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?
Is there a way to figure this out without using formulas?
A. John has X cards.
Zack has (X+45) cards.
X/(X+45) = 4/9,
Cross multiply:
4x+180 = 9x,
4x-9x = -180,
-5x = -180,
X = 36cards.
X+45 = 36+45 = 81 cards.
B. Zack has (81-X) cards.
John has (36+X) cards.
81-X = 6/7(36+X),
Multiply both sides by 7:
567-7X = 6(36+X),
567-7X = 216+6X,
-7X-6X = 216-567,
-13X = -351,
X = 27 cards given to John.
Yes, there is a way to figure this out without using formulas. Let's solve it step by step.
Let's assume that the number of postcards John had is 4x, where x is a positive integer.
Since the ratio of John's postcards to Zachary's postcards is 4:9, the number of postcards Zachary had is 9x.
Given that Zachary had 45 more postcards than John, we can set up the equation 9x = 4x + 45.
To solve this equation, we can subtract 4x from both sides: 5x = 45.
Dividing both sides by 5 gives us x = 9.
So, John had 4x = 4 * 9 = 36 postcards.
Now, we can find out how many postcards Zachary had in the beginning by substituting x = 9 into 9x: 9 * 9 = 81.
Therefore, Zachary had 81 postcards in the beginning. (Answer to (A))
To find out how many postcards Zachary gave to John, we can calculate the difference between the number of postcards Zachary had in the beginning (81) and the number of postcards he has after giving some to John.
Given that Zachary has 6/7 as many postcards as John after giving some to him, we can set up the equation (6/7) * 36 = 81 - x.
To solve this equation, we can multiply both sides by 7/6: 36 = (7/6) * (81 - x).
Simplifying this equation gives us 36 = (567/6) - (7/6)x.
Multiplying both sides by 6 gives us 216 = 567 - 7x.
Subtracting 567 from both sides gives us -351 = -7x.
Dividing both sides by -7 gives us x = 351/7 = 51.
So, Zachary gave 51 postcards to John. (Answer to (B))
Yes, there is a way to solve this problem without using formulas.
Let's break down the information step by step:
1. The ratio of the number of postcards John had to the number of postcards Zachary had was 4:9. This means that for every 4 postcards John had, Zachary had 9 postcards. We can represent this as:
Number of postcards John had = 4x
Number of postcards Zachary had = 9x
2. Zachary had 45 more postcards than John. This means that the difference in the number of postcards is 45. We can set up an equation based on this information:
9x - 4x = 45
Simplifying the equation, we get:
5x = 45
3. After giving some postcards to John, Zachary had 6/7 as many postcards as John. This means that the number of postcards Zachary had is 6/7 of what John has. We can set up another equation based on this information:
9x - y = (6/7) * (4x + y)
Now, let's solve the equations:
From equation 2, we found that 5x = 45. Dividing both sides by 5, we get:
x = 9
Now we can substitute this value back into the equations to find the number of postcards:
Using equation 1, we find:
Number of postcards John had = 4x = 4 * 9 = 36
Number of postcards Zachary had = 9x = 9 * 9 = 81
So, Zachary had 81 postcards in the beginning.
To find the number of postcards Zachary gave to John, we can use equation 3. Substituting the values we found:
9x - y = (6/7) * (4x + y)
81 - y = (6/7) * (36 + y)
Now we can solve this equation for y to find the number of postcards Zachary gave to John.
So, to summarize:
(A) Zachary had 81 postcards in the beginning.
(B) To find the number of postcards Zachary gave to John, solve the equation 81 - y = (6/7) * (36 + y)