XavierAlexander wrote:

A warehouse has \(n\) widgets to be packed in \(b\) boxes. Each box can hold \(x\) widgets. However, \(n\) is not evenly divisible by \(x\), so one of the boxes will contain fewer than \(x\) widgets. Which of the following expresses the number widgets in that box, assuming all other boxes are filled to their capacity of \(x\) widgets?

A. \(n−bx\)

B. \(n−bx+x\)

C. \(n−bx−x\)

D. \(nx−bx\)

E. \(n-x\frac{(bx)}{(b-1)}\)

Cheers !

Official solution from

Veritas Prep.

This word problem is essentially asking about the relationship between dividend, divisor, quotient, and remainder. The total number of widgets, \(n\), is the dividend (the number you start with and then divide). The number of widgets per box \((x)\) serves as the divisor (the number you're dividing by), and the number of FULL boxes will be the quotient. Since the last box will not be full with x widgets, that box is not part of the quotient, so the quotient will be \(b−1\) (the total number of boxes, minus the one box that isn't full). Then the remaining widgets, to go in the last box, will be the remainder.

Algebraically, you can set this up using the Quotient/Remainder relationship (or formula):

Dividend = Divisor * Quotient + RemainderSince you know which variables correspond to each value other than remainder, you can plug those in:

\(n=(b−1)(x)+R\)

And then solve for \(R\):

\(R=n−(b−1)(x)\)

Since this form does not match any answer choices, distribute the multiplication:

\(R=n−(bx−x)\)

Which then becomes (as you distribute the negative across parentheses):

\(R=n−bx+x\)

, which is answer choice B. Alternatively since this problem involves variables in all answer choices, you can simply pick numbers to test the relationship. If you were to say that there were 91 widgets to start, being packed in 10 boxes of 10 widgets each. then your variables are:

\(n=91\);\(x=10\);\(b=10\)

And you know that that will give you 9 boxes of 10 with one remaining for the last box.

Plug those in to the answer choices and you will see that only answer choice B will give you the correct remainder of \(1\), so \(B\) must be correct.

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