Mind-Boggling Puzzle: Solve How Many Parts Each Worker Got

[Wilmer]

Found this challenging:

A,B and C work at same speed.
When all 3 of them plant a field with D, the job gets done in 5 hours.
When all 3 of them plant the same field with E, the job gets done in 6 hours.
The field was divided between the 5 workers in proportion to their 5 work rates,
into a 2digit square number of parts, with E getting only 1 part.
A, B, C and D all got an integer number of parts.
How many parts did each get?

 

[Wilmer]

Well, that was a puzzle I found here:
Sunday Puzzle-19 - Welcome to the official website of Natalia Pogonina!

I changed the wording; answer same.
See my answer at bottom.

 

[I like Serena]

But... but... you changed the numbers in your solution.

Here's my solution.

Say $A,B,C,D,E$ are the number of parts each gets.
Let $a,b,c,d,e$ be their respective work rates in parts per hour.
And let $P$ be the square 2-digit number of parts.
Then it follows that:
\begin{array}{l}
A=B=C \\
a=b=c \\
A+B+C+D+E=P \\
E=1 \\
\frac Aa = \frac Bb = \frac Cc = \frac Dd = \frac Ee \\
5(a+b+c+d)=P \\
6(a+b+c+e)=P \\
\end{array}

We can simplify this to:
$$\left\{\begin{array}{l}
3A+D+1=P \\
\frac Aa = \frac Dd = \frac 1e \\
15a+5d=P \\
18a+6e=P \\
A,D \text{ whole numbers} \\
P \text{ square 2-digit number} \\
\end{array}\right.$$

By enumerating all square 2-digit numbers, we find $A=B=C=13,\ D=9,\ P=49$ as the only solution.
It will take them $\frac{234}{49} \approx 4 \text{ hours and }47\text{ minutes}$ to plant the field.

 

[Wilmer]

I have a good memory, but it's short(Nerd)

 

[CellCoree]

This is indeed a mind-boggling puzzle, but it can be solved using some basic principles of mathematics. Let's start by assigning variables to represent the work rates of each worker. Let A, B, and C be the work rates of workers A, B, and C respectively. Since all three workers have the same work rate, we can say that A = B = C = x.

Next, let's use the given information to form equations. We know that when all three workers plant the field with D, the job gets done in 5 hours. This can be represented as 1/x + 1/x + 1/x + 1/D = 1/5. Similarly, when all three workers plant the field with E, the job gets done in 6 hours, which can be represented as 1/x + 1/x + 1/x + 1/E = 1/6.

Now, we also know that the field was divided between the 5 workers in proportion to their work rates. This means that the number of parts each worker got is directly proportional to their work rates. So, we can set up the following proportion: A:B:C:D:E = x:x:x:1:1.

Since the field was divided into a 2-digit square number of parts, we can say that the total number of parts must be a perfect square. Let's represent this as n^2, where n is a positive integer. This means that the number of parts each worker got can be represented as nx, nx, nx, n, and n.

Now, using the proportion we set up earlier, we can write the following equations:
nx + nx + nx + n + n = n^2
3nx + 2n = n^2

We can then simplify this equation to get n(3x + 2) = n^2. Since n is a positive integer, we can divide both sides by n to get 3x + 2 = n. This means that n must be a multiple of 2, and since it is also a perfect square, it must be 4.

Substituting n = 4 into our previous equations, we get:
3x + 2 = 4
x = 2

This means that the work rate of each worker is 2, and the number of parts each worker got is: 2(2) = 4,