Question

ryan has a 20% off coupon to buy new clothes. he bought a shirt for $35 and socks for $7 per pair. he must spend more than $75 to apply the coupon. how many pairs of socks can he buy?

Explanation:

Step1: Define variables and inequality

Let \( x \) be the number of socks. The total cost is the cost of the shirt plus the cost of the socks. So the inequality is \( 35 + 7x>75 \).

Step2: Solve the inequality

Subtract 35 from both sides: \( 7x>75 - 35 \)
Simplify: \( 7x>40 \)
Divide both sides by 7: \( x>\frac{40}{7}\approx5.71 \)

Since \( x \) represents the number of socks, it must be a whole number. So the smallest whole number greater than \( 5.71 \) is 6. We can check: \( 35 + 7\times6 = 35+42 = 77>75 \), and if \( x = 5 \), \( 35+35 = 70<75 \). So \( x \) must be at least 6. But we need to find how many he can buy (the minimum number to apply the coupon, but actually the question is how many he can buy to meet the condition. Wait, the problem is to find the number of socks so that total is more than 75. So solving \( 35 + 7x>75 \) gives \( x>\frac{40}{7}\approx5.71 \), so \( x = 6,7,8,... \) But maybe the question is to find the minimum number? Wait, the problem says "How many socks can he buy?" to apply the coupon, so the minimum number. So \( x \) must be an integer greater than \( 5.71 \), so the minimum is 6. But let's do the steps again.

Wait, let's re - do the inequality:

Total cost \( C=35 + 7x \)

We need \( C>75 \)

So \( 35+7x>75 \)

Subtract 35: \( 7x>75 - 35=40 \)

Divide by 7: \( x>\frac{40}{7}\approx5.71 \)

Since \( x \) is the number of socks (a non - negative integer), the smallest integer \( x \) that satisfies this is 6. If we check \( x = 6 \), total cost is \( 35+7\times6 = 35 + 42=77>75 \). If \( x = 5 \), total is \( 35 + 35 = 70<75 \). So he needs to buy at least 6 socks. But the question is "How many socks can he buy?" to apply the coupon, so the minimum number is 6. But maybe the problem is to find the number, so the answer is that he can buy 6 or more socks. But since it's a math problem, likely the minimum number, so 6. Wait, but let's confirm the calculation.

Wait, \( \frac{40}{7}\approx5.71 \), so \( x \) must be greater than 5.71, so the number of socks must be 6,7,8,... But the problem is probably asking for the minimum number of socks to apply the coupon, so the answer is 6 (or more, but since it's a math problem, the minimum integer solution is 6).

Answer:

He can buy at least 6 socks (the minimum number to apply the coupon is 6, and he can buy 6,7,8,... socks as long as he has money, but based on the inequality solution, the minimum number is 6). If we consider the problem as finding the number of socks such that the total is more than 75, the smallest number of socks is 6.