Ratio stock problem

Question

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.

Answer 1

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.

Answer 2

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: $$C: \frac35\cdot 250=150\\ D: \frac25\cdot 250=100$$