To find out how many total pages Darryl needs to write for his history report, we can follow these logical steps:
Step 1: Write 60 percent as a ratio. The given ratio is:
[tex]\[ \frac{\text{part}}{\text{whole}} = \frac{60}{100} \][/tex]
Step 2: Write an equivalent ratio using the number of pages done so far, which is 12 pages:
[tex]\[ \frac{60}{100} = \frac{12}{?} \][/tex]
Step 3: To find the total number of pages, we need to set up the proportion correctly and solve for the unknown total number of pages. Let [tex]\( x \)[/tex] be the total number of pages Darryl needs to write. We can set up the following proportion:
[tex]\[ \frac{60}{100} = \frac{12}{x} \][/tex]
Step 4: Solve for [tex]\( x \)[/tex] by cross-multiplying and then isolating [tex]\( x \)[/tex]:
[tex]\[ 60x = 12 \times 100 \][/tex]
[tex]\[ 60x = 1200 \][/tex]
[tex]\[ x = \frac{1200}{60} \][/tex]
[tex]\[ x = 20 \][/tex]
Therefore, Darryl needs to write a total of 20 pages for his history report.
Now addressing Darryl's mistakes:
1. In Step 1, Darryl correctly wrote the ratio for 60 percent as [tex]\(\frac{60}{100}\)[/tex]; there is no mistake here.
2. In Step 2, Darryl correctly identified that 12 pages represent 60 percent of his report; so the mistake is not in putting 12 in the numerator versus the denominator.
3. Step 3 in Darryl's method was where the mistake occurred. He tried to divide the part (12) by something that didn't follow logically from the proportion.
Darryl should have set up the proportion correctly and solved it as shown in the solution above, which gives us the total number of pages as 20.
