Jane wishes to buy books for her children. At one shop a book costs sh.400 it is sold with 2 pens costing sh.10 each and another shop sells a book at sh.200 with a pen costing sh.40 each. She is prepared to spend not more than sh.3200 on books and sh.280 on pens. Find the largest number of books she can buy.
Before a math teacher looks at this, you need to show as much as you can toward solving this. What have you done so far?
To find the largest number of books Jane can buy, we need to determine the maximum number of books she can purchase within her budget constraints.
Let's assume Jane buys "x" books from the first shop and "y" books from the second shop.
From the first shop:
- The cost of each book is sh.400.
- In addition to each book, 2 pens are included, costing sh.10 each. So the cost of pens per book is sh.10 * 2 = sh.20.
From the second shop:
- The cost of each book is sh.200.
- In addition to each book, 1 pen is included, costing sh.40.
We can now set up the following equations to represent Jane's budget constraints:
400x + 20x ≤ 3200 (Considering the total cost of books from the first shop)
200y + 40y ≤ 3200 (Considering the total cost of books from the second shop)
20x + 40y ≤ 280 (Considering the total cost of pens)
Simplifying these equations, we get:
420x ≤ 3200 (Dividing both sides of the first equation by 20)
240y ≤ 3200 (Dividing both sides of the second equation by 20)
20x + 40y ≤ 280
Now, let's solve these inequalities simultaneously to find the largest possible values of x and y.
1) 420x ≤ 3200
Dividing both sides by 420:
x ≤ 3200/420
x ≤ 7.619
Since Jane cannot buy a fraction of a book, the maximum value for x can be 7.
2) 240y ≤ 3200
Dividing both sides by 240:
y ≤ 3200/240
y ≤ 13.333
Since Jane cannot buy a fraction of a book, the maximum value for y can be 13.
Now, we can calculate the total cost of buying 7 books from the first shop and 13 books from the second shop:
Total cost = (400 * 7) + (20 * 7) + (200 * 13) + (40 * 13)
Total cost = 2800 + 140 + 2600 + 520
Total cost = 6060
Since the total cost exceeds Jane's budget of sh.3200, she cannot buy this many books.
To find the largest number of books Jane can buy, we need to find a combination of x and y that satisfies her budget constraints. We can use a trial-and-error approach to determine the maximum value for x and y, considering the cost of pens as well.
Let's try different values for x and y until we find a combination that satisfies the budget:
- If x = 6 and y = 13:
Total cost = (400 * 6) + (20 * 6) + (200 * 13) + (40 * 13)
= 2400 + 120 + 2600 + 520
= 5640
Since 5640 is less than Jane's budget of sh.3200, she can buy 6 books from the first shop and 13 books from the second shop.
Therefore, the largest number of books Jane can buy is 6 + 13 = 19.