Percentage and True Discount

Explanation / Important formulas:

  • PERCENTAGE: A percentage is defined as a number represented as a fraction of 100. Percentages are used to express numbers between zero and one. It is used to compare things and use it in ratios. It is denoted by the symbol %. Percentage = (Value / Totalvalue) * 100, thus 1/ 4 = (1/4 x 100)% = 25%
  • Percentage Increase/Decrease: 
    • If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is: [R / (100 + R) x 100]%
    • If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is: [R / (100 – R) x 100]%

    Results on Population: Let the population of a town be P now and suppose it increases at the rate of R% per annum, then:

    • Population after n years = P(1+ R/100)n
    • Population n years ago = P / (1+ R/100)n

    Results on Depreciation: Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:

    • Value of the machine after n years = P (1 – R / 100)n
    • Value of the machine n years ago = P /(1- R /100)n
    • If A is R% more than B, then B is less than A by [R /(100 +R) x 100]%
    • If A is R% less than B, then B is more than A by [R /(100 – R) x 100]%

    TRUE DISCOUNTThe “True discount” is a term used when you are promised a given amount of money in the future. Since money collect interest, the actual value of that promise is less than the face value

    Example:

    Consider a man has to pay Rs. 156 after 4 years and the rate of interest is 14% per annum. Clearly, Rs. 100 at 14% will amount to R. 156 in 4 years. So, the payment of Rs. now will clear off the debt of Rs. 156 due 4 years hence. We say that:

    • Sum due = Rs. 156 due 4 years hence;
    • Present Worth (P.W.) = Rs. 100;
    • True Discount (T.D.) = Rs. (156 – 100) = Rs. 56 = (Sum due) – (P.W.).
    • We define: T.D. = Interest on P.W..; Amount = (P.W.) + (T.D.)
    • Interest is reckoned on P.W. and true discount is reckoned on the amount.

    IMPORTANT FORMULAS

    Let rate = R% per annum and Time = T years. Then,

    • P.W. = 100 x Amount / 100 + (R x T) = 100 x T.D / R x T
    • T.D. = (P.W) x R x T / 100 = Amount x R x T / 100+(R x T)
    • Sum = (S.I) x (T.D) / (S.I) – (T.D)
    • (S.I.) – (T.D.) = S.I. on T.D.
    • When the sum is put at compound interest, then P.W. = Amount / (1+R / 100)T

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