An item is available for $34000\$$ cash or $20000\$$ cash down payment together with $5$ equal monthly instalments. If the rate of interest charged under the instalment plan is $30\%$ per annum, calculate the amount of each instalment.
I have understood how to solve this question (by the help of an online pdf) and have posted its solution below. But I am looking for a more intutive and quicker solution.
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Remaining payment= $14000\$$. Let each monthly instalment be x. Monthly rate of interest=$\frac{30}{100*12}=\frac1{40}$
So, $14000+\frac{14000*5}{40}=(x+\frac{x*4}{40})+(x+\frac{x*3}{40})+(x+\frac{x*2}{40})+(x+\frac{x*1}{40})+x\implies 14000+1750=5x+\frac{10x}{40}\implies 15750=\frac{21x}{4}\implies x=3000$
Cash price = $34000\$$, Cash down payment = $20000\$$, Balance to be paid in $5$ equal instalments = $14000\$$, Let each instalment be x. So, interest charged under instalment plan = $(5x – 14000)$. The buyer owes to the seller for $1st$ month=$14000$, $2nd$ month=$(14000 –x)$, $3rd$ month=$(14000 –2x)$, $4th$ month=$(14000 –3x)$, $5th$ month= $(14000 –4x)$ Therefore, total principal for one month = $[70000 – 10x]$ So, $$(5x −14000)=(70000−10x)*\frac{30}{100}*\frac{1}{12}\implies 40 (5x – 14000) = 10(7000 – x)\implies 20x – 56000 = 7000 – x\implies 21x = 63000\implies x = 3000$$ Thus, the amount of each instalment = $3000\$$