SOLUTION: Hi Bob had to arrange chairs in a fixed number of rows. He estimated that there would be more than 30 but less than 70 chairs. If he puts 8 chairs in one row there would be 7ch

Question 1167407: Hi
Bob had to arrange chairs in a fixed number of rows. He estimated that there would be more than 30 but less than 70 chairs. If he puts 8 chairs in one row there would be 7chairs short . If he put 7 in a row there would be 2 left.how many chairs are there.
Thanks

Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.

This problem is to find an integer number n, 30 < n < 70 such that 1) it gives the remainder 1 when divided by 8 (7 chairs short), 2) it gives the remainder 2 when divided by 7. By trial and error, we check the numbers 33, 41, 49, 57, 65 from the interval (30,70) that satisfy the first condition. Of these numbers, only 65 satisfies the second condition. Thus the number is found: it is 65. ANSWER

Solved.

/\/\/\/\/\/\/\/

I looked into the solution by @Theo,

and after reading it, I can say that his approach is TOTALLY WRONG from the very beginning to the very end.

So, for your safety, you better ignore it.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
here's what i get and what i think is wrong with this problem.

let y equal the number of chairs needed.

let x equal the number of rows.

when you have 8 chairs in a row, then:

8x + 7 = y

this means you multiply 8 * x and get 7 less chairs than you need, so you have to add 8 to 8x.

when you have 7 chairs in a row, then:

7x - 2 = y

this means you multiply 7 * x and get 2 more chairs than you need, so you have to subtract 2 from 7x.

with 8 chairs in a row, multiplied by x, you need 7 more chairs to equal y.
with 7 chairs in a row, multiplied by x, you need 2 less chairs to equal y.

your two equations that you need to solve simultaneously are:

8x + 7 = y
7x - 2 = y

subtract the second equation from the first to get:

x + 9 = 0

solve for x to get:

x = -9

that's not good because x has to be greater than zero.

but, it does give you a solution, even though the solution is the wrong sign.

so, i said:

what makes the solution positive?

the answer was to assume the reverse.

8x + 7 = y becomes 8x - 7 = y
8x - 2 = y becomes 8x + 2 = y

solve these two equations simultaneously and i get:

8x - 7 = y
7x + 2 = y

subtract the second equation form the first and i get:

x - 9 = 0

solve for x to get x = 9.

the answer appears to be valid, but the conditions are reversed.

8 * 9 - 7 = 72 - 7 = 65
7 * 9 + 2 = 63 + 2 = 65

this says the number of chairs i needed was 65.

with 8 chairs in a row, i got 72 chairs which was 7 more than i needed.
with 7 chairs in a row, i got 63 chairs which was 2 less than i needed.

the number of chairs needed was between 30 and 70.
65 is between 30 and 70, so that requirement checked out ok.

with my original assumption, based on what i thought the problem was stating, i did not get a solution.

the following graphs show what i mean.

in the first graph, the equation graphed are y = 8x + 7 and y = 7x - 2.

as you can see there is no solution.

in the second graph, the equation graphed are y = 8x - 7 and y = 7x + 2.

as you can see there is a solution at x = 9.

the solution, in both cases, had to be between y = 30 and y = 70, as stated in the problem statement.

the only time i got a solution was when y = 8x - 7 and y = 7x = 2.

what did these equations mean?

since y = the number of chairs needed and x = the number of rows, then:

y = 8x - 7 means that the number of chairs needed is equal to 8 * the number of rows - 7 which means that you get 7 more chairs than you need.
if you add 7 to both sides of this equation, you get y + 7 = 8x.
this says that 8 chairs in a row times the number of rows gets you 7 more chairs than you need.

y = 7x + 2 means that the number of chairs needs is equal to 7 * he number of rows + 2 which means that you get 2 less chairs than you need.
if you subtract 2 from both sides of this equation, you get y - 2 = 8x.
this says that 7 chairs in a row time the number of rows gets you 2 less chairs than you need.

this is opposite what the problem statement says, which is: "If he puts 8 chairs in one row there would be 7 chairs short . If he put 7 in a row there would be 2 left."

what you actually get is the opposite, which is: "If he puts 8 chairs in one row there would be 7 chairs left. If he put 7 in a row there would be 2 short."

7 chairs left means he gets 7 more chairs than he needed.
2 chairs short means he gets 2 less chairs than he needed.

anyway, this is what i think.

i believe the solution to your problem is that the number of rows is 9.

with 9 rows:

9 * 8 = 72 and 9 * 7 = 63
72 is 7 rows more than you need.
63 is 2 rows less than you need.

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