Typical Kampong Problem..blergh!

During school holiday,Pak Hamid offers Hakimi to do a part time jobs to get an extra pocket money. He’ll pay RM 14 per hour for the painting his house and RM21 per hours to take care of his farm. By the end of the school holiday,Hakimi get a well paid RM147 from Pak Hamid. How many hours possible did Hakimi spend for each of the respective job?

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Solution.

The equation derived is 21x + 14y = 147

Let x represent the hours spent for taking care of Pak Hamid’s farm.
Let y represent the hours spent for the painting job.

Using Euclidean Algorithm we evaluate gcd(21,14)
21 = 1•14 + 7
14 = 2•7 + 0

Hence,the gcd(21,14) equals to 7.

Since gcd(21,14) equals to 7 and 7|147,hence the equation 21x+14y=147 has a solution.

To obtain the integer 7 as a linear combination of 21 and 14,we work backward from the previous computation as follows:
7 = 21 - 1•14

Hence,7=1•21- 1•14


Upon multiplying this equation by 21,we obtain
147 = 21•21+(-21)•14

Thus, and .
The general solution of 21x+14y=147 is given by the equation and

Since x and y represent time,hence it cannot be negative.

21+2t>0 and -21-3t>0
Hence,
t>-10.5 and t<-7.

The overlapped real value of t is -8,-9 and -10.
Substitute the value of t obtained to the general solution of x and y,we’ll get;
When t=-8,x=5 and y=3
When t=-9,x=3 and y=6
When t=-10,x=1 and y=9.

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