Liam is setting up folding chairs for a meeting. If he arranges the chairs in 5 rows of the same length, he has 5 chairs left over. If he arranges the chairs in 3 rows of that same length, he has 15 left over. How many chairs does Liam have?
Liam is setting up folding chairs for a meeting. If he arranges the chairs in 6 rows of the same length, he has 11 chairs left over. If he arranges the chairs in 4 rows of that same length, he has 17 left over. How many chairs does Liam have?
Let's solve this problem step by step.
Step 1: Given that Liam has 5 chairs left over when arranging them in 5 rows of the same length.
Let's assume the number of chairs in each row is x. So, the total number of chairs in 5 rows would be 5x. We are also given that there are 5 chairs left over, which means the total number of chairs must be evenly divisible by 5. Therefore, we can write the equation as:
5x + 5 = N, where N is the total number of chairs.
Step 2: Given that Liam has 15 chairs left over when arranging them in 3 rows of the same length.
Using the same logic as before, let's assume the number of chairs in each row is also x. So, the total number of chairs in 3 rows would be 3x. We are given that there are 15 chairs left over, which means the total number of chairs must be evenly divisible by 3. Therefore, we can write the equation as:
3x + 15 = N.
Step 3: Now we have two equations:
5x + 5 = N
3x + 15 = N
Step 4: Since both equations are equal to N, we can equate them to each other.
5x + 5 = 3x + 15
Step 5: Simplify and solve for x.
2x + 5 = 15
2x = 10
x = 5
Step 6: Now that we know the value of x, we can substitute it into any of the two equations to find N.
Substituting x = 5 into 5x + 5 = N, we get:
5(5) + 5 = 25 + 5 = N
N = 30.
Therefore, Liam has 30 chairs in total.