Solving ((-16/15) * (20/8)) - ((15/5) * (35/5)) A Step-by-Step Guide

In this comprehensive guide, we will delve into the step-by-step solution of the mathematical expression ( ${\frac{-16}{15} \times \frac{20}{8}}$ ) - ( ${\frac{15}{5} \times \frac{35}{5}}$ ). This problem involves the multiplication and subtraction of fractions, requiring a solid understanding of fraction arithmetic. We will break down each operation, providing clear explanations and intermediate steps to ensure clarity. By the end of this article, you will not only understand how to solve this particular problem but also gain a deeper insight into handling similar mathematical expressions. This article is designed to be accessible to learners of all levels, from those just beginning to grasp fraction arithmetic to those looking to refresh their skills.

To effectively solve the expression, we need to approach it systematically. The given expression is ( ${\frac{-16}{15} \times \frac{20}{8}}$ ) - ( ${\frac{15}{5} \times \frac{35}{5}}$ ). It consists of two main parts, each involving the multiplication of fractions, which are then subtracted from each other. The first part is ${\frac{-16}{15} \times \frac{20}{8}}$, and the second part is ${\frac{15}{5} \times \frac{35}{5}}$. We will tackle each part individually before performing the subtraction. This approach allows us to manage the complexity of the expression and reduces the chances of errors. Understanding the order of operations (PEMDAS/BODMAS) is crucial here, as multiplication must be performed before subtraction. Therefore, we will first compute the products of the fractions in each part and then find the difference between the results. This methodical approach is fundamental to solving mathematical problems accurately and efficiently.

Part 1: Multiplying ${\frac{-16}{15}}$ and ${\frac{20}{8}}$

First, let’s focus on the multiplication of the fractions ${\frac{-16}{15}}$ and ${\frac{20}{8}}$. To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we have:

${\frac{-16}{15} \times \frac{20}{8} = \frac{-16 \times 20}{15 \times 8}}$

This gives us ${\frac{-320}{120}}$. Now, we need to simplify this fraction. Both the numerator and the denominator are divisible by several common factors. To simplify, we can find the greatest common divisor (GCD) of 320 and 120. The GCD is 40. Dividing both the numerator and the denominator by 40, we get:

${\frac{-320 \div 40}{120 \div 40} = \frac{-8}{3}}$

So, the result of the first part of the expression is ${\frac{-8}{3}}$. This simplified fraction is easier to work with in the subsequent steps. Simplifying fractions is a crucial step in solving mathematical expressions, as it reduces the size of the numbers and makes further calculations simpler. It also helps in presenting the final answer in its most concise form.

Part 2: Multiplying ${\frac{15}{5}}$ and ${\frac{35}{5}}$

Next, we will address the second part of the expression, which involves multiplying the fractions ${\frac{15}{5}}$ and ${\frac{35}{5}}$. Similar to the first part, we multiply the numerators and the denominators:

${\frac{15}{5} \times \frac{35}{5} = \frac{15 \times 35}{5 \times 5}}$

This results in ${\frac{525}{25}}$. Now, we need to simplify this fraction. We can see that both 525 and 25 are divisible by 25. Dividing both the numerator and the denominator by 25, we get:

${\frac{525 \div 25}{25 \div 25} = 21}$

So, the result of the second part of the expression is 21. Note that 21 can also be written as a fraction ${\frac{21}{1}}$, which will be useful when we subtract it from the result of the first part. Simplifying fractions often involves identifying common factors and dividing both the numerator and denominator by these factors. In this case, recognizing that both 525 and 25 are divisible by 25 allowed us to simplify the fraction quickly and efficiently.

Part 3: Subtracting the Results

Now that we have the results from both parts of the expression, we can subtract them. We have ${\frac{-8}{3}}$ from the first part and 21 (or ${\frac{21}{1}}$) from the second part. The expression now looks like this:

${\frac{-8}{3} - 21}$

To subtract these, we need a common denominator. The common denominator for 3 and 1 is 3. So, we convert 21 to a fraction with a denominator of 3:

${21 = \frac{21 \times 3}{1 \times 3} = \frac{63}{3}}$

Now we can rewrite the subtraction as:

${\frac{-8}{3} - \frac{63}{3}}$

Subtracting the numerators, we get:

${\frac{-8 - 63}{3} = \frac{-71}{3}}$

So, the final result of the expression is ${\frac{-71}{3}}$. This fraction is already in its simplest form, as 71 and 3 have no common factors other than 1. The process of subtracting fractions requires finding a common denominator, which allows us to combine the fractions into a single term. In this case, converting 21 to a fraction with a denominator of 3 was crucial for performing the subtraction.

In conclusion, by systematically breaking down the expression ( ${\frac{-16}{15} \times \frac{20}{8}}$ ) - ( ${\frac{15}{5} \times \frac{35}{5}}$ ) into smaller, manageable parts, we have arrived at the final answer of ${\frac{-71}{3}}$. This process involved multiplying fractions, simplifying fractions, finding common denominators, and performing subtraction. Each step was carefully explained to provide a clear understanding of the mathematical operations involved. The key to solving complex expressions like this lies in understanding the order of operations and applying the rules of fraction arithmetic correctly. Simplifying fractions whenever possible helps to keep the calculations manageable and reduces the risk of errors.

This detailed solution not only answers the specific problem but also reinforces the fundamental principles of fraction arithmetic. By mastering these principles, you can confidently tackle a wide range of mathematical problems involving fractions. Remember, practice is essential for improving your skills in mathematics. Try solving similar problems to reinforce your understanding and build your confidence. The ability to work with fractions is a crucial skill in many areas of mathematics and beyond, making it a valuable asset for academic and professional pursuits. We hope this guide has been helpful in clarifying the steps involved in solving this expression and in enhancing your understanding of fraction arithmetic.