Solving Algebraic Expressions And Rounding To Nearest Centavo

In this mathematical problem, we are tasked with finding the difference between two algebraic expressions. The expressions are $10(2a - 3b + c)$ and $3(4a - 2b - 5c)$. To solve this, we will first distribute the constants outside the parentheses into the respective terms inside. Then, we will combine like terms to simplify the expression and find the final difference.

1.1 Distributing the Constants

Our first step involves distributing the constants outside the parentheses into each term inside. This means multiplying each term within the parentheses by the constant factor preceding it. For the first expression, $10(2a - 3b + c)$, we multiply each term by 10:

$10 * 2a = 20a$ $10 * -3b = -30b$ $10 * c = 10c$

So, the first expression expands to $20a - 30b + 10c$.

Next, we do the same for the second expression, $3(4a - 2b - 5c)$. We multiply each term by 3:

$3 * 4a = 12a$ $3 * -2b = -6b$ $3 * -5c = -15c$

Thus, the second expression expands to $12a - 6b - 15c$.

1.2 Subtracting the Expressions

Now that we have both expressions expanded, we need to find the difference between them. This means subtracting the second expression from the first. The expressions are:

First expression: $20a - 30b + 10c$ Second expression: $12a - 6b - 15c$

To subtract, we subtract the corresponding terms. We subtract the 'a' terms, the 'b' terms, and the 'c' terms separately:

$(20a - 30b + 10c) - (12a - 6b - 15c)$

This is equivalent to:

$20a - 30b + 10c - 12a + 6b + 15c$

Pay close attention to the signs when subtracting. Subtracting a negative term is the same as adding its positive counterpart. This is why $-(-6b)$ becomes $+6b$ and $-(-15c)$ becomes $+15c$.

1.3 Combining Like Terms

Now, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have 'a' terms, 'b' terms, and 'c' terms. We group them together:

$(20a - 12a) + (-30b + 6b) + (10c + 15c)$

Now, we perform the operations:

$20a - 12a = 8a$ $-30b + 6b = -24b$ $10c + 15c = 25c$

So, the simplified expression is:

$8a - 24b + 25c$

Comparing our result with the provided options:

A. $8a + 24b - 25c$ B. $8a - 24b + 25c$ C. $-8a - 36b + 5c$ D. $8a + 36b - 5c$

We find that our result, $8a - 24b + 25c$, matches option B.

1.4 Conclusion for Algebraic Expression

Thus, the difference between $10(2a - 3b + c)$ and $3(4a - 2b - 5c)$ is $8a - 24b + 25c$. This process involved distributing constants, careful subtraction, and combining like terms to arrive at the simplified expression. This problem emphasizes the importance of following the correct order of operations and paying attention to signs in algebraic manipulations. Understanding these steps is crucial for solving more complex algebraic problems.

Therefore, the correct answer is B. $8a - 24b + 25c$.

In this part, we are tasked with rounding the number 100.963 to the nearest centavo. The term centavo refers to the second decimal place in a currency system (like cents in a dollar). Therefore, we need to round 100.963 to two decimal places. Rounding is a fundamental mathematical operation used to simplify numbers while keeping them close to their original value. It is especially useful in financial calculations and everyday estimations.

2.1 Understanding Decimal Places

Before we begin rounding, let’s clarify what decimal places are. In the number 100.963:

• 100 is the whole number part.
• 9 is in the tenths place (the first decimal place).
• 6 is in the hundredths place (the second decimal place, also known as centavos).
• 3 is in the thousandths place (the third decimal place).

We want to round the number to the hundredths place, so we will focus on the digit in the thousandths place (3) to determine whether to round up or down.

2.2 The Rounding Rule

The standard rounding rule is as follows:

• If the digit to the right of the place you are rounding to is 5 or greater, you round up. This means you add 1 to the digit in the place you are rounding to.
• If the digit to the right is less than 5, you round down. This means the digit in the place you are rounding to stays the same, and you drop the digits to the right.

In our case, we are rounding to the hundredths place (the 6 in 100.963), and the digit to the right is 3. Since 3 is less than 5, we round down.

2.3 Applying the Rounding Rule

Applying the rounding rule to 100.963:

We look at the digit in the thousandths place, which is 3. Since 3 is less than 5, we round down. This means the digit in the hundredths place (6) remains the same, and we drop the digit 3.

So, 100.963 rounded to the nearest centavo is 100.96.

2.4 Practical Implications and Accuracy

Rounding to the nearest centavo is commonly used in financial transactions to ensure precision and avoid dealing with fractions of the smallest currency unit. It’s crucial for accurate accounting, billing, and payments. For example, in a store, prices are often rounded to the nearest centavo to simplify cash transactions and digital payments.

In a broader context, rounding is used in various fields, including science, engineering, and statistics, to present data in a more manageable form. However, it's essential to be aware that rounding can introduce a slight degree of error. While this error is typically small, it can accumulate in complex calculations, so it’s important to consider the level of precision required for the task at hand.

Therefore, Phn 100.963 rounded to the nearest centavo is 100.96. This process illustrates how simple rounding rules can be applied in practical situations to achieve accurate and meaningful results.