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hellen had to arrange some chairs in a fixed number of row. she estimated that there were more than 30 but fewer than 70chairs. if she put 8chairs in one row, she would be 7chairs short. if she put 7chairs in one row, she would have two chairs left. how many chairs were there?​

2 Answers

1 vote
There are a total of 65 chairs

Step by step

Let x be the amount of rows

For the 7 chairs in each row but two extra you can make and equation of 7x +2

For the 8 chairs each row but 7 short you can make the equation of 8x - 7

Set the equations equal to each other and solve for X which gives you the amount of rows.

Take X which would be 9 and then put it back into either of the equations and solve resulting in the answer if 65 chairs.
User Celal
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Let's use algebra to solve the problem.

Let's assume that there are x chairs in total.

From the first condition, we know that:

30 < x < 70

From the second condition, we know that if she put 8 chairs in one row, she would be 7 chairs short. This can be represented by the equation:

x = 8a - 7, where a is the number of rows.

Similarly, from the third condition, we know that if she put 7 chairs in one row, she would have two chairs left. This can be represented by the equation:

x = 7b + 2, where b is the number of rows.

Now we can set these two equations equal to each other, since they both represent the same number of chairs:

8a - 7 = 7b + 2

Simplifying this equation, we get:

8a - 7b = 9

We want to find a solution where a and b are both positive integers, since we are dealing with rows of chairs. We can start by trying different values of a and seeing if we get a corresponding integer value of b that satisfies the equation.

Let's try a = 2:

8(2) - 7b = 9

16 - 7b = 9

7b = 7

b = 1

This gives us a solution where there are 16 chairs (2 rows of 8 chairs) and 1 chair left over. However, we know that there are more than 30 chairs, so we need to try a larger value of a.

Let's try a = 5:

8(5) - 7b = 9

40 - 7b = 9

7b = 31

b = 31/7

This is not a whole number, so a = 5 does not work.

Let's try a = 6:

8(6) - 7b = 9

48 - 7b = 9

7b = 39

b = 39/7

This is also not a whole number, so a = 6 does not work.

Let's try a = 7:

8(7) - 7b = 9

56 - 7b = 9

7b = 47

b = 47/7

This is also not a whole number, so a = 7 does not work.

Let's try a = 8:

8(8) - 7b = 9

64 - 7b = 9

7b = 55

b = 55/7

This is also not a whole number, so a = 8 does not work.

Finally, let's try a = 9:

8(9) - 7b = 9

72 - 7b = 9

7b = 63

b = 9

This gives us a solution where there are 72 chairs (9 rows of 8 chairs) and 2 chairs left over when 7 chairs are put in each row. This satisfies all the conditions, so the answer is 72 chairs.

User Steve Quezadas
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