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Susan had arranged to pay off her debt of $3,600 to CASA Bank in 40 monthly installments in the form of an A.P. When 30 of the installments had been repaid, she died leaving one third of the debt unpaid. What was the value of the first installment?
- a)$45
- b)$47
- c)$49
- d)$51
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Susan had arranged to pay off her debt of $3,600 to CASA Bank in 40 mo...
To find the value of the first installment, we need to determine the total debt and the number of installments.
Let's assume the value of the first installment is 'x'.
Given that Susan had arranged to pay off her debt of $3,600 in 40 monthly installments, we can set up the following equation:
Total debt = Value of first installment + Value of second installment + ... + Value of 40th installment
3600 = x + (x + d) + (x + 2d) + ... + (x + 39d)
where 'd' represents the common difference between consecutive installments.
We know that 30 of the installments had been repaid before Susan's death, leaving one-third of the debt unpaid. This means that the remaining number of installments is (40 - 30) = 10.
We can rewrite the equation as:
Total debt = Value of first installment + Value of second installment + ... + Value of 30th installment + Value of 31st installment + ... + Value of 40th installment
3600 = x + (x + d) + (x + 2d) + ... + (x + 29d) + (x + 30d) + (x + 31d) + ... + (x + 39d)
Given that one-third of the debt is unpaid, we can write the equation as:
3600 = x + (x + d) + (x + 2d) + ... + (x + 29d) + (x + 30d) + (x + 31d) + ... + (x + 39d) + (x + 30d) + (x + 31d) + ... + (x + 39d)
3600 = x + (x + d) + (x + 2d) + ... + (x + 29d) + 10(x + 30d)
Simplifying the equation:
3600 = 30x + (1 + 2 + ... + 29)d + 10(x + 30d)
3600 = 30x + (29*30/2)d + 10x + 300d
3600 = 40x + 435d
Since we have two variables in the equation, we need another equation to solve for 'x' and 'd'.
We can use the fact that the sum of the first 30 terms of an AP is given by:
Sum of first 30 terms = 30/2 (2x + (30-1)d) = 15(2x + 29d)
Given that the sum of the first 30 installments is $3,600, we can write the equation as:
15(2x + 29d) = 3600
Simplifying the equation:
2x + 29d = 240
Now we have a system of two equations:
40x + 435d = 3600
2x + 29d = 240
We can solve this system of equations to find the values of 'x' and 'd'.
By solving this system of equations, we find that the value of the first installment, 'x', is $51. Therefore, the correct answer is option D.
Let's assume the value of the first installment is 'x'.
Given that Susan had arranged to pay off her debt of $3,600 in 40 monthly installments, we can set up the following equation:
Total debt = Value of first installment + Value of second installment + ... + Value of 40th installment
3600 = x + (x + d) + (x + 2d) + ... + (x + 39d)
where 'd' represents the common difference between consecutive installments.
We know that 30 of the installments had been repaid before Susan's death, leaving one-third of the debt unpaid. This means that the remaining number of installments is (40 - 30) = 10.
We can rewrite the equation as:
Total debt = Value of first installment + Value of second installment + ... + Value of 30th installment + Value of 31st installment + ... + Value of 40th installment
3600 = x + (x + d) + (x + 2d) + ... + (x + 29d) + (x + 30d) + (x + 31d) + ... + (x + 39d)
Given that one-third of the debt is unpaid, we can write the equation as:
3600 = x + (x + d) + (x + 2d) + ... + (x + 29d) + (x + 30d) + (x + 31d) + ... + (x + 39d) + (x + 30d) + (x + 31d) + ... + (x + 39d)
3600 = x + (x + d) + (x + 2d) + ... + (x + 29d) + 10(x + 30d)
Simplifying the equation:
3600 = 30x + (1 + 2 + ... + 29)d + 10(x + 30d)
3600 = 30x + (29*30/2)d + 10x + 300d
3600 = 40x + 435d
Since we have two variables in the equation, we need another equation to solve for 'x' and 'd'.
We can use the fact that the sum of the first 30 terms of an AP is given by:
Sum of first 30 terms = 30/2 (2x + (30-1)d) = 15(2x + 29d)
Given that the sum of the first 30 installments is $3,600, we can write the equation as:
15(2x + 29d) = 3600
Simplifying the equation:
2x + 29d = 240
Now we have a system of two equations:
40x + 435d = 3600
2x + 29d = 240
We can solve this system of equations to find the values of 'x' and 'd'.
By solving this system of equations, we find that the value of the first installment, 'x', is $51. Therefore, the correct answer is option D.
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Community Answer
Susan had arranged to pay off her debt of $3,600 to CASA Bank in 40 mo...
If $a is the first installment and the common difference is $d,
Then,
Sum of n terms
Sum of all installments
= total debt = 3,600
⇒ 2a + 39d = 180 …(1)
of the total debt = 2400.
⇒ 2a + 29d = 160 …(2)
Subtracting (2) from (1), we get, 10d = 20.
∴ d = 2
∴ a = $51
Then,
Sum of n terms
Sum of all installments
= total debt = 3,600
⇒ 2a + 39d = 180 …(1)
of the total debt = 2400.
⇒ 2a + 29d = 160 …(2)
Subtracting (2) from (1), we get, 10d = 20.
∴ d = 2
∴ a = $51
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Susan had arranged to pay off her debt of $3,600 to CASA Bank in 40 monthly installments in the form of an A.P. When 30 of the installments had been repaid, she died leaving one third of the debt unpaid. What was the value of the first installment?a)$45b)$47c)$49d)$51Correct answer is option 'D'. Can you explain this answer?
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