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?

Question

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

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 N_initial)
2. J = number of game cards Joshua had at first (to compare to the final values, we can also express this variable as J_initial)

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. J_new = J_initial + 20
2. Since we know that J_initial = 2/3 the amount of N_initial, we can also model this equation as: J_new = 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. N_new = N_initial - 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 = J_new = J_initial + 20 = 2/3 N + 20
3. Nat's new number of cards = N_new = N_initial - 29
4. We’re told that after these changes, Joshua has 4/5 as many cards as Nat, so:
5. Joshua’s new total, J_new, = 4/5 Nat’s new total, or 4/5 N_new
6. Since we know N_new = 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.

Answer 2

Let J = # cards Joshua started with

N = #cards Nat started with

From the first sentence we know:

J = (2/3)N

After Joshua bought an additional 20 cards he now has:

J + 20

Nat lost 29 cards so now has:

N - 29

After the gain and loss:

J + 20 = (4/5)(N - 29) ; Josh now has 4/5 of the number of cards held by Nat. Use the substitution method to rewrite the last equation in terms of N only (we know that J = (2/3) N

(2/3)N + 20 = (4/5)(N - 29)

Since I do not like to work with fractions if at all possible, I would multiply every term by 15 (the LCM of 3 and 5) to clear the equation of fractions:

10N + 300 = 12(N - 29)

10N + 300 = 12N -12(29)

300 + 12(29) = 2N

150 + 6(29) = N

N = 324 (number of cards held by Nat initially)

J = 2/3 (324) = 216

Check:

Josh gains 20: 236

Nat loses 29: 295

And: 4/5 (295) = 4(59) = 236