51. What annual instalment will discharge a debt of ₹ 6450 due in 4 years at 5% simple interest ?
(a) ₹ 1500
(b) ₹ 1835
(c) ₹ 1935
(d) ₹ 1950
(e) ₹ 1980
Solution: (a)
Let each instalment be x Then,
\(\displaystyle (x+\frac{{x\times 5\times 1}}{{100}})+(x+\frac{{x\times 5\times 2}}{{100}})+(x+\frac{{x\times 5\times 3}}{{100}})+x=6450\)
\(\displaystyle (x+\frac{x}{{20}})+(x+\frac{x}{{10}})+(x+\frac{{3x}}{{20}})+x=6450\)
\(\displaystyle \frac{{21x}}{{20}}+\frac{{11x}}{{10}}+\frac{{23x}}{{20}}+x=6450\)
\(\displaystyle \frac{{21x+22x+23x+20x}}{{20}}=6450\)
\(\displaystyle \frac{{86x}}{{20}}=6450\)
X = \(\displaystyle \frac{{6450\times 20}}{{86}}=1500\)
Alternate method,
Equal instalment = \(\displaystyle \frac{{6450\times 200}}{{4[200+(4-1)\times 5]}}\)
\(\displaystyle \frac{{6450\times 200}}{{4(215)}}\)
\(\displaystyle \frac{{6450\times 50}}{{215}}=1500\)
52. In what time will 72 become ₹ 81 at \(\displaystyle 6\frac{1}{4}\%\) per annum simple interest ?
(a) 2 years
(b) 3 years
(c) 2 years 6 months
(d) 3 years 4 months
(e) None of these
Solution: (a)
Interest = ₹ (81–72)= ₹ 9
Let the time be t years.
Then, 9 = \(\displaystyle \frac{{72\times 25\times t}}{{4\times 100}}\)
\(\displaystyle \Rightarrow \) t = \(\displaystyle \frac{{9\times 400}}{{72\times 25}}=2years\)
53. The simple interest on ₹ 7,300 from 11 May, 1987 to 10 September, 1987 (both days included) at 5% per annum is
(a) ₹ 103
(b) ₹ 123
(c) ₹ 200
(d) ₹ 223
(e) ₹ 285
Solution: (b)
Time from 11 May to 10 September, 1987
= 21 + 30 + 31 + 31 + 10
= 123 days
Therefore, Time = 123 days = \(\displaystyle \frac{{123}}{{365}}year\)
\(\displaystyle \Rightarrow \)S.I. = \(\displaystyle \frac{{7300\times 123\times 5}}{{365\times 100}}=123\)
54. A person borrows ₹ 5,000 for 2 years at 4% per annum simple interest. He immediately lends it to another person at \(\displaystyle 6\frac{1}{4}\)% per annum simple interest for 2 years. His gain in the transaction is
(a) ₹ 112.50
(b) ₹ 450
(c) ₹ 225
(d) ₹ 150
(e) ₹ 340
Solution: (c)
Case I :
S.I. = \(\displaystyle \frac{{5000\times 2\times 4}}{{100}}=400\)
Case II :
S.I. = \(\displaystyle \frac{{5000\times 25\times 2}}{{100\times 4}}=625\)
\(\displaystyle \Rightarrow \)Gain = ₹ (625 – 400) = ₹ 225
55. A man had ₹ 16,000, part of which he lent at 4% and the rest at 5% per annum simple interest. If the total interest received was ₹ 700 in one year, the money lent at 4% per annum was
(a) ₹ 12,000
(b) ₹ 8,000
(c) ₹ 10,000
(d) ₹ 6,000
(e) ₹ 7,000
Solution: (c)
Let the sum lent at 4% = Rs.x
Therefore, Amount at 5%= (16000 – x )
According to the question,
\(\displaystyle \frac{{x\times 4\times 1}}{{100}}+\frac{{(16000-x)\times 5\times 1}}{{100}}=700\)
\(\displaystyle \Rightarrow \) 4x + 80000 – 5x = 70000
\(\displaystyle \Rightarrow \) x = 80000 – 70000
= ₹ 10000
56. ₹ 1,000 is invested at 5% per annum simple interest. If the interest is added to the principal after every 10 years, the amount will become ₹ 2,000 after
(a) 15 years
(b) 18 \(\displaystyle \frac{2}{3}\) years
(c) 20 years
(d) 16 \(\displaystyle \frac{2}{3}\) years
(e) 22 years
Solution: (d)
After 10 years
SI = \(\displaystyle \frac{{1000\times 5\times 10}}{{100}}=500\)
Principal for 11th year = 1000 + 500 = ₹ 1500
SI = ₹ (2000 – 1500) = ₹ 500
\(\displaystyle \Rightarrow \) \(\displaystyle \frac{{SI\times 100}}{{P\times R}}=\frac{{500\times 100}}{{1500\times 5}}\)
= \(\displaystyle \frac{{20}}{3}years=6\frac{2}{3}years\)
\(\displaystyle \Rightarrow \) Total time = \(\displaystyle 10+6\frac{2}{3}=16\frac{2}{3}years\)
57. A sum of money amounts to ₹ 5,200 in 5 years and to ₹ 5,680 in 7 years at simple interest. The rate of interest per annum is
(a) 3%
(b) 4%
(c) 5%
(d) 6%
(e) 8%
Solution: (d)
P + S.I. for 5 years = 5200 ..(i)
P + SI for 7 years = 5680 …(ii)
On subtracting equation (i ) from (ii),
SI for 2 years = 480
\(\displaystyle \Rightarrow \)SI for 1 year = ₹ 240
\(\displaystyle \Rightarrow \) From equation (i),
P + 5 × 240 = 5200
\(\displaystyle \Rightarrow \) P = 5200 – 1200 = ₹ 4000
\(\displaystyle \Rightarrow \) R = \(\displaystyle \frac{{SI\times 100}}{{T\times P}}\)
= \(\displaystyle \frac{{240\times 100}}{{1\times 4000}}=6\%\)
Alternate method,
R = \(\displaystyle (\frac{{{{A}_{1}}-{{A}_{2}}}}{{{{A}_{2}}{{T}_{1}}-{{A}_{1}}{{T}_{2}}}})\times 100\)
\(\displaystyle (\frac{{5200-5680}}{{5680\times 5-5200\times 7}})\times 100\)
\(\displaystyle \frac{{-480}}{{-8000}}\times 100=6\%\)
58. ₹ 800 becomes ₹ 956 in 3 years at a certain rate of simple interest. If the rate of interest is increased by 4%, what amount will ₹ 800 become in 3 years ?
59. A person deposited ₹ 400 for 2 years, ₹ 550 for 4 years and ₹ 1,200 for 6 years. He received the total simple interest of ₹ 1,020. The rate of interest per annum is
(a) 10%
(b) 5%
(c) 15%
(d) 20%
(e) 25%
Solution: (a)
Let the rate of interest be R percent per annum
\(\displaystyle \frac{{400\times 2\times R}}{{100}}+\frac{{550\times 4\times R}}{{100}}+\frac{{1200\times 6\times R}}{{100}}=1020\)
\(\displaystyle \Rightarrow \) 8P+22P+72P=1020
\(\displaystyle \Rightarrow \) 102P=1020
\(\displaystyle \Rightarrow \) \(\displaystyle R=\frac{{1020}}{{102}}=10\%\)
60. Manoj deposited ₹ 29400 for 6 years at a simple interest. He got ₹ 4200 as interest after 6 years. The annual rate of interest was
