The ratio of the number of red pens to the number of black pens in a
bookstore was 3: 7. An equal number of red pens and black pens were
sold. In the end, the ratio of the number of red pens left to the number
of black pens left was 5: 13.
There were 20 red pens in the end.
How
many red pens were sold?
started with 3 k red and 7 k black, total 10 k
s of each sold
so
red left = 3 k - s = 20 so s = 3 k - 20
black left = 7 k - s
total left = 10 k - 2 s
5/13 = red left / black left = (3k-s) / (7k-s)
5/13 = (3 k - 3k+20) / (7k -3k+20)
5/13 = 20 / (4k + 20)
260 = 20 k + 100
20 k = 160
k = 80
so s = 3k-20 = 240-0 = 220
Step 1: Let's say the number of red pens and black pens in the bookstore initially was 3x and 7x respectively.
Step 2: We are given that an equal number of red and black pens were sold. Therefore, the number of red pens left is 3x - x = 2x, and the number of black pens left is 7x - x = 6x.
Step 3: It is also given that the ratio of red pens left to black pens left is 5:13. We can set up the following equation:
2x / 6x = 5 / 13
Step 4: Cross-multiplying the equation, we get:
13 * 2x = 6x * 5
Step 5: Simplifying the equation:
26x = 30x
Step 6: Dividing both sides of the equation by x, we get:
26 = 30
Step 7: Since this equation is not true, it means our initial assumption that an equal number of red and black pens were sold is incorrect.
Conclusion: Based on the given information, we cannot determine how many red pens were sold. Further information is needed.
To solve this problem, let's break it down step by step.
Step 1: Determine the starting ratio of red pens to black pens.
The given information states that the ratio of red pens to black pens in the bookstore is 3:7. This means that for every 3 red pens, there are 7 black pens.
Step 2: Determine the number of red pens sold.
The problem states that an equal number of red pens and black pens were sold. Let's assume that x red pens were sold. Since an equal number of black pens were sold, x black pens were also sold.
Step 3: Determine the number of red pens remaining.
To find the number of red pens remaining, we need to subtract the number of red pens sold (x) from the total number of starting red pens. Since we know that there were 20 red pens left in the end, we can write the equation:
Total number of starting red pens - Number of red pens sold = Number of red pens remaining
3/10 (total number of pens) - x = 20
Step 4: Solve the equation.
To solve the equation, we first need to find the total number of pens in the bookstore. Since the ratio of red pens to black pens is given as 3:7, we can assume the total number of pens to be a multiple of 10 (3+7). Let's assume the total number of pens is 10k, where k is a positive integer.
Therefore, the equation becomes:
(3/10) * (10k) - x = 20
3k - x = 20
Step 5: Determine the number of red pens sold.
Substituting the equation from Step 4 into Step 3, we have:
3k - x = 20
3k - 20 = x
Now, we know that x (the number of red pens sold) is equal to 3k - 20.
Step 6: Determine the value of k.
To find the value of k, we can use the fact that there were 20 red pens left in the end. Substituting x = 20 into the equation from Step 5, we have:
3k - 20 = 20
3k = 40
k = 40/3
Step 7: Calculate the number of red pens sold.
Substituting the value of k into the equation from Step 5, we have:
x = 3k - 20
x = 3(40/3) - 20
x = 40 - 20
x = 20
Therefore, 20 red pens were sold.