Abuela Tomasa's Balloon Bonanza

by Tom Lembong 32 views
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Hey mathletes, gather 'round because we've got a fun little puzzle on our hands today, all thanks to Abuela Tomasa and her massive stash of balloons! You know how grandmas are – always thinking of ways to spoil their grandkids. Well, Abuela Tomasa is no different. She's got a whopping 32 balloons tucked away, just waiting for her six adorable grandkids to arrive. Now, here's where it gets interesting, guys. Abuela Tomasa isn't just about handing out a few balloons; she wants each of her six grandkids to get more than six balloons each. That's a generous grandma right there! So, the big question we need to help her solve is: how many dozens of balloons does she need to buy to make sure every single grandchild gets the exact same, generous amount? This isn't just about counting; it's about planning, a little bit of math magic, and making sure everyone gets a fair share of the colorful fun. We're going to dive deep into this, break it down step-by-step, and figure out just how many balloons Abuela Tomasa needs to add to her collection. Get ready, because we're about to become balloon-buying experts!

The Balloon Situation: What We Know So Far

Alright, let's break down the scenario with Abuela Tomasa and her balloon dilemma. We know she's starting with 32 balloons. That's a pretty good start, right? But then her six beloved grandchildren are on their way, and she wants to make sure they all have a blast. The key constraint here is that each grandchild must receive more than 6 balloons. This is super important because it sets a minimum for how many balloons each kid needs. If each of the 6 grandkids gets more than 6 balloons, the minimum total balloons needed would be 6 grandkids * 7 balloons/grandchild = 42 balloons. So, right off the bat, we can see that her current stash of 32 balloons isn't going to cut it. She definitely needs to buy more. The problem then asks us to figure out the number of dozens she needs to purchase. Remember, a dozen is simply 12 items. So, when we talk about buying balloons by the dozen, we're talking about buying them in sets of 12. This little detail is crucial for our final answer. We need to find the smallest number of dozens she can buy so that when added to her existing 32 balloons, she has enough to give each of her 6 grandkids more than 6 balloons each, with the added condition that everyone gets the same amount. This means the total number of balloons she ends up with must be divisible by 6 (so she can share equally) and also be greater than 42 (to satisfy the 'more than 6 balloons each' rule). Let's get our thinking caps on, folks!

Step 1: Calculating the Minimum Total Balloons Needed

First things first, let's figure out the absolute minimum number of balloons Abuela Tomasa needs to have in total to meet her generous goal. We know she has 6 grandchildren, and she wants to give each of them more than 6 balloons. The smallest whole number that is 'more than 6' is 7. So, to find the minimum total balloons required, we multiply the number of grandchildren by this minimum number of balloons per grandchild:

  • Minimum balloons per grandchild = 7
  • Number of grandchildren = 6
  • Minimum total balloons needed = 7 balloons/grandchild * 6 grandchildren = 42 balloons

This tells us that Abuela Tomasa needs to have at least 42 balloons in total to satisfy the condition of giving each grandchild more than 6 balloons. Her current 32 balloons are just not enough. She needs to acquire at least 10 more balloons (42 - 32 = 10).

Step 2: Determining How Many More Balloons She Needs to Buy

Now that we know she needs a grand total of at least 42 balloons, and she already has 32, we can figure out exactly how many more she needs to purchase.

  • Total balloons needed = 42
  • Balloons Abuela Tomasa already has = 32
  • Additional balloons required = Total balloons needed - Balloons she has = 42 - 32 = 10 balloons

So, Abuela Tomasa needs to buy at least 10 more balloons. However, there's a twist! Balloons are usually sold in packs, and in this problem, we're told she needs to buy them by the dozen. This means she can't just buy exactly 10 balloons. She has to buy in multiples of 12.

Step 3: Working with Dozens - The Real Challenge!

This is where the 'dozen' part comes into play, and it makes things a bit trickier and more realistic, right? Abuela Tomasa can't just pop down to the shop and pick up exactly 10 balloons. She has to buy them in packs of 12. So, she needs to buy enough dozens to get at least 10 more balloons.

Let's think about how many dozens she could buy:

  • If she buys 1 dozen, that's 12 balloons. This is more than the 10 she needs. Let's see if this works out for the 'equal sharing' part. If she buys 1 dozen (12 balloons), her new total will be 32 (original) + 12 (bought) = 44 balloons. Can 44 balloons be divided equally among 6 grandchildren such that each gets more than 6? 44 / 6 = 7 with a remainder of 2. This means she can't give everyone the exact same number of balloons if she buys just one dozen. So, 1 dozen is not enough.

  • What if she buys 2 dozens? That's 2 * 12 = 24 balloons. Her new total would be 32 (original) + 24 (bought) = 56 balloons. Now, let's check if 56 balloons can be divided equally among her 6 grandchildren. 56 divided by 6 is 9 with a remainder of 2. Again, she can't give everyone the exact same amount. So, 2 dozens isn't the answer either.

We need a total number of balloons that is divisible by 6 (for equal sharing) AND is greater than 42 (for more than 6 balloons each).

Let's re-evaluate. We need to find a total number of balloons, let's call it 'T', such that:

  1. T >= 42 (to give more than 6 balloons to each of the 6 grandkids)
  2. T is divisible by 6 (to give the same amount to each grandchild).
  3. T = 32 + (Number of dozens * 12)

Let's look for multiples of 6 that are greater than 42:

  • 48 is a multiple of 6 (48 / 6 = 8). Can Abuela Tomasa reach 48 balloons? She has 32. She needs 48 - 32 = 16 more balloons. Can she buy 16 balloons in dozens? No, because she'd have to buy 2 dozens (24 balloons) to get at least 16.

  • 54 is a multiple of 6 (54 / 6 = 9). Can Abuela Tomasa reach 54 balloons? She has 32. She needs 54 - 32 = 22 more balloons. Can she buy 22 balloons in dozens? No, she'd have to buy 2 dozens (24 balloons) to get at least 22.

  • 60 is a multiple of 6 (60 / 6 = 10). Can Abuela Tomasa reach 60 balloons? She has 32. She needs 60 - 32 = 28 more balloons. Can she buy 28 balloons in dozens? No, she'd have to buy 3 dozens (36 balloons) to get at least 28.

  • 66 is a multiple of 6 (66 / 6 = 11). Can Abuela Tomasa reach 66 balloons? She has 32. She needs 66 - 32 = 34 more balloons. Can she buy 34 balloons in dozens? No, she'd have to buy 3 dozens (36 balloons) to get at least 34.

  • 72 is a multiple of 6 (72 / 6 = 12). Can Abuela Tomasa reach 72 balloons? She has 32. She needs 72 - 32 = 40 more balloons. Can she buy 40 balloons in dozens? No, she'd have to buy 4 dozens (48 balloons) to get at least 40.

This approach of finding the target total first and then figuring out how many dozens is getting a bit complicated. Let's try thinking about the number of dozens she buys first.

Step 4: Testing the Number of Dozens Directly

Let's go back to testing the number of dozens she buys and see what total number of balloons she ends up with. Remember, she needs enough balloons so that when divided by 6, each grandchild gets more than 6 balloons, and the division is equal. So, the total number of balloons must be a multiple of 6, and the result of the division (balloons per grandchild) must be greater than 6.

  • If she buys 1 dozen (12 balloons):

    • Total balloons = 32 (original) + 12 (bought) = 44 balloons.
    • Can 44 be divided equally by 6? No (44 / 6 = 7 remainder 2).

  • If she buys 2 dozens (24 balloons):

    • Total balloons = 32 (original) + 24 (bought) = 56 balloons.
    • Can 56 be divided equally by 6? No (56 / 6 = 9 remainder 2).

  • If she buys 3 dozens (36 balloons):

    • Total balloons = 32 (original) + 36 (bought) = 68 balloons.
    • Can 68 be divided equally by 6? No (68 / 6 = 11 remainder 2).

  • If she buys 4 dozens (48 balloons):

    • Total balloons = 32 (original) + 48 (bought) = 80 balloons.
    • Can 80 be divided equally by 6? No (80 / 6 = 13 remainder 2).

Wait a minute, guys! I think I might have spotted a pattern here. It seems like no matter how many dozens she buys (which is always a multiple of 12), when we add it to her original 32, the total always leaves a remainder of 2 when divided by 6. Let's check why:

  • 32 divided by 6 is 5 with a remainder of 2 (32 = 6 * 5 + 2).
  • Any number of dozens is a multiple of 12. And 12 is a multiple of 6 (12 = 6 * 2).
  • So, if you buy 'N' dozens, you buy N * 12 balloons. N * 12 is always divisible by 6.
  • Total balloons = 32 + (N * 12) = (6 * 5 + 2) + (N * 6 * 2) = 6 * 5 + 2 + 6 * 2N = 6 * (5 + 2N) + 2.

This means that any total number of balloons Abuela Tomasa can achieve by buying whole dozens will always have a remainder of 2 when divided by 6. This implies she can never divide the balloons perfectly equally among her 6 grandchildren if she only buys whole dozens and starts with 32 balloons!

Is there a mistake in my understanding or the problem itself? Let me re-read carefully. "decide repartir los globos, pero desea darles más de 6 globos a cada uno ¿Cuántas docenas debe comprar para dar la misma cantidad de globos a cada nieto?"

Ah, the phrasing! "dar la misma cantidad de globos a cada nieto". This strongly suggests that the total must be divisible by 6. The remainder of 2 means she can't give them the exact same amount if she uses all the balloons. Perhaps the question implies she should buy enough dozens so that the resulting total is divisible by 6, and she can then distribute them equally, potentially with a few left over that she can't distribute equally. Or, more likely, the starting number of balloons should have been different to make the math work cleanly.

Let's assume the problem intends for a clean mathematical solution where the total number of balloons is divisible by 6. This means the initial 32 balloons must be adjusted or the requirement must be interpreted differently.

However, if we must stick to the numbers given and the