Joshua had 2/3 as many game cards as Nat. After Joshua bought another 20 game cards and Nat lost 29 game cards, Joshua now has 4/5 as many game cards as Nat. How many game cards did Nat have at first?

Answer 1

n=324

j = 2n/3

j+20 = (4/5)(n-29)) multiply by 5

5j + 100 = 4n -116

5(2n/3) + 100 =4n -116

10n/3 = 4n -216, multiply by 3

10n = 12n -648

2n =648

n = 324

Answer 2

Answer 3

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Answer 6

Hi! For this problem, I’ll guide you through an efficient process that can be followed to work through the problem step-by-step through substitution and solving for the unknown.

Provided Information:

Given:

1. Joshua had 2/3 as many game cards as Nat at first.
2. After Joshua bought 20 more cards, and Nat lost 29 cards,
3. Joshua now has 4/5 as many cards as Nat.

Find:

How many game cards did Nat have at first?

Step 1: Define variables

Let:

1. N = number of game cards Nat had at first (to compare to the final values, we can also express this variable as Ninitial)
2. J = number of game cards Joshua had at first (to compare to the final values, we can also express this variable as Jinitial)

From the problem, we know Joshua initially has 2/3 times as many cards as Nat. This means that, for any number of cards Nat has, Joshua will have 2/3 of that amount (Because 2/3 is less than 1, we can infer that Joshua begins with less cards than Nat).

1. Therefore, J = 2/3 * N, or simply, J = 2/3 N

Step 2: Express the situation after changes

After change #1 (Joshua buys 20 more cards):

1. Jnew = Jinitial + 20
2. Since we know that Jinitial = 2/3 the amount of Ninitial, we can also model this equation as: Jnew = 2/3 N + 20
3. This format will allow us to solve the problem by comparing ratios in Step 3 using a single variable, N, which relates our solution to the target person, Nat. 

After change #2 (Nat loses 29 cards):

1. Nnew = Ninitial - 29

Step 3: Use the new ratio given (4/5)

According to the problem, “After Joshua bought another 20 game cards and Nat lost 29 game cards, Joshua now has 4/5 as many game cards as Nat.” This tells us the new ratio between Joshua's and Nat’s card totals after the changes:

1. From our previous step, we found that:
2. Joshua's new number of cards = Jnew = Jinitial + 20 = 2/3 N + 20
3. Nat's new number of cards = Nnew = Ninitial - 29
4. We’re told that after these changes, Joshua has 4/5 as many cards as Nat, so:
5. Joshua’s new total, Jnew, = 4/5 Nat’s new total, or 4/5 Nnew
6. Since we know Nnew = N - 29, we can rewrite the equation as 4/5*(N-29)
7. By combining our 2 equations, we find that 2/3 N + 20 = 4/5 * (N-29)

Step 5: Evaluate and solve the algebraic equation

1. Multiply both sides by 15 (the least common multiple of 3 and 5) to clear denominators. This will simplify our equation to 10N+300=12(N−29)
2. Distribute 12 on the right: 10N+300=12N−348
3. Rearrange terms by subtracting 10N from both sides and adding 348 to both sides. This will put all of our constants on one side of the equation, and all of our terms with variable “N” on the other. This will give us 2N = 648
4. Divide both sides by coefficient 2
5. Solution: N = 324

Answer: Nat originally had 324 game cards at the start of the game.