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QUESTION: "The total number of panes lack 32 of being 9 times the L.C.M. of the sum of the panes in 3 windows, one of each type."

I think this means if 32 panes were added, then that number would equal the number calculated from the remainder of statement.

ANSWER: "A building is constructed with 3 floors. The windows on each floor have several panes of glass. First floor - 24 panes of glass in each window. Second floor - 20 panes of glass in each window. Third floor - 16 panes of glass in each window. Ratio of number of windows of the 3 types are 4:2:1."

The first, second, and third floors have 4n, 2n, and n windows, respectively.

total number of panes = 24(4n) + 20(2n) + 16(n) = 152n

"The total number of panes lack 32 of being 9 times the L.C.M. of the sum of the panes in 3 windows, one of each type."

Windows have 24, 20, or 16 panes.

The least common multiple of 24, 20, and 16 is 240.

152n + 32 = 9·240 = 2160

152n = 2160 - 32 = 2128

n = 2128/152 = 14

total number of windows = 4n+2n+n = 98

---------- FOLLOW-UP ----------

QUESTION: "The total number of panes lack 32 of being 9 times the L.C.M. of the sum of the panes in 3 windows, one of each type."

Confusing statement in the problem- Is the wording supposed to make you think more or just a poorly worded part of the problem ?

I discovered the problem in a BEGINNING algebra book from 1902.

"Is the wording supposed to make you think more or just a poorly worded part of the problem ?"

Both, I think.

It was actually the last half of the statement that I found confusing, because it was poorly worded:

...the L.C.M. of the sum of the panes in 3 windows, one of each type.

"The sum" implies a single quantity, but LCM is based on two or more numbers. It would be more accurate to say "the LCM of the numbers of panes in each type of window."

I also believe the wording was supposed to make you think more, because otherwise they would have simply said "the LCM of 24, 20, and 16."

As for the first part of the statement:

The total number of panes lack 32 of being 9 times...

The phrasing sounds strange, but it may simply reflect the different style of writing in that time and locale.

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