Applying The Associative Property In Mathematics Exercises

In the realm of mathematics, the associative property stands as a cornerstone principle, particularly crucial in simplifying complex arithmetic operations. This property, applicable to both addition and multiplication, asserts that the grouping of numbers does not alter the final result. This means that when you add or multiply three or more numbers, the way you group them using parentheses will not change the sum or product. For instance, in addition, (a + b) + c is equivalent to a + (b + c), and similarly, in multiplication, (a × b) × c is the same as a × (b × c). This flexibility is not just a theoretical concept but a practical tool that streamlines calculations, making it easier to solve problems mentally or on paper. By strategically regrouping numbers, one can often identify combinations that lead to simpler arithmetic, thus reducing the chances of errors and enhancing efficiency. In the subsequent sections, we will delve into specific exercises that demonstrate the power and utility of the associative property, providing a clear understanding of how it can be applied in various mathematical contexts.

Exercise A: $25 imes 11 imes 18$

In this first exercise, we aim to apply the associative property to the expression $25 imes 11 imes 18$. The goal is to rearrange the grouping of these numbers in a way that simplifies the multiplication process. At first glance, multiplying these numbers in their given order might seem straightforward, but by leveraging the associative property, we can discover a more efficient path to the solution. The key here is to identify pairs of numbers that, when multiplied together, yield a result that is easy to work with. For instance, multiples of 10 or 100 are generally simpler to handle in subsequent calculations. So, let's explore how regrouping the numbers can make this multiplication easier. One approach could be to first multiply 25 by 18, as this will result in a multiple of 100. This strategic regrouping allows us to transform a potentially cumbersome calculation into a more manageable one. Let's delve into the step-by-step solution to see this principle in action and appreciate how it simplifies the arithmetic.

Step-by-Step Solution

1. Original Expression: $25 imes 11 imes 18$
2. Apply Associative Property: We can regroup the numbers as $(25 imes 18) imes 11$. This is where the associative property comes into play, allowing us to change the grouping without affecting the outcome.
3. Multiply 25 by 18: $25 imes 18 = 450$. This step is crucial as it simplifies the calculation by creating a larger, more manageable number.
4. Multiply 450 by 11: Now we have $450 imes 11$. Multiplying by 11 can be done by adding the number to itself shifted by one decimal place: $450 + 4500 = 4950$.
5. Final Result: Therefore, $25 imes 11 imes 18 = 4950$.

By strategically applying the associative property, we transformed a seemingly complex multiplication problem into a series of simpler steps. This approach not only makes the calculation easier but also reduces the likelihood of errors. The key takeaway here is that identifying opportunities for strategic regrouping can significantly streamline mathematical operations, especially in multiplication.

Exercise B: 83 imes (oxed{ lacksquare } imes oxed{ lacksquare }) = oxed{ lacksquare }

Exercise B presents a slightly different challenge compared to the previous one. Here, we are given an incomplete equation and tasked with filling in the blanks to demonstrate the associative property. The structure 83 imes (oxed{ lacksquare } imes oxed{ lacksquare }) = oxed{ lacksquare } prompts us to think creatively about how to introduce numbers that, when multiplied, will simplify the overall calculation. The goal is not just to find any numbers that fit but to select numbers that make the multiplication process as straightforward as possible. This often involves looking for numbers that, when multiplied together, result in round numbers like 10, 100, or 1000, as these are easier to work with. Furthermore, this exercise encourages a deeper understanding of how the associative property works, as it requires us to think about the relationship between the numbers and how different groupings can lead to the same result. Let's explore potential solutions and the reasoning behind them, highlighting the flexibility and utility of the associative property in mathematical problem-solving.

Finding Suitable Numbers

To effectively complete this exercise, we need to introduce numbers that simplify the multiplication. A common strategy is to look for pairs of numbers that multiply to a power of 10, such as 10, 100, or 1000. However, in this case, finding such a direct combination might be challenging. Instead, let's focus on making the calculation with 83 easier.

One approach could be to choose numbers that, when multiplied, result in a number close to a multiple of 100. This will help in simplifying the final multiplication. For instance, if we could make the product inside the parentheses equal to a number close to 100, multiplying by 83 would become more manageable.

Let's consider the numbers 10 and a fraction. If we choose 10 as one of the numbers, we need another number that, when multiplied by 10, will give us a manageable product when multiplied by 83. This requires some trial and error, but it's a valuable exercise in number sense.

A Possible Solution

After some consideration, let's try the numbers 10 and 10. Then the equation becomes:

83 imes (10 imes 10) = oxed{ lacksquare }

1. Multiply inside the parentheses: $10 imes 10 = 100$
2. Multiply 83 by 100: $83 imes 100 = 8300$
3. Final Result: Therefore, $83 imes (10 imes 10) = 8300$

This solution demonstrates how choosing appropriate numbers can simplify the calculation. By selecting 10 and 10, we created a product of 100, which made the final multiplication straightforward. This exercise underscores the importance of strategic number selection when applying the associative property, highlighting its role in making complex calculations more accessible.

Exercise C: $16 imes 15 imes 12$

Moving onto Exercise C, we encounter another multiplication problem where the associative property can be effectively applied. The expression $16 imes 15 imes 12$ presents an opportunity to rearrange the grouping of these numbers in order to simplify the calculation. Similar to the first exercise, our aim here is to identify pairs of numbers that, when multiplied together, yield a result that is easier to work with in subsequent steps. This often means looking for combinations that produce multiples of 10 or 100, as these are generally simpler to handle. However, other considerations might also come into play, such as identifying numbers that have common factors or that are easy to multiply mentally. The key is to explore different groupings and assess which one leads to the most efficient solution. Let's delve into the process of applying the associative property to this particular problem, highlighting the strategic thinking involved in choosing the optimal grouping.

Step-by-Step Solution

1. Original Expression: $16 imes 15 imes 12$
2. Apply Associative Property: There are a couple of ways we can regroup these numbers. Let's try $(16 imes 12) imes 15$ first.
3. Multiply 16 by 12: $16 imes 12 = 192$. This multiplication might seem a bit challenging at first, but it's a manageable calculation.
4. Multiply 192 by 15: Now we have $192 imes 15$. This can be broken down further: $(192 imes 10) + (192 imes 5) = 1920 + 960 = 2880$.
5. Final Result: Therefore, $16 imes 15 imes 12 = 2880$.

Alternatively, we could have regrouped the numbers as $(15 imes 12) imes 16$:

1. Multiply 15 by 12: $15 imes 12 = 180$. This is a simpler multiplication compared to $16 imes 12$.
2. Multiply 180 by 16: Now we have $180 imes 16$. This can also be broken down: $(180 imes 10) + (180 imes 6) = 1800 + 1080 = 2880$.
3. Final Result: Therefore, $16 imes 15 imes 12 = 2880$.

Both groupings lead to the same result, but the second approach, multiplying 15 by 12 first, might be considered slightly more efficient due to the simpler initial multiplication. This exercise illustrates that while the associative property allows for flexibility in grouping, some groupings can lead to easier calculations than others. The key is to identify the most strategic approach based on the specific numbers involved.

Exercise D: $17 imes 45 imes 9$

Finally, in Exercise D, we are presented with the expression $17 imes 45 imes 9$, another opportunity to harness the power of the associative property. As with the previous exercises, the core idea is to regroup these numbers in a way that simplifies the multiplication process. The challenge lies in identifying which grouping will lead to the most manageable calculations. This often involves looking for pairs of numbers that, when multiplied, result in multiples of 10, 100, or other easily workable numbers. However, sometimes the most efficient approach involves recognizing other relationships between the numbers, such as common factors or numbers that are easy to multiply mentally. Let's explore how we can strategically apply the associative property to this particular problem, paying close attention to the reasoning behind our choice of grouping.

Step-by-Step Solution

1. Original Expression: $17 imes 45 imes 9$
2. Apply Associative Property: Let's consider regrouping the numbers as $(45 imes 9) imes 17$. This grouping might be advantageous because multiplying 45 by 9 results in a number that is relatively easy to work with.
3. Multiply 45 by 9: $45 imes 9 = 405$. This multiplication can be done mentally by thinking of 9 as $(10 - 1)$, so $45 imes 9 = (45 imes 10) - (45 imes 1) = 450 - 45 = 405$.
4. Multiply 405 by 17: Now we have $405 imes 17$. This multiplication is a bit more involved, but we can break it down: $(405 imes 10) + (405 imes 7) = 4050 + (400 imes 7 + 5 imes 7) = 4050 + (2800 + 35) = 4050 + 2835 = 6885$.
5. Final Result: Therefore, $17 imes 45 imes 9 = 6885$.

Alternatively, we could have regrouped the numbers as $(17 imes 9) imes 45$:

1. Multiply 17 by 9: $17 imes 9 = 153$. This multiplication can also be done mentally: $(17 imes 10) - 17 = 170 - 17 = 153$.
2. Multiply 153 by 45: Now we have $153 imes 45$. This multiplication is also a bit involved, and it might be argued that this grouping doesn't simplify the calculation as much as the first approach.

Comparing the two approaches, it seems that the first grouping, $(45 imes 9) imes 17$, leads to a slightly more manageable calculation, as it results in a larger intermediate number (405) that is then multiplied by a smaller number (17). This exercise reinforces the idea that while the associative property provides flexibility, the choice of grouping can significantly impact the ease of calculation. Strategic thinking and careful consideration of the numbers involved are key to effectively applying the associative property.

In conclusion, the exercises presented here serve as a practical demonstration of the associative property in mathematics. This property, which applies to both addition and multiplication, allows us to regroup numbers without changing the final result. Throughout these exercises, we've seen how strategically applying the associative property can significantly simplify complex calculations. By identifying pairs of numbers that, when multiplied together, yield easily workable results, such as multiples of 10 or 100, we can transform seemingly daunting problems into manageable steps. The exercises also highlight the importance of number sense and strategic thinking in mathematical problem-solving. While the associative property provides flexibility in grouping, the choice of grouping can greatly impact the efficiency of the calculation. Therefore, careful consideration of the numbers involved and their relationships is crucial for effectively applying this property. Ultimately, mastering the associative property not only enhances computational skills but also fosters a deeper understanding of mathematical principles and their practical applications.