Question

a clothing store is donating socks to various charities. the store gave a certain number of regular packs and a certain number of value packs, which contained a total of 228 pairs of socks. for another charity, the store donated 4 regular packs and 5 value packs, which equaled 258 pairs. how many pairs of socks are in each regular pack and how many pairs in each value pack?

Explanation:

Step1: Define variables

Let \( x \) be the number of pairs of socks in each regular pack and \( y \) be the number of pairs in each value pack.

Step2: Set up equations

From the first donation: \( 4x + 5y = 228 \)
From the second donation: \( 5x + 5y = 258 \)

Step3: Subtract the first equation from the second

\( (5x + 5y) - (4x + 5y) = 258 - 228 \)
\( 5x + 5y - 4x - 5y = 30 \)
\( x = 30 \)

Step4: Substitute \( x = 30 \) into the second equation

\( 5(30) + 5y = 258 \)
\( 150 + 5y = 258 \)
\( 5y = 258 - 150 \)
\( 5y = 108 \) Wait, no, 258 - 150 is 108? Wait, 258 - 150 = 108? No, 150 + 108 = 258? Wait, 150 + 100 = 250, 250 + 8 = 258. So 5y = 108? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the second donation: 5 regular packs and 5 value packs equal 258. So 5(x + y) = 258? Wait, no, 5 regular and 5 value. So 5x + 5y = 258. Then divide both sides by 5: x + y = 51.6? That can't be, since the number of pairs should be an integer. Wait, maybe the problem was transcribed incorrectly. Wait, let's check again. Wait, the first donation: 4 regular, 5 value: 228. Second: 5 regular, 5 value: 258. So subtracting, (5x +5y) - (4x +5y) = 258 -228 => x = 30. Then substitute x=30 into 4x +5y=228: 4*30 +5y=228 => 120 +5y=228 => 5y=108 => y=21.6. That's not an integer, which is odd. Maybe the second donation was 5 regular and 6 value? Or maybe the numbers are different. Alternatively, maybe the second donation is 5 regular and 5 value, but 258 divided by 5 is 51.6, which is not integer. So perhaps there's a typo. But assuming the problem is correct as given, then x=30, y=21.6. But that's unusual. Alternatively, maybe the second donation is 5 regular and 6 value? Or maybe the first is 4 regular and 6 value? Wait, maybe I misread. Let me re-express the problem.

Wait, the user's image: "A clothing store is donating socks to various charities. The store gave a number of packs to one charity which contained a total of 228 pairs of socks. For another charity, they donated 4 regular packs and 5 value packs, which equaled 258 pairs. How many... pairs of socks in each regular pack and... pairs in each value pack." Wait, no, maybe the first donation is 4 regular and 5 value: 228, second is 5 regular and 5 value: 258. So as above, x=30, y=21.6. But that's not possible. Maybe the second donation is 5 regular and 6 value? Or maybe the numbers are 225 and 255? Let's assume that maybe it's a typo, and the second donation is 255. Then 5x +5y=255 => x + y=51. Then 4x +5y=228. Subtract: (4x +5y) - (4x +4y)=228 -4*51 => y=228 -204=24. Then x=51-24=27. But that's just a guess. Alternatively, maybe the original problem has different numbers. But based on the given numbers, the solution would be x=30, y=21.6, which is non-integer. So perhaps there's a mistake in the problem. But assuming the problem is correct, then:

Regular pack: 30 pairs, Value pack: 21.6 pairs. But that's odd. Maybe the second donation is 5 regular and 6 value? Let's try that. If second donation is 5x +6y=258, first is 4x +5y=228. Then multiply first by 5: 20x +25y=1140, second by 4: 20x +24y=1032. Subtract: y=108. Then x=(228 -5*108)/4=(228-540)/4=(-312)/4=-78. No, that's worse. Alternatively, maybe the first donation is 5 regular and 4 value. Let's try. First: 5x +4y=228, second:4x +5y=258. Then add:9x +9y=486 =>x + y=54. Subtract: (5x +4y)-(4x +5y)=228-258 =>x - y= -30. Then solve x + y=54 and x - y=-30. Add:2x=24 =>x=12, then y=42. That gives integer values. So maybe the problem was transcribed with the packs reversed. So perhaps the first donation is 5 regular, 4 value: 228, second is 4 regular, 5 value:258. Then x=12, y=42. That makes sense. So maybe a transcription error. But based on the given problem as written, with first donation as 4 regular,5 value:228, second as 5 regular,5 value:258, the solution is x=30, y=21.6. But since that's not integer, likely a typo. However, following the given numbers:

Answer:

Regular pack: 30 pairs, Value pack: 21.6 pairs (though non-integer, based on given numbers)

But since the problem likely has a typo, but assuming the numbers are correct, the answer is regular pack: 30, value pack: 21.6. But that's unusual. Alternatively, maybe the second donation is 5 regular and 5 value, total 255. Then x + y=51, 4x +5y=228. Then 4x +5(51 -x)=228 =>4x +255 -5x=228 =>-x= -27 =>x=27, y=24. That's integer. So maybe the second donation is 255. Then regular:27, value:24. But since the user's problem says 258, I'll go with the calculation as per the given numbers, even if non-integer.

Wait, maybe I made a mistake in subtraction. 258 -228 is 30, correct. 4x +5y=228, 5x +5y=258. Subtract: x=30. Then 4*30=120, 228-120=108, 108/5=21.6. So yes, that's the result. So the regular pack has 30 pairs, value pack has 21.6 pairs.