61. A man lent ₹ 60,000, partly at 5% and the rest at 4% simple interest. If the total annual interest is ₹ 2560, the money lent at 4% was
(a) ₹ 40000
(b) ₹ 44000
(c) ₹ 30000
(d) ₹ 45000
(e) ₹ 48000
Solution: (b)
Let the amount lent at 4% be x
Therefore, Amount lent at 5% = (60000 – x )
According to the question,
\(\displaystyle \frac{{(60000-x)\times 5\times 1}}{{100}}+\frac{{x\times 4\times 1}}{{100}}\)
= 2560
\(\displaystyle \Rightarrow \) 300000 – 5x + 4x = 256000
\(\displaystyle \Rightarrow \) x = 300000 – 256000 = ₹ 44000
62. A sum of money at some rate of simple interest amounts to ₹ 2,900 in 8 years and to ₹ 3,000 in 10 years. The rate of interest per annum is
(a) 4%
(b) \(\displaystyle 2\frac{1}{2}\)
(c) 3%
(d) 2%
(e) 5%
Solution: (d)
Principal + interest for 8 years= ₹ 2900… (i)
Principal + interest for 10 years = ₹ 3000 … (ii)
Subtracting equation (i) from (ii)
Interest for 2 years = ₹ 100
Therefore, Interest for 8 years = \(\displaystyle \frac{{100}}{2}\times 8=400\)
From equation (i),
Principal = (2900 – 400) = ₹ 2500
\(\displaystyle \Rightarrow \)Rate = \(\displaystyle \frac{{SI\times 100}}{{Time\times \Pr incipal}}\)
\(\displaystyle \frac{{400\times 100}}{{8\times 2500}}=2\%\)
Alternate method,
R = \(\displaystyle (\frac{{{{A}_{1}}-{{A}_{2}}}}{{{{A}_{2}}{{T}_{1}}-{{A}_{1}}{{T}_{2}}}})\times 100\)
\(\displaystyle (\frac{{2900-3000}}{{3000\times 8-2900\times 1000}})\times 100\)
\(\displaystyle (\frac{{-100}}{{24000-29000}})\times 100\)
\(\displaystyle \frac{{-100}}{{-500}}\times 100=2\%\)
63. In how many years will a sum of ₹ 3,000 yield a simple interest of ₹ 1,080 at 12% per annum ?
64. A sum of money amounts to ₹ 850 in 3 years and to ₹ 925 in 4 years at some rate of simple interest. The sum is :
(a) ₹ 550
(b) ₹ 600
(c) ₹ 625
(d) ₹ 700
(e) ₹ 750
Solution: (c)
Interest for 1 year = ₹ (925 – 850) = ₹ 75
Therefore, If a sum becomes \(\displaystyle {{a}_{1}}\) in \(\displaystyle {{t}_{1}}\) years and \(\displaystyle {{a}_{2}}\) in \(\displaystyle {{t}_{2}}\) years then rate of interest
= \(\displaystyle \frac{{100({{a}_{2}}-{{a}_{1}})}}{{({{a}_{1}}{{t}_{2}}-{{a}_{2}}{{t}_{1}})}}\%\)
= \(\displaystyle \frac{{100(952-850)}}{{850\times 4-3\times 925}}=\frac{{7500}}{{625}}=12\%\)
\(\displaystyle \Rightarrow \) Principal = \(\displaystyle \frac{{SI\times 100}}{{time\times rate}}=\frac{{75\times 100}}{{1\times 12}}=625\)
Alternate method,
P = \(\displaystyle \frac{{{{A}_{2}}{{T}_{1}}-{{A}_{1}}{{T}_{2}}}}{{{{T}_{1}}-{{T}_{2}}}}\)
= \(\displaystyle \frac{{925\times 3-850\times 4}}{{3-4}}\)
= \(\displaystyle \frac{{2775-3400}}{{-1}}=\frac{{-625}}{{-1}}=625\)
65. The sum lent at 5% per annum (i.e. 365 days) simple interest, that produces interest, of ₹ 2.00 a day, is
66. A sum of money becomes 3 times in 5 years at simple interest. In how many years will the same sum become 6 times at the same rate of simple interest?
(a) 10 years
(b) 12 years
(c) 12.5 years
(d) 10.5 years
(e) None of these
Solution: (c)
A sum of money becomes 3 times in 5 years at simple interest.
3 times \(\displaystyle \Rightarrow \) 100% to 300%
6 times \(\displaystyle \Rightarrow \) 100% to 600%
Therefore, 3 times in 5 years ⇒ (300 – 100)% in 5 years
\(\displaystyle \Rightarrow \) 40% in 1 year
\(\displaystyle \Rightarrow \) Rate of interest per annum = 40%.
Now, Time taken to become 6 times
\(\displaystyle \Rightarrow \) (600 – 100)%/40%
\(\displaystyle \Rightarrow \) 12.5
Therefore, the sum of money becomes 6 times in 12.5 years
67. How much amount given for 4 years at 5% simple interest will be equal to Rs. 500 loan given for 4 years at 6% simple interest?
(a) 550
(b) 570
(c) 600
(d) 650
(e) None of these
Solution: (c)
If ‘p’ be the principal and ‘r’ be the rate of interest and ‘t’ be the time then simple interest = \(\displaystyle \frac{{PRT}}{{100}}\)
Rs. 500 loans given for 4 years at 6% simple interest
Interest = (\(\displaystyle \frac{{500\times 6\times 4}}{{100}}\)
\(\displaystyle \Rightarrow =120\)
In other case interest should be 120
Let principal be p
Then,
\(\displaystyle \begin{array}{l}\frac{{p\times 5\times 4}}{{100}}=120\\\Rightarrow \frac{p}{5}=120\\\Rightarrow p=120\times 5\\=600\end{array}\)
68. The principal is four times the simple interest and the number of years is same as the rate of interest per annum. What is the time?
(a) 6 years
(b) 5 years
(c) 4 years
(d) 2 years
(e) None of these
Solution: (b)
Principal = 4 × simple interest
Simple interest = Principal/4
Time (T) = Rate (R)
Simple Interest = \(\displaystyle \frac{{PRT}}{{100}}\)
Let the principal be ‘4P’
\(\displaystyle \Rightarrow \)Simple Interest= \(\displaystyle \frac{1}{4}\times 4P\)=P
Let the rate of interest be ‘R’
So, Time = R
Simple Interest =\(\displaystyle \frac{{PRT}}{{100}}\)
\(\displaystyle \begin{array}{l}\Rightarrow P=\frac{{4P\times R\times R}}{{100}}\\\Rightarrow 100=4{{R}^{2}}\\\Rightarrow {{R}^{2}}=\frac{{100}}{4}=25\\\Rightarrow R=5\end{array}\)
69. A sum was invested at simple interest at a certain interest for 2 years. It would have fetched Rs. 60 more had it been invested at 2% higher rate. What was the sum?
(a) Rs. 1000
(b) Rs. 1200
(c) Rs. 1300
(d) Rs. 1500
(e) Rs. 2000
Solution: (d)
Let the rate be R at which Principal P is invested for 2 years.
According to question,
{Interest at Rate (R+2)%} – (Interest at rate R%)=Rs. 60
\(\displaystyle \begin{array}{l}\frac{{P\times 2\times (R+2)}}{{100}}-\frac{{P\times 2\times R}}{{100}}=60\\\frac{{2PR+4P-2PR}}{{100}}=60\\4P=60\times 100\\P=\frac{{60\times 100}}{4}=1500\end{array}\)
70. Ajay lent Rs. 400 to Raju for 2 years and Rs. 100 to Kumar for 4 years and received together from both Rs. 60 as interest. Find the rate of interest, simple interest being calculated.
(a) 5%
(b) 6%
(c) 8%
(d) 9%
(e) 10%
Solution: (a)
Let rate is R%
According to the question.
\(\displaystyle \begin{array}{l}\frac{{R\times 400\times 2}}{{100}}+\frac{{R\times 100\times 4}}{{100}}=60\\\frac{{1200R}}{{100}}=60\\12R=60\\R=\frac{{60}}{{12}}=5\end{array}\)
