Question

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?

Answer 1

40 packing and 80 moving

Answer 2

9514 1404 393

Answer:

• packing: $50 per hour
• loading: $75 per hour
• packing LCM: 12; loading LCM: 15

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.

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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