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# A trust fund has Rs.30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Determine how to divide Rs.30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of(a) Rs.1800(b) Rs.2000 Time for investment=1 year.  Hint: In this type of question, where total principal is given but it has to be divided into two parts such that on calculating the simple interest on the two principal at two given rates the required simple interest is obtained. Here we proceed by assuming the first variable as an unknown value and second is obtained as total minus the unknown value .

In the question, it is given that the total money invested =Rs.30,000.
Let us assume that the money invested in first type of bond= Rs.x
$\therefore$ The money invested in another type of bond = Rs.(30,000-x).
Given that:
Rate of interest for first fund = 5% per annum.
Rate of interest for second year= 7% per annum.
Time for investment= 1 year.
Now, we know that the formula for calculating simple interest is given by:
${\text{Simple interest = }}\dfrac{{{\text{P}} \times {\text{R}} \times {\text{T}}}}{{100}}$
Where P is principal, R is rate per annum and T is time in year.
For bond 1:
Principal(P) = Rs.x.
Rate of interest = 5% per annum.
Time of investment = 1year.
So, using the formula for calculating simple interest, we get:
${\text{Simple interest = (SI}}{{\text{)}}_1}{\text{ = }}\dfrac{{{\text{P}} \times {\text{R}} \times {\text{T}}}}{{100}} = \dfrac{{{\text{x}} \times {\text{5}} \times {\text{1}}}}{{100}} = \dfrac{{5x}}{{100}}.$ -----(1)
For bond 2:
Principal = Rs.(30,000-x)
Rate of interest = 7% per annum.
Time of investment = 1 year.
So, using the formula of simple interest, we get:
${\text{Simple interest = (SI}}{{\text{)}}_2}{\text{ = }}\dfrac{{{\text{P}} \times {\text{R}} \times {\text{T}}}}{{100}} = \dfrac{{(30000 - x) \times 7 \times 1}}{{100}}.$ ----(2)

(a) It is given that total simple interest = Rs.1800.
Therefore, we can write:
${({\text{SI)}}_1} + {({\text{SI)}}_2} = 1800$
Putting the values from Equation 1 and 2, we get:
$\dfrac{{5x}}{{100}} + \dfrac{{7 \times (30000 - {\text{x}})}}{{100}} = 1800 \\ \Rightarrow 5{\text{x}} + 210000 - 7{\text{x}} = 180000. \\ \Rightarrow 5{\text{x}} - 7{\text{x}} = 180000 - 210000 \\ \Rightarrow - 2{\text{x}} = - 30000 \\ \Rightarrow {\text{x}} = \dfrac{{30000}}{2} = 15000 \\$
So, amount invested in bond 1=Rs.15,000.
And amount invested in bond 1=Rs.(30,000-15000)=Rs.15,000.
(b) It is given that total simple interest=Rs.2000.
Therefore, we can write:
${({\text{SI)}}_1} + {({\text{SI)}}_2} = 2000$ .
Putting the values from Equation 1 and 2, we get:
$\dfrac{{5x}}{{100}} + \dfrac{{7 \times (30000 - {\text{x}})}}{{100}} = 2000 \\ \Rightarrow 5{\text{x}} + 210000 - 7{\text{x}} = 200000. \\ \Rightarrow 5{\text{x}} - 7{\text{x}} = 200000 - 210000 \\ \Rightarrow - 2{\text{x}} = - 10000 \\ \Rightarrow {\text{x}} = \dfrac{{10000}}{2} = 5000 \\$
So, the money invested in bond 1=Rs.5000.
And the money invested in bond 2=Rs(30,000-5,000)=Rs.25,000.
Note: In this type of question, you have to first consider the first principal as an unknown variable x, and then the second principal becomes Rs.(30,000-x). Now, calculate the individual simple interest at the given rate and given time and then add them up and equate to given simple interest and finally solve the linear equation to get the two principals.
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