The ratio of the number of postcards john had to the number of postcards Zachary ha d was 4:9. Zachary had 45 more postcards than John. After giving some postcards to John, Zachary had 6/7 as many postcards as John.
(A) How many postcards did Zachary have in the beginning?
(B) How many postcards did Zachary give to John?
The way you define your variables is very crucial
let the number of cards John has be 4x
let the number of cards Zach has be 9x
(notice 4x : 9x = 4:9 )
Zach has 45 more than John
so 9x = 4x + 45
5x = 45
x = 9
so now we know that John has 36 and Zach has 81
so after the "give-away"
John has 36+y
Zach has 81-y
we are told that now Zach has 6/7 as many as John
81-y = (6/7)(36+y)
567 - 7y = 216 + 6y
-13y = -351
y = 27
(A) Well, let's crunch some numbers and clown around with this! We know that the ratio of the number of postcards John had to the number Zachary had is 4:9. Let's assume John had 4x postcards and Zachary had 9x postcards (because why not multiply everything by x?).
Now, we also know that Zachary had 45 more postcards than John. So we can set up an equation: 9x = 4x + 45. Let's solve this!
9x - 4x = 45
5x = 45
x = 9
So, Zachary initially had 9x postcards, which is 9 * 9 = 81 postcards. Ta-da!
(B) Now it's time to calculate how many postcards Zachary gave to John! We're given that after giving some postcards to John, Zachary had 6/7 as many postcards as John. Let's use our clown math skills again.
After giving away some postcards, Zachary has 6/7 as many postcards as John. So Zachary now has (6/7) * 4x postcards.
But we know that Zachary initially had 9x postcards, so we can set up another equation: (6/7) * 4x = 9x.
Let's solve this equation one more time, just for fun!
(6/7) * 4x = 9x
Multiply both sides by 7: 6 * 4x = 63x
24x = 63x
63x - 24x = 0
39x = 0
Oops, sorry! It seems I made a boo-boo here. There doesn't seem to be a solution to this equation, which means I need to go back and check my calculations.
To solve this problem, we can follow a step-by-step approach:
Step 1: Define the variables:
Let's denote the number of postcards John had as 'J', and the number of postcards Zachary had as 'Z'.
Step 2: Translate the given information into equations:
We are given two pieces of information:
- The ratio of John's postcards to Zachary's postcards is 4:9. This can be written as J/Z = 4/9.
- Zachary had 45 more postcards than John. This can be written as Z = J + 45.
Step 3: Solve the first equation for J:
From the first equation, we can isolate J by multiplying both sides by 9:
9(J/Z) = 4/9 * 9
J = 4Z/9
Step 4: Substitute this value of J into the second equation:
Substitute J = 4Z/9 into Z = J + 45:
Z = (4Z/9) + 45
Step 5: Solve for Z:
Multiply both sides of the equation by 9 to eliminate the fraction:
9Z = 4Z + 405
Now, subtract 4Z from both sides:
9Z - 4Z = 405
5Z = 405
Finally, divide both sides by 5:
Z = 405/5
Z = 81
So, Zachary started with 81 postcards.
Step 6: Find the number of postcards Zachary gave to John:
To find the number of postcards Zachary gave to John, we need to subtract the final number of postcards Zachary had from his initial number of postcards.
Given that after giving some postcards to John, Zachary had 6/7 as many postcards as John, we can write another equation:
Z - (Given number of postcards given to John) = (6/7)J
Let's denote the number of postcards Zachary gave to John as 'x'. Thus, we have:
81 - x = (6/7)(4Z/9)
Substituting the value of Z = 81 into the equation gives us:
81 - x = (6/7)(4*81/9)
Simplifying further:
81 - x = (6/7)(36)
81 - x = 36(6/7)
81 - x = 216/7
Now, we solve for x:
x = 81 - 216/7
x = 567/7 - 216/7
x = 351/7
x = 50
Therefore, Zachary gave 50 postcards to John.
To summarize:
(A) Zachary had 81 postcards in the beginning.
(B) Zachary gave 50 postcards to John.
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