Question

solve.
1
beatrice is finishing her social studies project using 20 newspaper clippings and 4 pictures. she wants each page in her project to be set up the same. as a requirement of the assignment, she must use all the items. what is the greatest number of pages beatrice can create?
2
the sum of three consecutive multiples of eight is 96. find the multiples.

Explanation:

Sub - Question 1

Step 1: Find GCD of 20 and 4

To find the greatest number of pages, we need the greatest common divisor (GCD) of the number of newspaper clippings (20) and pictures (4). The factors of 20 are \(1, 2, 4, 5, 10, 20\) and the factors of 4 are \(1, 2, 4\). The greatest common factor is 4.

Step 2: Verify

If we have 4 pages, for newspaper clippings: \(20\div4 = 5\) clippings per page. For pictures: \(4\div4=1\) picture per page. And we use all items.

Step 1: Define the multiples

Let the first multiple of 8 be \(8x\). Then the next consecutive multiple is \(8(x + 1)\) and the third is \(8(x+2)\).

Step 2: Set up the equation

The sum of these three multiples is 96, so we have the equation:
\[8x+8(x + 1)+8(x + 2)=96\]
Factor out 8:
\[8(x+(x + 1)+(x + 2))=96\]
Simplify inside the parentheses:
\[8(3x + 3)=96\]
Divide both sides by 8:
\[3x+3 = 12\]
Subtract 3 from both sides:
\[3x=9\]
Divide by 3:
\[x = 3\]

Step 3: Find the multiples

The first multiple: \(8x=8\times3 = 24\)
The second multiple: \(8(x + 1)=8\times4=32\)
The third multiple: \(8(x + 2)=8\times5 = 40\)
Check: \(24+32 + 40=96\)

Answer:

4

Sub - Question 2