Question

A fruit company delivers its fruit in two types of boxes: large and small. A delivery of 2 large boxes and 12 small boxes has a total weight of 223 kilograms. A delivery of 5 large boxes and 3 small boxes has a total weight of 139 kilograms. How much does each type of box weigh?

Answer 1

A large box weighs 18.5 kilograms and a small box weighs 15.5 kilograms

Explanation:

Create a system of equations where l is the weight of one large box and s is the weight of one small box.

2l + 12s = 223

5l + 3s = 139

Solve by elimination by multiplying the bottom equation by -4:

2l + 12s = 223

-20l - 12s = -556

Add these together and solve for l:

-18l = -333

l = 18.5

So, a large box weighs 18.5 kilograms. Plug in 18.5 as l into one of the equations, and solve for s:

5l + 3s = 139

5(18.5) + 3s = 139

92.5 + 3s = 139

3s = 46.5

s = 15.5

A large box weighs 18.5 kilograms and a small box weighs 15.5 kilograms.

Answer 2

9514 1404 393

Answer:

• large: 18.5 kg
• small: 15.5 kg

Explanation:

Two equations for the total weight can be written in general form as ...

2L +12S -223 = 0

5L +3S -139 = 0

These can be solved using the "cross multiplication method" by computing three differences:

d1 = (2)(3) -(5)(12) = 6 -60 = -54

d2 = (12)(-139) -(3)(-223) = -1668 +669 = -999

d3 = (-223)(5) -(-139)(2) = -1115 +278 = -837

Then the weighs of the boxes are ...

1/d1 = L/d2 = S/d3

L = d2/d1 = -999/-54 = 18.5

S = d3/d1 = -837/-54 = 15.5

The large box weighs 18.5 kg; the small box weighs 15.5 kg.

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Additional comment

If the coefficients are written in two rows of four numbers, with the first repeated at the end, the pattern of differences can be seen to be (ad -cb) for [a, b] in the first row and [c, d] in the second row of a pair of adjacent columns. This method of solution is fully equivalent to solution using Cramer's Rule for a system of equations represented by a matrix equation.