At first, Ann had some pens while Ben had some markers. Ann then gave 1/2 of her pens to Ben while Ben gave 50% of his markers to Ann. However, Ann lost 12 pens and Ben lost 18 markers. In the end, the ratio of the number of pens to the number of markers became 1:7 for Ann and 1:4 for Ben.
(a) How many pens did Ann have in the end?
(b) How many markers did Ben have in the end?
Let's start by using variables to represent the initial number of pens and markers that Ann and Ben had. Let's use "A" for Ann's pens and "B" for Ben's markers.
After Ann gave 1/2 of her pens to Ben, she would have (1/2)A pens left, and Ben would have B + (1/2)A pens.
After Ben gave 50% of his markers to Ann, he would have (1/2)B markers left, and Ann would have (1/2)B + (1/4)B markers.
Now we know that Ann lost 12 pens and Ben lost 18 markers. So Ann's final number of pens is:
(1/2)A - 12
And Ben's final number of markers is:
(1/2)B - 18
Finally, we know that the ratio of pens to markers is 1:7 for Ann and 1:4 for Ben. So we can set up two equations:
(1/7)(1/2)B + (1/2)B + (1/4)B = (1/2)A - 12
and
(1/4)(1/2)A + (1/2)A = (1/2)B - 18
Simplifying these equations:
(9/28)B = (1/2)A - 12
(3/8)A = (1/2)B - 18
Now we can use substitution to solve for either A or B. Let's solve for A:
(9/28)B = (1/2)A - 12
multiply both sides by 28:
9B = 14A - 336
(3/4)A = (1/2)B - 18
multiply both sides by 4:
3A = 2B - 72
Now we can substitute 2B - 72 for 3A in the other equation:
9B = 14(2B - 72) - 336
simplify:
9B = 28B - 1008
B = 36
Substituting B = 36 into the equation we just used to solve for B, we get:
3A = 2(36) - 72
3A = 0
A = 0
Uh-oh, this doesn't work! If Ann had 0 pens, then she couldn't give any to Ben. Let's go back and check our work. The problem says that Ann gave 1/2 of her pens to Ben, so that means she started with more than 0.
Let's try again, but this time, let's use a different variable for the initial number of pens:
Let's use "P" for the initial number of pens that Ann had.
After Ann gave 1/2 of her pens to Ben, she would have (1/2)P pens left, and Ben would have B + (1/2)P pens.
After Ben gave 50% of his markers to Ann, he would have (1/2)B markers left, and Ann would have (1/2)B + (1/4)B markers.
Now we know that Ann lost 12 pens and Ben lost 18 markers. So Ann's final number of pens is:
(1/2)P - 12
And Ben's final number of markers is:
(1/2)B - 18
Finally, we know that the ratio of pens to markers is 1:7 for Ann and 1:4 for Ben. So we can set up two equations:
(1/7)(1/2)B + (1/2)B + (1/4)B = (1/2)P - 12
and
(1/4)(1/2)P + (1/2)P = (1/2)B - 18
Simplifying these equations:
(9/28)B = (1/2)P - 12
(3/8)P = (1/2)B - 18
Now we can use substitution to solve for either P or B. Let's solve for B:
(3/8)P = (1/2)B - 18
multiply both sides by 8:
3P = 4B - 144
Substituting 3P/4 + 12 for B in the other equation:
(9/28)(3P/4 + 12) = (1/2)P - 12
simplify:
27P/98 + 27/7 = (1/2)P - 12
multiply both sides by 14 (to get rid of the fractions):
27P/7 + 54 = 7P - 168
simplify:
13P = -240
P = -240/13
Uh-oh again! This doesn't work either, because P should be a positive integer.
What went wrong? It looks like there is no solution that satisfies all of the conditions in the problem. Here's what we can conclude:
- The problem is inconsistent or contradictory. There is no combination of whole numbers that satisfies all of the conditions in the problem.
- It's possible that one of the conditions in the problem is incorrect or unclear. For example, it's possible that the ratio of pens to markers is supposed to be 7:1 for Ann instead of 1:7.
- It's also possible that there is a typo or error in the wording of the problem.
Without additional information or clarification, we cannot solve the problem as stated.
Ann gave x/2 pens to Ben and Ben gave y/2 magnets to Ann.
However, Ann lost 12 pens and Ben lost 18 magnets.
So Ann had x/2 - 12 pens and Ben had y/2 - 18 markers
For Ann, the ratio of pens to markers is 1:7
=> (x/2 - 12)/y/2 = 1/7
=> (x - 24)/y = 1/7
=> 7x - 168 = y
=> 7x - y = 168 - - - - (1)
Similarly for Ben, (x/2)/(y/2 - 18) = 14
=> x/(y - 36) = 1/4
=> 4x = y - 36
=> 4x - y = -36 - - - - (2)
Now eq (1) - eq (2) gives 3x = 204.
=> x = 68, setting x = 68 in eq(2) we get y = 4(68) + 36 = 308
=> y = 308
Finally x/2 - 12 = 68/2 - 12 = 22 and y/2 - 18 = 308/2 - 18 = 136
∴ In the end (a) Ann had 22 pens and (b) Ben had 136 markers.
Correct! Good job on solving the problem.
Let's solve this step by step:
Step 1: Introduce variables
Let's say Ann had 'p' pens and Ben had 'm' markers initially.
Step 2: Actions performed by Ann and Ben
According to the information given:
- Ann gave 1/2 of her pens to Ben, so she gave p/2 pens to Ben.
- Ben gave 50% of his markers to Ann, so he gave m/2 markers to Ann.
Step 3: Calculate remaining pens and markers
After the above actions, Ann and Ben both lost some pens and markers.
- Ann lost 12 pens, so Ann had p - p/2 - 12 pens left.
- Ben lost 18 markers, so Ben had m - m/2 - 18 markers left.
Step 4: Calculate ratios
At the end, the ratio of Ann's pens to markers is 1:7, and the ratio of Ben's pens to markers is 1:4.
- Ann had p - p/2 - 12 pens and m - m/2 - 18 markers.
- Ben had p/2 pens and m - m/2 - 18 markers.
According to the ratios, we can set up the following equations:
Ann's ratio: (p - p/2 - 12) / (m - m/2 - 18) = 1/7
Ben's ratio: (p/2) / (m - m/2 - 18) = 1/4
Step 5: Solve the equations
We have two equations and two variables (p and m). We can solve these equations to find the values of p and m.
Using the first ratio equation, we can simplify it as follows:
7(p - p/2 - 12) = m - m/2 - 18
Multiply each term by 2 to clear the fraction:
14p - 7p - 84 = 2m - m - 36
Simplify the equation:
7p - 84 = m - 36
Using the second ratio equation, we can simplify it as follows:
4(p/2) = m - m/2 - 18
Simplify the equation:
2p = m - m/2 - 18
Now we can substitute the value of m - 36 from the first equation into the second equation:
2p = 7p - 84 - 18
Simplify the equation:
-5p = -66
Divide both sides by -5:
p = 13.2
Step 6: Find the value of m
Substitute the value of p into one of the equations:
7p - 84 = m - 36
7(13.2) - 84 = m - 36
92.4 - 84 = m - 36
8.4 = m - 36
m = 8.4 + 36
m = 44.4
(a) Ann had 13.2 pens in the end.
(b) Ben had 44.4 markers in the end.