Evaluating Algebraic Expressions With Fractions Step-by-Step

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Hey guys! Let's dive into the world of fraction evaluation! In this article, we're going to tackle some expressions involving fractions and variables. We'll be given values for the variables a, b, and c, and our mission is to substitute these values into the expressions and simplify the results. Don't worry, we'll break it down step by step so it's super easy to follow. So, grab your pencils and let's get started!

Problem Setup

We're given the following values for our variables:

  • a = - rac{5}{6}

  • b = -3 rac{3}{8}

  • c = rac{7}{10}

Our goal is to evaluate the following expressions using these values. We need to write the product in simplest form for each one. This means that after we perform the multiplication, we'll need to reduce the fraction to its lowest terms. Remember, simplifying fractions involves dividing both the numerator and the denominator by their greatest common factor (GCF). This ensures that our final answer is in the most concise form. Let's get to it!

  1. bcb c

  2. aca c

  3. 4 rac{2}{5} c

  4. −2abc-2 a b c

Now, let's evaluate each of these expressions one by one.

1. Evaluating $b c$

Fraction multiplication is super fun, especially when you know the tricks. Our first expression is bc, which means b multiplied by c. We know that $b = -3 rac{3}{8}$ and $c = rac{7}{10}$. Before we can multiply, we need to convert the mixed number b into an improper fraction. To do this, we multiply the whole number part (3) by the denominator (8) and add the numerator (3). This gives us (3 * 8) + 3 = 27. So, the improper fraction form of b is $- rac{27}{8}$. Remember to keep the negative sign!

Now we can rewrite our expression as: $b c = - rac{27}{8} imes rac{7}{10}$. When multiplying fractions, we simply multiply the numerators together and the denominators together. So, we have:

- rac{27}{8} imes rac{7}{10} = - rac{27 imes 7}{8 imes 10} = - rac{189}{80}

Now, we have an improper fraction, meaning the numerator is larger than the denominator. We need to convert this back into a mixed number. To do this, we divide 189 by 80. 80 goes into 189 two times (2 * 80 = 160) with a remainder of 29. So, we can write the mixed number as $-2 rac{29}{80}$.

Finally, we need to check if we can simplify the fraction $ rac{29}{80}$. The factors of 29 are 1 and 29 (it's a prime number), and the factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80. The only common factor is 1, so the fraction is already in its simplest form.

Therefore, the final answer for $b c$ is $-2 rac{29}{80}$. It's crucial to understand the process of converting mixed numbers to improper fractions and back, as this is a fundamental skill when dealing with fraction operations. Additionally, always remember to check if your final fraction can be simplified. Reducing fractions to their simplest form ensures your answer is both accurate and presented in the most concise way. In this case, we successfully converted the mixed number to an improper fraction, performed the multiplication, and then converted the resulting improper fraction back to a simplified mixed number. This methodical approach is key to mastering fraction manipulations.

2. Evaluating $a c$

Next up, we have the expression ac, which means a multiplied by c. We know that $a = - rac{5}{6}$ and $c = rac{7}{10}$. Let's multiply these fractions together. Remember, multiplying fractions is straightforward: multiply the numerators and multiply the denominators:

a c = - rac{5}{6} imes rac{7}{10} = - rac{5 imes 7}{6 imes 10} = - rac{35}{60}

Now, we need to simplify this fraction. Both 35 and 60 are divisible by 5. So, we can divide both the numerator and the denominator by 5:

- rac{35}{60} = - rac{35 ext{ ÷ } 5}{60 ext{ ÷ } 5} = - rac{7}{12}

The factors of 7 are 1 and 7 (it's a prime number), and the factors of 12 are 1, 2, 3, 4, 6, and 12. The only common factor is 1, so the fraction is already in its simplest form.

Therefore, the final answer for $a c$ is $- rac{7}{12}$. Simplifying fractions after multiplication, as we did here, is a crucial step in arriving at the correct answer. By identifying and dividing out common factors, you ensure the fraction is in its most reduced form. In this particular case, recognizing that both 35 and 60 share a common factor of 5 allowed us to simplify the fraction considerably. Always look for these opportunities to simplify; it not only makes your answer cleaner but also demonstrates a strong understanding of fraction manipulation.

3. Evaluating $4 rac{2}{5} c$

Our third expression is $4 rac{2}{5} c$. We know that $c = rac{7}{10}$. First, we need to convert the mixed number $4 rac{2}{5}$ into an improper fraction. To do this, we multiply the whole number part (4) by the denominator (5) and add the numerator (2). This gives us (4 * 5) + 2 = 22. So, the improper fraction form is $ rac{22}{5}$.

Now we can rewrite our expression as: $4 rac{2}{5} c = rac{22}{5} imes rac{7}{10}$. Let's multiply these fractions together:

rac{22}{5} imes rac{7}{10} = rac{22 imes 7}{5 imes 10} = rac{154}{50}

Now, we need to simplify this fraction. Both 154 and 50 are divisible by 2. So, we can divide both the numerator and the denominator by 2:

rac{154}{50} = rac{154 ext{ ÷ } 2}{50 ext{ ÷ } 2} = rac{77}{25}

We have an improper fraction again. Let's convert it to a mixed number. 25 goes into 77 three times (3 * 25 = 75) with a remainder of 2. So, the mixed number is $3 rac{2}{25}$.

The factors of 2 are 1 and 2, and the factors of 25 are 1, 5, and 25. The only common factor is 1, so the fraction is already in its simplest form.

Therefore, the final answer for $4 rac{2}{5} c$ is $3 rac{2}{25}$. Converting mixed numbers to improper fractions and back, as demonstrated in this step, is a vital skill for working with fractions. This process allows for easier multiplication and division. After performing the multiplication, it's equally important to simplify the resulting fraction. In this case, we simplified the improper fraction by dividing both numerator and denominator by their greatest common divisor, and then converted it back into a mixed number for a cleaner, more understandable final answer. This cycle of conversion, multiplication, and simplification is a common pattern in fraction-based calculations and worth mastering.

4. Evaluating $-2 a b c$

Finally, let's tackle the expression $-2 a b c$. We know that $a = - rac{5}{6}$, $b = -3 rac{3}{8}$, and $c = rac{7}{10}$. We already converted b to an improper fraction in the first problem, so we know $b = - rac{27}{8}$. Now, let's substitute these values into the expression:

-2 a b c = -2 imes - rac{5}{6} imes - rac{27}{8} imes rac{7}{10}

Let's multiply these fractions together. Remember, a negative times a negative is a positive, and a positive times a negative is a negative. In this case, we have three negative numbers being multiplied, so the result will be negative.

-2 imes - rac{5}{6} imes - rac{27}{8} imes rac{7}{10} = - rac{2 imes 5 imes 27 imes 7}{6 imes 8 imes 10} = - rac{1890}{480}

Now, we need to simplify this fraction. Both 1890 and 480 are divisible by 10:

- rac{1890}{480} = - rac{1890 ext{ ÷ } 10}{480 ext{ ÷ } 10} = - rac{189}{48}

Both 189 and 48 are divisible by 3:

- rac{189}{48} = - rac{189 ext{ ÷ } 3}{48 ext{ ÷ } 3} = - rac{63}{16}

Now, let's convert this improper fraction to a mixed number. 16 goes into 63 three times (3 * 16 = 48) with a remainder of 15. So, the mixed number is $-3 rac{15}{16}$.

The factors of 15 are 1, 3, 5, and 15, and the factors of 16 are 1, 2, 4, 8, and 16. The only common factor is 1, so the fraction is already in its simplest form.

Therefore, the final answer for $-2 a b c$ is $-3 rac{15}{16}$. Handling multiple negative signs, as we did here, is a crucial skill in fraction multiplication. Keeping track of the signs and correctly applying the rules of multiplication ensures an accurate final answer. Additionally, this problem highlights the importance of simplifying fractions in multiple stages. We first divided by 10, then by 3, demonstrating that simplification can sometimes be a step-by-step process. This approach can make the process more manageable, especially when dealing with larger numbers. Remember, the goal is to get the fraction to its simplest form, regardless of how many steps it takes.

Conclusion

Great job, guys! We've successfully evaluated all four expressions. We tackled converting mixed numbers to improper fractions, multiplying fractions, simplifying fractions, and keeping track of negative signs. Remember, practice makes perfect, so keep working on these skills, and you'll become a fraction master in no time!