Question

solve.
1
roman is having people over to watch football. he has 12 crackers and 8 slices of cheese. if each plate must be identical, with no food left over, what is the greatest number of plates he could make? what would be on each plate?
2
three consecutive integers are such that when they are added in increasing order and multiplied by 2, 3 and 4 respectively, they equal 74. find these numbers.

Explanation:

Problem 1

Step1: Find GCD of 12 and 8

To find the greatest number of identical plates, we need the greatest common divisor (GCD) of 12 and 8. The factors of 12 are \(1, 2, 3, 4, 6, 12\) and the factors of 8 are \(1, 2, 4, 8\). The GCD of 12 and 8 is 4.

Step2: Calculate food per plate

For crackers: \(\frac{12}{4} = 3\) crackers per plate. For cheese: \(\frac{8}{4} = 2\) slices of cheese per plate.

Step1: Define the integers

Let the three consecutive integers be \(n\), \(n + 1\), and \(n + 2\).

Step2: Set up the equation

According to the problem, \(2n+3(n + 1)+4(n + 2)=74\).

Step3: Simplify the equation

First, expand the terms: \(2n+3n + 3+4n + 8 = 74\). Then combine like terms: \(9n+11 = 74\).

Step4: Solve for \(n\)

Subtract 11 from both sides: \(9n=74 - 11=63\). Then divide by 9: \(n=\frac{63}{9}=7\).

Step5: Find the integers

The first integer is \(n = 7\), the second is \(n + 1=8\), and the third is \(n + 2 = 9\).

Answer:

The greatest number of plates is 4. Each plate will have 3 crackers and 2 slices of cheese.

Problem 2