Certainly! Let's approach and solve each part of the problem step by step.
### 1. a) A printing press prints 10,000 textbooks in 3 days if it is operated 10 hours everyday.
#### (i) In what proportion are the number of textbooks and the printing days?
To find the proportion of textbooks to the printing days, we consider the given values: 10,000 textbooks in 3 days.
[tex]\[ \text{Proportion of textbooks to days} = \frac{\text{Number of textbooks}}{\text{Number of days}} \][/tex]
[tex]\[ \text{Proportion} = \frac{10000}{3} = 3333.3333333333335 \][/tex]
Thus, the proportion of textbooks to printing days is approximately 3333.33 textbooks per day.
#### (ii) How long does the press take to print 40,000 books in the same operating hours?
Let's determine the total days needed to print 40,000 textbooks.
First, consider how many textbooks are printed per day:
[tex]\[ 10000 \text{ textbooks in } 3 \text{ days} \implies 3333.3333333333335 \text{ textbooks per day} \][/tex]
Now, we want to calculate the number of days required to print 40,000 textbooks:
[tex]\[ \text{Days required} = \frac{40000 \text{ textbooks}}{3333.3333333333335 \text{ textbooks per day}} \][/tex]
[tex]\[ \text{Days required} = 12.0 \][/tex]
Thus, it takes 12 days to print 40,000 textbooks when operating for 10 hours everyday.
#### (iii) In what proportion are the everyday printing hours and the printing days?
To find the proportion of everyday printing hours to the number of printing days, we consider the given values: 10 hours per day over 3 days.
[tex]\[ \text{Proportion of printing hours to days} = \frac{\text{Everyday printing hours}}{\text{Number of days}} \][/tex]
[tex]\[ \text{Proportion} = \frac{10}{3} = 3.3333333333333335 \][/tex]
Thus, the proportion of everyday printing hours to the printing days is approximately 3.33.
#### (iv) How long does the press take to print 40,000 books if it is operated 8 hours a day?
Let's determine the total days needed to print 40,000 textbooks if the press operates 8 hours a day.
Firstly, calculate the rate of textbooks printed when operating for 8 hours compared to 10 hours:
[tex]\[ \text{Textbooks printed per day at 10 hours} = 3333.3333333333335 \][/tex]
[tex]\[ \text{Textbooks printed per day at 8 hours} = \left( \frac{8}{10} \right) \times 3333.3333333333335 \][/tex]
[tex]\[ \text{Textbooks printed per day at 8 hours} = 2666.6666666666665 \][/tex]
Now, calculate the number of days required to print 40,000 textbooks at this rate:
[tex]\[ \text{Days required} = \frac{40000 \text{ textbooks}}{2666.6666666666665 \text{ textbooks per day}} \][/tex]
[tex]\[ \text{Days required} = 14.999999999999998 \][/tex]
Thus, it takes approximately 15 days to print 40,000 textbooks when operating for 8 hours everyday.
