Vicky has 50 more cards than Stella . Thomas has 180 cards less than the sum of Vick”S and Stella “s. If Vicky , Stella ,are Thomas have 2200 cards altogether,how many cards does Stella have?
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Let's solve this card problem step by step. We have three people: Vicky, Stella, and Thomas. Vicky has 50 more cards than Stella. Thomas has 180 cards less than the sum of Vicky's and Stella's cards. Together, all three have 2200 cards. We need to find how many cards Stella has. Let's call Stella's cards S, which makes this our unknown variable to solve for.
Now let's set up our variables and translate each relationship into mathematical expressions. Let S represent Stella's cards, which is our unknown variable. Since Vicky has 50 more cards than Stella, Vicky's cards equal S plus 50. For Thomas, he has 180 cards less than the sum of Vicky's and Stella's cards. The sum of Vicky's and Stella's cards is S plus S plus 50, which equals 2S plus 50. Subtracting 180 from this gives us 2S plus 50 minus 180, which simplifies to 2S minus 130.
Now we create our equation by combining all the expressions using the total constraint. We know that the total number of cards is 2200. This means Stella's cards plus Vicky's cards plus Thomas's cards equals 2200. Substituting our expressions, we get S plus S plus 50 plus 2S minus 130 equals 2200. This single equation captures all the given relationships and can be solved to find our unknown variable S.
Now let's solve the equation step by step using algebraic manipulation. Starting with S plus S plus 50 plus 2S minus 130 equals 2200, we first remove the parentheses. Next, we combine like terms: S plus S plus 2S gives us 4S, and 50 minus 130 gives us minus 80. So we have 4S minus 80 equals 2200. Adding 80 to both sides gives us 4S equals 2280. Finally, dividing both sides by 4, we get S equals 570. Therefore, Stella has 570 cards.
Now that we know S equals 570, let's calculate each person's cards by working backwards from our solution. Stella has S cards, which is 570 cards. Vicky has S plus 50 cards, so 570 plus 50 equals 620 cards. Thomas has 2S minus 130 cards, so 2 times 570 minus 130. That's 1140 minus 130, which equals 1010 cards. So our final results are: Stella has 570 cards, Vicky has 620 cards, and Thomas has 1010 cards.