Zane received a quote to move his belongings to his new house. The estimate from the mover is $675 for 6 hours of packing and 5 hours of loading on the weekend. The
cost during the week is $425 for 4 hours of packing and 3 hours of loading. If the
mover charges a set hourly rate for packing and another set hourly rate for loading,
what are these hourly rates?
What is the LCM for packing? What is the LCM for loading?

9514 1404 393

Step-by-step explanation:

We can let P and L represent the hourly costs of packing and loading, respectively. The two estimates can be represented by the equations ...

  6P +5L = 675

  4P +3L = 425

The LCM of the coefficients of P is LCM(6, 4) = 12.

The LCM of the coefficients of L is LCM(5, 3) = 15.

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Each LCM is the product of the numbers, divided by their greatest common factor. For 4 and 6, the product is 24, and both even numbers have a factor of 2, so the LCM of 4 and 6 is 24/2 = 12. The numbers 3 and 5 have no common factors, so the LCM is simply their product.

The LCM is useful if you're going to solve the equations by "elimination". Here, the LCM of 12 means we can eliminate P by making its coefficient be 12 in both equations. Multiplying the first equation by 2, we can subtract 3 times the second equation to eliminate P:

  2(6P +5L) -3(4P +3L) = 2(675) -3(425)

  12P +10L -12P -9L = 1350 -1275

  L = 75 . . . . . simplify

Similarly, we can eliminate L by making its coefficients be 15.

  5(4P +3L) -3(6P +5L) = 5(425) -3(675)

  20P +15L -18P -15L = 2125 -2025

  2P = 100 . . . . simplify

  P = 50 . . . . . . divide by 2

The hourly rates are ...

  $50 per hour for packing

  $75 per hour for loading

40 packing and 80 moving