

S.No	Disc @ 0.75%	Discounting Factor
1	1/1.0075	0.9926
2	1/( 1.0075)^2	0.9852
3	1/( 1.0075)^3	0.9778
4	1/( 1.0075)^4	0.9706
5	1/( 1.0075)^5	0.9633
6	1/( 1.0075)^6	0.9562
7	1/( 1.0075)^7	0.9490
8	1/( 1.0075)^8	0.9420
9	1/( 1.0075)^9	0.9350
10	1/( 1.0075)^10	0.9280
11	1/( 1.0075)^11	0.9211
12	1/( 1.0075)^12	0.9142
13	1/( 1.0075)^13	0.9074
14	1/( 1.0075)^14	0.9007
15	1/( 1.0075)^15	0.8940
16	1/( 1.0075)^16	0.8873
17	1/( 1.0075)^17	0.8807
18	1/( 1.0075)^18	0.8742
19	1/( 1.0075)^19	0.8676
20	1/( 1.0075)^20	0.8612
21	1/( 1.0075)^21	0.8548
22	1/( 1.0075)^22	0.8484
23	1/( 1.0075)^23	0.8421
24	1/( 1.0075)^24	0.8358
25	1/( 1.0075)^25	0.8296
26	1/( 1.0075)^26	0.8234
27	1/( 1.0075)^27	0.8173
28	1/( 1.0075)^28	0.8112
29	1/( 1.0075)^29	0.8052
30	1/( 1.0075)^30	0.7992
31	1/( 1.0075)^31	0.7932
32	1/( 1.0075)^32	0.7873
33	1/( 1.0075)^33	0.7815
34	1/( 1.0075)^34	0.7757
35	1/( 1.0075)^35	0.7699
36	1/( 1.0075)^36	0.7641
37	1/( 1.0075)^37	0.7585
38	1/( 1.0075)^38	0.7528
39	1/( 1.0075)^39	0.7472
40	1/( 1.0075)^40	0.7416
41	1/( 1.0075)^41	0.7361
42	1/( 1.0075)^42	0.7306
43	1/( 1.0075)^43	0.7252
44	1/( 1.0075)^44	0.7198
45	1/( 1.0075)^45	0.7145
46	1/( 1.0075)^46	0.7091
47	1/( 1.0075)^47	0.7039
48	1/( 1.0075)^48	0.6986
49	1/( 1.0075)^49	0.6934
50	1/( 1.0075)^50	0.6883
51	1/( 1.0075)^51	0.6831
52	1/( 1.0075)^52	0.6780
53	1/( 1.0075)^53	0.6730
54	1/( 1.0075)^54	0.6680
55	1/( 1.0075)^55	0.6630
56	1/( 1.0075)^56	0.6581
57	1/( 1.0075)^57	0.6532
58	1/( 1.0075)^58	0.6483
59	1/( 1.0075)^59	0.6435
60	1/( 1.0075)^60	0.6387
61	1/( 1.0075)^61	0.6339
62	1/( 1.0075)^62	0.6292
63	1/( 1.0075)^63	0.6245
64	1/( 1.0075)^64	0.6199
65	1/( 1.0075)^65	0.6153
66	1/( 1.0075)^66	0.6107
67	1/( 1.0075)^67	0.6062
68	1/( 1.0075)^68	0.6016
69	1/( 1.0075)^69	0.5972
70	1/( 1.0075)^70	0.5927
71	1/( 1.0075)^71	0.5883
72	1/( 1.0075)^72	0.5839
73	1/( 1.0075)^73	0.5796
74	1/( 1.0075)^74	0.5753
75	1/( 1.0075)^75	0.5710
76	1/( 1.0075)^76	0.5667
77	1/( 1.0075)^77	0.5625
78	1/( 1.0075)^78	0.5583
79	1/( 1.0075)^79	0.5542
80	1/( 1.0075)^80	0.5500
81	1/( 1.0075)^81	0.5459
82	1/( 1.0075)^82	0.5419
83	1/( 1.0075)^83	0.5378
84	1/( 1.0075)^84	0.5338
85	1/( 1.0075)^85	0.5299
86	1/( 1.0075)^86	0.5259
87	1/( 1.0075)^87	0.5220
88	1/( 1.0075)^88	0.5181
89	1/( 1.0075)^89	0.5143
90	1/( 1.0075)^90	0.5104
91	1/( 1.0075)^91	0.5066
92	1/( 1.0075)^92	0.5029
93	1/( 1.0075)^93	0.4991
94	1/( 1.0075)^94	0.4954
95	1/( 1.0075)^95	0.4917
96	1/( 1.0075)^96	0.4881
97	1/( 1.0075)^97	0.4844
98	1/( 1.0075)^98	0.4808
99	1/( 1.0075)^99	0.4772
100	1/( 1.0075)^100	0.4737
101	1/( 1.0075)^101	0.4702
102	1/( 1.0075)^102	0.4667
103	1/( 1.0075)^103	0.4632
104	1/( 1.0075)^104	0.4597
105	1/( 1.0075)^105	0.4563
106	1/( 1.0075)^106	0.4529
107	1/( 1.0075)^107	0.4496
108	1/( 1.0075)^108	0.4462
109	1/( 1.0075)^109	0.4429
110	1/( 1.0075)^110	0.4396
111	1/( 1.0075)^111	0.4363
112	1/( 1.0075)^112	0.4331
113	1/( 1.0075)^113	0.4298
114	1/( 1.0075)^114	0.4266
115	1/( 1.0075)^115	0.4235
116	1/( 1.0075)^116	0.4203
117	1/( 1.0075)^117	0.4172
118	1/( 1.0075)^118	0.4141
119	1/( 1.0075)^119	0.4110
120	1/( 1.0075)^120	0.4079
	Total	78.9417

X * PVAF( 0.75%,120) = $ 42000

78.9417 X = $ 42000

X = $ 42000/78.9417

X = $ 532.0382

a)  Hence the Monthly installment is $ 532.0382

Amortization Schedule:

S.No	Opening Balance	Interest @ 0.75%	Total Balance	Amount paid	Principal Component ( Amount paid - Interest)	Cummulative Principal Repayments	Closing Balance
1	$42,000	$315.00	$42,315.00	$532.0382	$217.0382	$217.0382	$41,782.96
2	$41,782.96	$313.37	$42,096.33	$532.0382	$218.6660	$435.7042	$41,564.30
3	$41,564.30	$311.73	$41,876.03	$532.0382	$220.3060	$656.0102	$41,343.99
4	$41,343.99	$310.08	$41,654.07	$532.0382	$221.9583	$877.9684	$41,122.03
5	$41,122.03	$308.42	$41,430.45	$532.0382	$223.6230	$1,101.5914	$40,898.41
6	$40,898.41	$306.74	$41,205.15	$532.0382	$225.3001	$1,326.8915	$40,673.11
7	$40,673.11	$305.05	$40,978.16	$532.0382	$226.9899	$1,553.8814	$40,446.12
8	$40,446.12	$303.35	$40,749.46	$532.0382	$228.6923	$1,782.5737	$40,217.43
9	$40,217.43	$301.63	$40,519.06	$532.0382	$230.4075	$2,012.9812	$39,987.02
10	$39,987.02	$299.90	$40,286.92	$532.0382	$232.1356	$2,245.1168	$39,754.88
11	$39,754.88	$298.16	$40,053.04	$532.0382	$233.8766	$2,478.9934	$39,521.01
12	$39,521.01	$296.41	$39,817.41	$532.0382	$235.6307	$2,714.6240	$39,285.38
13	$39,285.38	$294.64	$39,580.02	$532.0382	$237.3979	$2,952.0219	$39,047.98
14	$39,047.98	$292.86	$39,340.84	$532.0382	$239.1784	$3,191.2003	$38,808.80
15	$38,808.80	$291.07	$39,099.87	$532.0382	$240.9722	$3,432.1725	$38,567.83
16	$38,567.83	$289.26	$38,857.09	$532.0382	$242.7795	$3,674.9520	$38,325.05
17	$38,325.05	$287.44	$38,612.49	$532.0382	$244.6003	$3,919.5523	$38,080.45
18	$38,080.45	$285.60	$38,366.05	$532.0382	$246.4348	$4,165.9872	$37,834.01
19	$37,834.01	$283.76	$38,117.77	$532.0382	$248.2831	$4,414.2703	$37,585.73
20	$37,585.73	$281.89	$37,867.62	$532.0382	$250.1452	$4,664.4155	$37,335.58
21	$37,335.58	$280.02	$37,615.60	$532.0382	$252.0213	$4,916.4368	$37,083.56
22	$37,083.56	$278.13	$37,361.69	$532.0382	$253.9115	$5,170.3483	$36,829.65
23	$36,829.65	$276.22	$37,105.87	$532.0382	$255.8158	$5,426.1641	$36,573.84
24	$36,573.84	$274.30	$36,848.14	$532.0382	$257.7344	$5,683.8985	$36,316.10
25	$36,316.10	$272.37	$36,588.47	$532.0382	$259.6674	$5,943.5660	$36,056.43
26	$36,056.43	$270.42	$36,326.86	$532.0382	$261.6149	$6,205.1809	$35,794.82
27	$35,794.82	$268.46	$36,063.28	$532.0382	$263.5771	$6,468.7580	$35,531.24
28	$35,531.24	$266.48	$35,797.73	$532.0382	$265.5539	$6,734.3118	$35,265.69
29	$35,265.69	$264.49	$35,530.18	$532.0382	$267.5455	$7,001.8574	$34,998.14
30	$34,998.14	$262.49	$35,260.63	$532.0382	$269.5521	$7,271.4095	$34,728.59
31	$34,728.59	$260.46	$34,989.05	$532.0382	$271.5738	$7,542.9833	$34,457.02
32	$34,457.02	$258.43	$34,715.44	$532.0382	$273.6106	$7,816.5939	$34,183.41
33	$34,183.41	$256.38	$34,439.78	$532.0382	$275.6627	$8,092.2565	$33,907.74
34	$33,907.74	$254.31	$34,162.05	$532.0382	$277.7301	$8,369.9866	$33,630.01
35	$33,630.01	$252.23	$33,882.24	$532.0382	$279.8131	$8,649.7997	$33,350.20
36	$33,350.20	$250.13	$33,600.33	$532.0382	$281.9117	$8,931.7114	$33,068.29
37	$33,068.29	$248.01	$33,316.30	$532.0382	$284.0260	$9,215.7375	$32,784.26
38	$32,784.26	$245.88	$33,030.14	$532.0382	$286.1562	$9,501.8937	$32,498.11
39	$32,498.11	$243.74	$32,741.84	$532.0382	$288.3024	$9,790.1961	$32,209.80
40	$32,209.80	$241.57	$32,451.38	$532.0382	$290.4647	$10,080.6608	$31,919.34
41	$31,919.34	$239.40	$32,158.73	$532.0382	$292.6432	$10,373.3039	$31,626.70
42	$31,626.70	$237.20	$31,863.90	$532.0382	$294.8380	$10,668.1419	$31,331.86
43	$31,331.86	$234.99	$31,566.85	$532.0382	$297.0493	$10,965.1912	$31,034.81
44	$31,034.81	$232.76	$31,267.57	$532.0382	$299.2771	$11,264.4683	$30,735.53
45	$30,735.53	$230.52	$30,966.05	$532.0382	$301.5217	$11,565.9900	$30,434.01
46	$30,434.01	$228.26	$30,662.27	$532.0382	$303.7831	$11,869.7731	$30,130.23
47	$30,130.23	$225.98	$30,356.20	$532.0382	$306.0615	$12,175.8346	$29,824.17
48	$29,824.17	$223.68	$30,047.85	$532.0382	$308.3570	$12,484.1916	$29,515.81
49	$29,515.81	$221.37	$29,737.18	$532.0382	$310.6696	$12,794.8612	$29,205.14
50	$29,205.14	$219.04	$29,424.18	$532.0382	$312.9997	$13,107.8609	$28,892.14
51	$28,892.14	$216.69	$29,108.83	$532.0382	$315.3472	$13,423.2081	$28,576.79
52	$28,576.79	$214.33	$28,791.12	$532.0382	$317.7123	$13,740.9203	$28,259.08
53	$28,259.08	$211.94	$28,471.02	$532.0382	$320.0951	$14,061.0154	$27,938.98
54	$27,938.98	$209.54	$28,148.53	$532.0382	$322.4958	$14,383.5112	$27,616.49
55	$27,616.49	$207.12	$27,823.61	$532.0382	$324.9145	$14,708.4258	$27,291.57
56	$27,291.57	$204.69	$27,496.26	$532.0382	$327.3514	$15,035.7772	$26,964.22
57	$26,964.22	$202.23	$27,166.45	$532.0382	$329.8065	$15,365.5837	$26,634.42
58	$26,634.42	$199.76	$26,834.17	$532.0382	$332.2801	$15,697.8638	$26,302.14
59	$26,302.14	$197.27	$26,499.40	$532.0382	$334.7722	$16,032.6359	$25,967.36
60	$25,967.36	$194.76	$26,162.12	$532.0382	$337.2830	$16,369.9189	$25,630.08
61	$25,630.08	$192.23	$25,822.31	$532.0382	$339.8126	$16,709.7315	$25,290.27
62	$25,290.27	$189.68	$25,479.95	$532.0382	$342.3612	$17,052.0927	$24,947.91
63	$24,947.91	$187.11	$25,135.02	$532.0382	$344.9289	$17,397.0216	$24,602.98
64	$24,602.98	$184.52	$24,787.50	$532.0382	$347.5159	$17,744.5375	$24,255.46
65	$24,255.46	$181.92	$24,437.38	$532.0382	$350.1222	$18,094.6597	$23,905.34
66	$23,905.34	$179.29	$24,084.63	$532.0382	$352.7481	$18,447.4078	$23,552.59
67	$23,552.59	$176.64	$23,729.24	$532.0382	$355.3938	$18,802.8016	$23,197.20
68	$23,197.20	$173.98	$23,371.18	$532.0382	$358.0592	$19,160.8608	$22,839.14
69	$22,839.14	$171.29	$23,010.43	$532.0382	$360.7447	$19,521.6055	$22,478.39
70	$22,478.39	$168.59	$22,646.98	$532.0382	$363.4502	$19,885.0557	$22,114.94
71	$22,114.94	$165.86	$22,280.81	$532.0382	$366.1761	$20,251.2318	$21,748.77
72	$21,748.77	$163.12	$21,911.88	$532.0382	$368.9224	$20,620.1543	$21,379.85
73	$21,379.85	$160.35	$21,540.19	$532.0382	$371.6894	$20,991.8436	$21,008.16
74	$21,008.16	$157.56	$21,165.72	$532.0382	$374.4770	$21,366.3206	$20,633.68
75	$20,633.68	$154.75	$20,788.43	$532.0382	$377.2856	$21,743.6062	$20,256.39
76	$20,256.39	$151.92	$20,408.32	$532.0382	$380.1152	$22,123.7215	$19,876.28
77	$19,876.28	$149.07	$20,025.35	$532.0382	$382.9661	$22,506.6876	$19,493.31
78	$19,493.31	$146.20	$19,639.51	$532.0382	$385.8384	$22,892.5260	$19,107.47
79	$19,107.47	$143.31	$19,250.78	$532.0382	$388.7321	$23,281.2581	$18,718.74
80	$18,718.74	$140.39	$18,859.13	$532.0382	$391.6476	$23,672.9057	$18,327.09
81	$18,327.09	$137.45	$18,464.55	$532.0382	$394.5850	$24,067.4907	$17,932.51
82	$17,932.51	$134.49	$18,067.00	$532.0382	$397.5444	$24,465.0351	$17,534.96
83	$17,534.96	$131.51	$17,666.48	$532.0382	$400.5260	$24,865.5611	$17,134.44
84	$17,134.44	$128.51	$17,262.95	$532.0382	$403.5299	$25,269.0910	$16,730.91
85	$16,730.91	$125.48	$16,856.39	$532.0382	$406.5564	$25,675.6474	$16,324.35
86	$16,324.35	$122.43	$16,446.79	$532.0382	$409.6056	$26,085.2529	$15,914.75
87	$15,914.75	$119.36	$16,034.11	$532.0382	$412.6776	$26,497.9305	$15,502.07
88	$15,502.07	$116.27	$15,618.34	$532.0382	$415.7727	$26,913.7032	$15,086.30
89	$15,086.30	$113.15	$15,199.44	$532.0382	$418.8910	$27,332.5942	$14,667.41
90	$14,667.41	$110.01	$14,777.41	$532.0382	$422.0327	$27,754.6268	$14,245.37
91	$14,245.37	$106.84	$14,352.21	$532.0382	$425.1979	$28,179.8247	$13,820.18
92	$13,820.18	$103.65	$13,923.83	$532.0382	$428.3869	$28,608.2116	$13,391.79
93	$13,391.79	$100.44	$13,492.23	$532.0382	$431.5998	$29,039.8114	$12,960.19
94	$12,960.19	$97.20	$13,057.39	$532.0382	$434.8368	$29,474.6482	$12,525.35
95	$12,525.35	$93.94	$12,619.29	$532.0382	$438.0981	$29,912.7462	$12,087.25
96	$12,087.25	$90.65	$12,177.91	$532.0382	$441.3838	$30,354.1300	$11,645.87
97	$11,645.87	$87.34	$11,733.21	$532.0382	$444.6942	$30,798.8242	$11,201.18
98	$11,201.18	$84.01	$11,285.18	$532.0382	$448.0294	$31,246.8536	$10,753.15
99	$10,753.15	$80.65	$10,833.79	$532.0382	$451.3896	$31,698.2432	$10,301.76
100	$10,301.76	$77.26	$10,379.02	$532.0382	$454.7750	$32,153.0182	$9,846.98
101	$9,846.98	$73.85	$9,920.83	$532.0382	$458.1858	$32,611.2041	$9,388.80
102	$9,388.80	$70.42	$9,459.21	$532.0382	$461.6222	$33,072.8263	$8,927.17
103	$8,927.17	$66.95	$8,994.13	$532.0382	$465.0844	$33,537.9107	$8,462.09
104	$8,462.09	$63.47	$8,525.55	$532.0382	$468.5725	$34,006.4832	$7,993.52
105	$7,993.52	$59.95	$8,053.47	$532.0382	$472.0868	$34,478.5700	$7,521.43
106	$7,521.43	$56.41	$7,577.84	$532.0382	$475.6275	$34,954.1975	$7,045.80
107	$7,045.80	$52.84	$7,098.65	$532.0382	$479.1947	$35,433.3922	$6,566.61
108	$6,566.61	$49.25	$6,615.86	$532.0382	$482.7886	$35,916.1808	$6,083.82
109	$6,083.82	$45.63	$6,129.45	$532.0382	$486.4096	$36,402.5904	$5,597.41
110	$5,597.41	$41.98	$5,639.39	$532.0382	$490.0576	$36,892.6480	$5,107.35
111	$5,107.35	$38.31	$5,145.66	$532.0382	$493.7331	$37,386.3811	$4,613.62
112	$4,613.62	$34.60	$4,648.22	$532.0382	$497.4361	$37,883.8171	$4,116.18
113	$4,116.18	$30.87	$4,147.05	$532.0382	$501.1668	$38,384.9840	$3,615.02
114	$3,615.02	$27.11	$3,642.13	$532.0382	$504.9256	$38,889.9096	$3,110.09
115	$3,110.09	$23.33	$3,133.42	$532.0382	$508.7125	$39,398.6221	$2,601.38
116	$2,601.38	$19.51	$2,620.89	$532.0382	$512.5279	$39,911.1499	$2,088.85
117	$2,088.85	$15.67	$2,104.52	$532.0382	$516.3718	$40,427.5218	$1,572.48
118	$1,572.48	$11.79	$1,584.27	$532.0382	$520.2446	$40,947.7664	$1,052.23
119	$1,052.23	$7.89	$1,060.13	$532.0382	$524.1464	$41,471.9128	$528.09
120	$528.09	$3.96	$532.05	$532.0382	$528.0775	$41,999.9904	$0.01

b) From the Above table we can see that interest Component for 20th Repayment is $ 281.89

c) From the Above table, we can say that balance after 8 years ( i.e 96 months ) is $ 11645.87

d) From the above table we can say that the Cummulative Principal repaiid upto 9th year is ( i.e 108 months ) is $ 35916.1808

Option 2:

We know that Present value of the future cash outfllows is equal to the loan amount

Total Tenure = 3.5+1+3.5 = 8 years

After 3.5 Years there is no repayment of loan for 1 year. After the Expiry of 1 year again borrower has to pay double of installment amount

Converting 3.5 Years into Months = 3.5*12= 42 Months

S.No	Disc @ 0.75%	Discounting Factor	Cummulative Discount factor
1	1/1.0075	0.9926	0.9926
2	1/( 1.0075)^2	0.9852	1.9777
3	1/( 1.0075)^3	0.9778	2.9556
4	1/( 1.0075)^4	0.9706	3.9261
5	1/( 1.0075)^5	0.9633	4.8894
6	1/( 1.0075)^6	0.9562	5.8456
7	1/( 1.0075)^7	0.9490	6.7946
8	1/( 1.0075)^8	0.9420	7.7366
9	1/( 1.0075)^9	0.9350	8.6716
10	1/( 1.0075)^10	0.9280	9.5996
11	1/( 1.0075)^11	0.9211	10.5207
12	1/( 1.0075)^12	0.9142	11.4349
13	1/( 1.0075)^13	0.9074	12.3423
14	1/( 1.0075)^14	0.9007	13.2430
15	1/( 1.0075)^15	0.8940	14.1370
16	1/( 1.0075)^16	0.8873	15.0243
17	1/( 1.0075)^17	0.8807	15.9050
18	1/( 1.0075)^18	0.8742	16.7792
19	1/( 1.0075)^19	0.8676	17.6468
20	1/( 1.0075)^20	0.8612	18.5080
21	1/( 1.0075)^21	0.8548	19.3628
22	1/( 1.0075)^22	0.8484	20.2112
23	1/( 1.0075)^23	0.8421	21.0533
24	1/( 1.0075)^24	0.8358	21.8891
25	1/( 1.0075)^25	0.8296	22.7188
26	1/( 1.0075)^26	0.8234	23.5422
27	1/( 1.0075)^27	0.8173	24.3595
28	1/( 1.0075)^28	0.8112	25.1707
29	1/( 1.0075)^29	0.8052	25.9759
30	1/( 1.0075)^30	0.7992	26.7751
31	1/( 1.0075)^31	0.7932	27.5683
32	1/( 1.0075)^32	0.7873	28.3557
33	1/( 1.0075)^33	0.7815	29.1371
34	1/( 1.0075)^34	0.7757	29.9128
35	1/( 1.0075)^35	0.7699	30.6827
36	1/( 1.0075)^36	0.7641	31.4468
37	1/( 1.0075)^37	0.7585	32.2053
38	1/( 1.0075)^38	0.7528	32.9581
39	1/( 1.0075)^39	0.7472	33.7053
40	1/( 1.0075)^40	0.7416	34.4469
41	1/( 1.0075)^41	0.7361	35.1831
42	1/( 1.0075)^42	0.7306	35.9137
43	1/( 1.0075)^43	0.7252
44	1/( 1.0075)^44	0.7198
45	1/( 1.0075)^45	0.7145
46	1/( 1.0075)^46	0.7091
47	1/( 1.0075)^47	0.7039
48	1/( 1.0075)^48	0.6986
49	1/( 1.0075)^49	0.6934
50	1/( 1.0075)^50	0.6883
51	1/( 1.0075)^51	0.6831
52	1/( 1.0075)^52	0.6780
53	1/( 1.0075)^53	0.6730
54	1/( 1.0075)^54	0.6680
55	1/( 1.0075)^55	0.6630	0.6630
56	1/( 1.0075)^56	0.6581	1.3211
57	1/( 1.0075)^57	0.6532	1.9743
58	1/( 1.0075)^58	0.6483	2.6226
59	1/( 1.0075)^59	0.6435	3.2661
60	1/( 1.0075)^60	0.6387	3.9048
61	1/( 1.0075)^61	0.6339	4.5387
62	1/( 1.0075)^62	0.6292	5.1679
63	1/( 1.0075)^63	0.6245	5.7925
64	1/( 1.0075)^64	0.6199	6.4124
65	1/( 1.0075)^65	0.6153	7.0277
66	1/( 1.0075)^66	0.6107	7.6384
67	1/( 1.0075)^67	0.6062	8.2445
68	1/( 1.0075)^68	0.6016	8.8461
69	1/( 1.0075)^69	0.5972	9.4433
70	1/( 1.0075)^70	0.5927	10.0360
71	1/( 1.0075)^71	0.5883	10.6243
72	1/( 1.0075)^72	0.5839	11.2082
73	1/( 1.0075)^73	0.5796	11.7878
74	1/( 1.0075)^74	0.5753	12.3631
75	1/( 1.0075)^75	0.5710	12.9341
76	1/( 1.0075)^76	0.5667	13.5008
77	1/( 1.0075)^77	0.5625	14.0633
78	1/( 1.0075)^78	0.5583	14.6216
79	1/( 1.0075)^79	0.5542	15.1758
80	1/( 1.0075)^80	0.5500	15.7258
81	1/( 1.0075)^81	0.5459	16.2718
82	1/( 1.0075)^82	0.5419	16.8137
83	1/( 1.0075)^83	0.5378	17.3515
84	1/( 1.0075)^84	0.5338	17.8854
85	1/( 1.0075)^85	0.5299	18.4152
86	1/( 1.0075)^86	0.5259	18.9412
87	1/( 1.0075)^87	0.5220	19.4632
88	1/( 1.0075)^88	0.5181	19.9813
89	1/( 1.0075)^89	0.5143	20.4956
90	1/( 1.0075)^90	0.5104	21.0060
91	1/( 1.0075)^91	0.5066	21.5127
92	1/( 1.0075)^92	0.5029	22.0155
93	1/( 1.0075)^93	0.4991	22.5146
94	1/( 1.0075)^94	0.4954	23.0101
95	1/( 1.0075)^95	0.4917	23.5018
96	1/( 1.0075)^96	0.4881	23.9898

X * PVAF ( 0.75% ,42) + 2X * 23.9898 = $ 42000

X * 35.9137 + 2X * 23.9898 = $ 42000

35.9137X + 47.9796X = $ 42000

83.8933X = $ 42000

X = $ 42000/83.8933

X = $ 500.6359

e) Hence X = $ 500.6359

If you have any doubts,please post a comment.

Thank you.Please rate it.