Maths-
General
Easy

Question

A Saving account has a balance of $1 . Savings plan A will add $1000 to an account each month , and plan B will double the amount each month ?
a. Which plan is better in the short run ? for How long , Explain.
b. Which plan is better in the long run ? Explain.

hintHint:

1. When the difference between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant. i.e. y(n)- y(n-1) is constant for any value of n, the function is known as a linear function.
2. When the ratio between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant i.e. y(n)/y(n-1) is constant for any value of n, the function is known as an exponential function.

The correct answer is: Plan A is better in the short run for around 13 months. After which, Plan B is better in the long run.


    Step-by-step solution:-
    From the given information, we get-
    Opening balance in Savings account = $ 1.
    Let the function be represented by 'y'.
    For Savings plan A-
    Rate of change = +1,000 every month.
    Since every month the same amount is added to the balance, the first difference = constant = 1,000.
    Hence, Savings plan A can be represented by a linear function.
    A linear function is always represented in the form-
                                                Y = a + bX ....................................... (where a represents the starting point and b represents the common difference)
                                             ∴ Y = 1 + 1,000X ................................ (starting point is $ 1 and rate of increase is 1,000) .................. (Equation i)
    For Savings plan B-
    Rate of change = *2 every month.
    Since ratio of balance in every consecutive month will remain the same i.e. 2 because every month balance will be doubled, Savings plan B can be represented by an exponential function.
    A linear function is always represented in the form-
                                               Y = aX ....................................... (where a represents the rate of change)
                                            ∴ Y = 2X ....................................... (Rate of change is 2 times every month) .......................................... (Equation ii)
    We can plot the above equations on a graph and observe the movement.
    a). As we observe from the adjacent graph, in the short run, Savings plan A is better.
    This trend can be observed till around the 13th month. From the 14th month, Savings in Plan B exceeds the savings from Plan A.
    b). As we observe from the adjacent graph, in the long run, Savings plan B is better.
    Final Answer:-
    ∴ Plan A is better in the short run for around 13 months. After which, Plan B is better in the long run.

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