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CHAPTER VI-MIXED PROBLEMS.

gave in discharge of the debt the sum of a series in arithmetical progression consisting of ${\tfrac {80}{8}}$ terms and gave also the interest accruing on those multiples of 8. The debt amount (corresponding to the sum of the series), the interest (which he paid), and the time of discharge (of that debt)-tell me, friend, after calculating, what the (respective) value of those quantities is.

The rule for arriving at the average common interest:-

77 and $77{\tfrac {1}{2}}$.^[*] Divide the sum of the (various accruing) interests by the sum of the (various corresponding) interests due for a month; the resulting quotient is the required time. The product of the (assumed) rate-time and the rate-capital is divided by this required time then multiplied by the sum of the (various accruing) interests and then divided again by the sum of the (various given) capital amounts. This gives rise to the (required) rate-interest.

An example in illustration thereof.

$78{\tfrac {1}{2}}$. In this problem, four hundreds were (separately) invested at the (respective) rates of 2, 3, 5 and 4 per cent (per mensem) for 5, 4, 2 and 3 months (respectively). What is the average common time of investment, and what the average common rate of interest?

Thus end the problems bearing on interest in this chapter on mixed problems.

 

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77 and $77{\tfrac {1}{2}}$^  The various accruing interests are the various amounts of interests accruing on the several amounts at the various rates for their respective periods.

Symbolically, ${\Big \{}{\tfrac {c_{1}\times t_{1}\times I_{1}}{T\times C}}+{\tfrac {c_{2}\times t_{2}\times I_{2}}{T\times C}}...{\Big \}}\div {\Big \{}+{\tfrac {C_{1}\times 1\times I_{1}}{T\times C}}+...{\Big \}}=t_{a}$ or average time;
and ${\tfrac {T\times C}{t_{a}}}\times {\Big \{}{\tfrac {c_{1}\times t_{1}\times I_{1}}{T\times C}}+{\tfrac {c_{2}\times t_{2}\times I_{2}}{T\times C}}+...{\Big \}}\div (c_{1}+c_{2}+...)=i_{a}$ or average interest.