Numerical Expressions: Sums And Values

Hey guys! Let's dive into the world of numerical expressions and how to find their values. We'll break down a few examples step-by-step, so you'll be a pro in no time. Get ready to sharpen those pencils and flex your math muscles!

a) Adding the number $13$ to the sum of the numbers $15$ and $-40$

Okay, let's tackle the first part of our problem. We need to create a numerical expression and then find its value. The key here is to follow the order of operations. First, we need to find the sum of $15$ and $-40$. Think of it like this: you have $15$, but you owe $40$. After paying what you have, you'll still owe $25$. So, $15 + (-40) = -25$. Now, we need to add $13$ to this sum. So, our expression looks like this:

(15 + (-40)) + 13

Remember, we already found that $15 + (-40) = -25$. Now we just need to add $13$ to $-25$. Imagine you owe $25$, but you have $13$. After paying some of your debt, how much will you still owe? You'll still owe $12$. Therefore, $-25 + 13 = -12$. So, the final answer is:

(15 + (-40)) + 13 = -12

Numerical expressions are a fundamental part of mathematics, allowing us to represent and solve problems involving numbers and operations. In this particular problem, we focused on the order of operations, which is crucial for obtaining the correct result. The expression $(15 + (-40)) + 13$ involves both addition and the handling of negative numbers. The initial step requires calculating the sum of $15$ and $-40$, which introduces the concept of adding numbers with different signs. This step is a practical application of understanding how positive and negative numbers interact. When we add a negative number, it's the same as subtracting its absolute value. So, $15 + (-40)$ can be thought of as $15 - 40$. The result is $-25$, which represents a value that is $25$ units less than zero on the number line. The second part of the expression involves adding $13$ to the result we just obtained, which is $-25$. This is another instance of adding numbers with different signs, but this time, we are adding a positive number to a negative number. The calculation $-25 + 13$ is similar to starting at $-25$ on the number line and moving $13$ units to the right. The final result, $-12$, is obtained by understanding that moving $13$ units towards zero from $-25$ still leaves us $12$ units away from zero on the negative side. This process reinforces the understanding of how negative numbers work in conjunction with positive numbers, and how the number line can be a visual aid in understanding these operations. By breaking down the expression into smaller parts and focusing on each operation individually, we ensure that we follow the correct order and avoid common pitfalls. This methodical approach is invaluable in solving mathematical problems accurately and efficiently.

b) Adding the sum of the numbers $3.2$ and $-10$ to the number $9.4$

Alright, let's move on to the next one! This time, we're dealing with decimals, but don't worry, the process is the same. First, we need to find the sum of $3.2$ and $-10$. Think of this as having $3.2$ and owing $10$. After you use what you have to pay off some of the debt, you'll still owe $6.8$. So, $3.2 + (-10) = -6.8$. Now, we need to add this sum to $9.4$. Our expression looks like this:

9.4 + (3.2 + (-10))

We already figured out that $3.2 + (-10) = -6.8$. So now we have $9.4 + (-6.8)$. Imagine you have $9.4$, and you need to pay $6.8$. After you pay, how much will you have left? You'll have $2.6$ left. Therefore, $9.4 + (-6.8) = 2.6$. So, the final answer is:

9.4 + (3.2 + (-10)) = 2.6

This part of the problem involves the application of arithmetic operations to decimal numbers, introducing an additional layer of complexity. The expression $9.4 + (3.2 + (-10))$ requires us to perform addition with decimals and handle both positive and negative numbers, making it a practical exercise in number manipulation. The initial step within the parentheses is to add $3.2$ and $-10$. This operation combines a positive decimal with a negative integer, which requires an understanding of how decimal and integer arithmetic interact. Adding a negative number is equivalent to subtracting its absolute value, so $3.2 + (-10)$ can be rewritten as $3.2 - 10$. To perform this subtraction, we are essentially finding the difference between $10$ and $3.2$, and then applying the negative sign because the larger number ($10$) is negative. The result is $-6.8$, indicating that the sum is $6.8$ units less than zero on the number line. The next step is to add this result, $-6.8$, to $9.4$. This is another example of adding numbers with different signs, but this time, both numbers are decimals. The operation $9.4 + (-6.8)$ can also be thought of as $9.4 - 6.8$. To perform this subtraction, we align the decimal points and subtract the numbers column by column, borrowing if necessary. The result is $2.6$, which represents the value obtained after combining the two numbers. This final step is crucial as it demonstrates the practical application of adding and subtracting decimals, a skill that is widely used in everyday calculations. By breaking down the expression into manageable steps and focusing on each operation individually, we can accurately solve the problem and gain a deeper understanding of decimal arithmetic. This methodical approach is not only helpful in solving mathematical problems but also in building confidence and fluency in number manipulation.

c) Increasing the sum of the numbers $-0.8$ and $-0.08$ by $1$

Last one, guys! This one looks a bit tricky with those decimals, but we've got this. First, we need to find the sum of $-0.8$ and $-0.08$. Both numbers are negative, so we're essentially adding two debts together. To make it easier, let's line up the decimal points: $-0.80 + (-0.08) = -0.88$. Now, we need to increase this sum by $1$. Our expression looks like this:

(-0.8 + (-0.08)) + 1

We know that $-0.8 + (-0.08) = -0.88$. So now we have $-0.88 + 1$. Imagine you owe $0.88$, but you have $1$. After paying your debt, how much will you have left? You'll have $0.12$ left. Therefore, $-0.88 + 1 = 0.12$. So, the final answer is:

(-0.8 + (-0.08)) + 1 = 0.12

This final part of the problem tests our understanding of decimal arithmetic and operations involving negative numbers, culminating in a problem that requires careful consideration of place values and the rules of addition. The expression $(-0.8 + (-0.08)) + 1$ involves both adding negative decimals and then adding a positive integer to the result. The first step is to calculate the sum of $-0.8$ and $-0.08$. Adding two negative numbers is conceptually similar to combining two debts. To accurately add these decimals, it's important to align the decimal points and add the numbers column by column. This ensures that we are adding hundredths to hundredths, tenths to tenths, and so on. So, $-0.8 + (-0.08)$ can be visualized as $-0.80 + (-0.08)$. When we add these, we get $-0.88$, which represents the total negative value obtained by combining the two decimals. The second step is to add $1$ to the result we just obtained, which is $-0.88$. This operation is an example of adding a positive integer to a negative decimal. To perform this addition, we can think of it as starting at $-0.88$ on the number line and moving $1$ unit to the right. Alternatively, we can subtract the absolute value of $-0.88$ from $1$. So, the calculation becomes $1 - 0.88$. To perform this subtraction, we need to align the decimal points and subtract the numbers column by column, borrowing if necessary. The result is $0.12$, which represents the value obtained after adding $1$ to $-0.88$. This final answer is positive because the positive number ($1$) has a greater absolute value than the negative number ($-0.88$). By meticulously breaking down the expression and performing each operation with attention to decimal places and sign conventions, we can accurately solve the problem and solidify our understanding of arithmetic operations with decimals and negative numbers. This step-by-step approach reinforces the importance of precision in mathematical calculations and the application of fundamental rules to complex problems.

Hope that helps you guys understand how to create and solve numerical expressions! Keep practicing, and you'll become a math whiz in no time!