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This problem is for children and I found it while reading Ray's Intellectual Arithmetic. It is under the ratios topic.

Problem :

$C$ and $D$ join their stocks in trade; $C$ puts in $50\$$ for 4 months, and $D$ $60\$$ for 5 months. They gain $45\$$. What is the share of each?

Solution :

$C$'s $50\$$ for 4 months = $200\$$ for 1 month. $D$'s $60\$$ for 5 months = $300\$$ for 1 month. $200\$ + 300\$ = 500\$$. $C$ has $\frac{2}{5}$ of $45\$ = 18\$$ and $D$ has $\frac{3}{5}$ of $45\$ = 27\$$.

What is vague to me here is $C$'s $50\$$ for 4 months is the same as $200\$$ for 1 month. Why is this the same and how should children know this?

EDIT

Here is a similar question from the same book:

Problem :

At the beginning of the year $C$ went into business with a capital of $600\$$, four months after $D$ formed a partnership with $C$ and put in $600\$$. The gain for the year was $250\$$. What was each one's share?

Solution :

$C$'s capital = $600\$$ for 12 months, or $7200\$$ for 1 month. $D$'s $600\$$ for 8 months, or $4800\$$ for 1 month. $7200\$ + 4800\$ = 12000\$$

Rest of the solution with fractions is the same.

Michael Munta
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2 Answers2

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There are concepts of simple and compound interest, whose formulas are: $$FV=PV(1+rt);\\ FV=PV(1+r)^t,$$ respectively, where $FV$ - future value, $PV$ - present value, $r$ - interest rate, $t$ - time (years, months).

For example, $\$100$ invested under simple interest of $10\%$ for $4$ years is: $$FV=100(1+4\cdot 0.1)=140.$$ Think of the shares earning simple interest. Then: $$C: FV=50(1+r\cdot 4)=50+200r;\\ D: FV=60(1+r\cdot 5)=60+300r.$$ They together earned $\$45$, so: $$200r+300r=45 \Rightarrow r=\frac{45}{500}.$$ So: $$C:200\cdot \frac {45}{500}=\frac25\cdot 45=18;\\ D: 300\cdot \frac {45}{500}=\frac35\cdot 45=27.$$

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Addendum after edit.

Another way to look at it. The amount of investment and the time are inversely proportional: $$A\cdot T=k$$ So, in problem $2$: $C$ invests $\$600$ for $12$ months, which is the same as investing $\$7200$ for $1$ month: $$600\cdot 12=7200\cdot 1$$ Similarly, $D$ invests $\$600$ for $8$ months, which is the same as investing $\$4800$ for $1$ month: $$600\cdot 8=4800\cdot 1$$ Joint investment of $C$ and $D$ for $1$ month is: $$7200+4800=12000$$ The ratio of each: $$C: \frac{7200}{12000}=\frac35;\\ D: \frac{4800}{12000}=\frac25.$$ The joint earning $\$250$ must be divided as: $$C: \frac35\cdot 250=150\\ D: \frac25\cdot 250=100$$

farruhota
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  • Ok, but the core concept that should be clear before solving this is that one needs to understand that when you put money in the bank the money will grow by some small percentage. That is the only way one can think of these equalities for different months? – Michael Munta Aug 22 '19 at 13:56
  • Both interest and proportion methods are shown. Yes, money deposited once grows a certain period at a certain rate and is collected at the end with accumulated interest. Note that money is not deposited every month, otherwise it will be an annuity and calculation of interest gets complicated. Good luck! – farruhota Aug 22 '19 at 18:12
  • Yes, children at this point don't know that and the problem should have at least noted how money behaves so they know how to think about the solution. – Michael Munta Aug 22 '19 at 19:33
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Imagine this problem:

Two plumbing companies work jointly on a large construction project. Company C sends 4 workers for 50 hours, and Company D sends 5 workers for 60 hours. If they receive $45,000 for the job, how should they split it?

In this case, it's a fairly clear argument that C dedicated 200 worker-hours and D 300 worker-hours, so a 40-60 split of the payment is appropriate. The problem you bring up has the same concept, but it is measuring the opportunity cost of cash in the units of dollar-months. I've never heard of this notion before and I'm not sure that it's actually a "thing" in finance, but it's not utterly implausible.