Question

Liam is setting up folding chairs for a meeting. If he arranges the chairs in 9 rows of the same length, he has 7 chairs left over. If he arranges the chairs in 7 rows of that same length, he has 17 left over. How many chairs does Liam have?

Answer 1

367

Explanation:

Let there be x chair in each row of 9 rows.

So, the number of chair in 9 rows = 9x

As 7 chair left-over, so, the number of chair in a row is more than 7, i.e

`x>7\cdots(i)`

Total numbers of the chair
`= 9x+7 ...(ii)`

Similarly, let there be y chair in each row of 7 rows.

So, the number of chair in 7 rows = 7x

As 17 chair left-over, so, the number of chair in a row is more than 17, i.e

`y>17\cdots(iii)`

so total numbers of the chair
`= 7x+17 ...(iv)`

From equation (ii) and (iv),

`9x+7=7x+17`

`\Rightarrow 9x-7y=10\cdots(v)`

As x and y are the numbers of chairs in a row, so it must be a counting number.

So, to satisfy the equation (iv), the possible values of (x, y) are

`(x,y)=(5,5),(12,14), (40,50), (110,140),\cdots` and so on.

Now, from equations (i) and (iii), the possible values are

(x,y)= (40,50), (110,140), ...

Now, to get the minimum number of chairs, put x=40 in the equation (ii) or put y=50 in the equation (iv),

So, total number of chair=

`9x+7=9*40 +7 =367.`

For (x,y)=(110,140)

Total number of chair=

`9x+7=9*110 +7 =997.`

Similarly, the other possibility can be determined.

Hence, the total (minimum) number of chair is 367

Answer 2

Answer: Liam has 52 chairs.

Explanation:

Let x= Number of chairs in each row.

If he arranges the chairs in 9 rows and 7 chairs left over.

Then total chairs = 9x+7

If he arranges the chairs in 7 rows and 17 left over.

Total chairs = 7x+17

Now,
`9x+7=7x+17`

`\Rightarrow\ 9x-7x=17-7\\\\\Rightarrow\ 2x=10\\\\\Rightarrow\ x=5`

Total chairs = 9(5)+7 = 45+7= 52

Hence, Liam has 52 chairs.