Multiply Sums, Find Differences, And Multiplicative Inverses A Math Guide

In this section, we embark on a mathematical journey to multiply the sums of two distinct pairs of fractions. Our initial task involves finding the sum of $\frac{3}{4}$ and $\frac{7}{8}$, followed by calculating the sum of $\frac{11}{13}$ and $\frac{5}{26}$. Finally, we will multiply these two sums to arrive at our final result. This exercise not only reinforces our understanding of fraction arithmetic but also highlights the importance of order of operations in mathematical calculations.

To begin, let's focus on the first pair of fractions, $\frac{3}{4}$ and $\frac{7}{8}$. To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 4 and 8 is 8. Therefore, we need to convert $\frac{3}{4}$ into an equivalent fraction with a denominator of 8. We can do this by multiplying both the numerator and the denominator of $\frac{3}{4}$ by 2, resulting in $\frac{6}{8}$. Now, we can add the two fractions:

$$\frac{3}{4} + \frac{7}{8} = \frac{6}{8} + \frac{7}{8} = \frac{6+7}{8} = \frac{13}{8}$$

Next, we turn our attention to the second pair of fractions, $\frac{11}{13}$ and $\frac{5}{26}$. Again, we need to find a common denominator to add these fractions. The LCM of 13 and 26 is 26. To convert $\frac{11}{13}$ into an equivalent fraction with a denominator of 26, we multiply both the numerator and the denominator of $\frac{11}{13}$ by 2, resulting in $\frac{22}{26}$. Now, we can add the two fractions:

$$\frac{11}{13} + \frac{5}{26} = \frac{22}{26} + \frac{5}{26} = \frac{22+5}{26} = \frac{27}{26}$$

Now that we have the sums of both pairs of fractions, we can multiply them together:

$$\frac{13}{8} \times \frac{27}{26} = \frac{13 \times 27}{8 \times 26}$$

Before we perform the multiplication, we can simplify the expression by canceling out common factors. Notice that 13 is a factor of both 13 and 26. We can divide both 13 and 26 by 13, which gives us 1 and 2, respectively. So, the expression becomes:

$$\frac{1 \times 27}{8 \times 2} = \frac{27}{16}$$

Therefore, the product of the sums of the two pairs of fractions is $\frac{27}{16}$. This fraction is an improper fraction, meaning that the numerator is greater than the denominator. We can convert this improper fraction into a mixed number by dividing 27 by 16. The quotient is 1, and the remainder is 11. So, the mixed number equivalent of $\frac{27}{16}$ is $1 \frac{11}{16}$. This comprehensive breakdown not only provides the solution but also explains each step in detail, ensuring a clear understanding of the process involved in multiplying the sums of fractions. Understanding these fundamentals is crucial for more advanced mathematical concepts.

In this section, our objective is to find the difference between 1 and the product of three fractions: $\frac{2}{3}$, $\frac{15}{27}$, and $1 \frac{19}{26}$. This problem involves a combination of fraction multiplication and subtraction, requiring us to first calculate the product of the fractions and then subtract the result from 1. This exercise emphasizes the importance of converting mixed numbers to improper fractions before performing multiplication and the correct application of the order of operations.

First, let's focus on the multiplication of the fractions. We have $\frac{2}{3} \times \frac{15}{27} \times 1 \frac{19}{26}$. Before we can multiply, we need to convert the mixed number $1 \frac{19}{26}$ into an improper fraction. To do this, we multiply the whole number part (1) by the denominator (26) and add the numerator (19), then place the result over the original denominator:

$$1 \frac{19}{26} = \frac{(1 \times 26) + 19}{26} = \frac{26 + 19}{26} = \frac{45}{26}$$

Now we can rewrite the multiplication problem as:

$$\frac{2}{3} \times \frac{15}{27} \times \frac{45}{26}$$

Before we multiply, we can simplify the fractions by canceling out common factors. Notice that 15 and 27 have a common factor of 3, and 45 and 27 also have a common factor of 9, while 2 and 26 share a factor of 2. Furthermore, we can simplify the fraction $\frac{15}{27}$ by dividing both numerator and denominator by their greatest common divisor, which is 3. This gives us $\frac{5}{9}$. We can also simplify $\frac{45}{27}$ by dividing both numerator and denominator by 9, which results in $\frac{5}{3}$. The expression can be simplified further by dividing 2 and 26 by their common factor, 2, resulting in $\frac{1}{13}$. Thus, we rewrite the equation as follows:

$$\frac{2}{3} \times \frac{15}{27} \times \frac{45}{26} = \frac{2}{3} \times \frac{5}{9} \times \frac{45}{26} = \frac{1}{3} \times \frac{5}{9} \times \frac{45}{13}$$

Now, we can multiply the numerators and the denominators:

$$\frac{2 \times 15 \times 45}{3 \times 27 \times 26}$$

Instead of multiplying directly, let's continue to simplify by canceling common factors. We can divide 15 and 27 by 3 (giving 5 and 9), and 45 and 27 by 9 (giving 5 and 3). Also, 2 and 26 share a factor of 2 (giving 1 and 13). This simplifies the expression to:

$$\frac{2}{3} \times \frac{5}{9} \times \frac{5}{26} = \frac{1 \times 5 \times 15}{1 \times 9 \times 13}$$

Further simplifying, we divide 15 and 9 by 3 (giving 5 and 3):

$$\frac{1 \times 5 \times 5}{1 \times 3 \times 13} = \frac{25}{39}$$

So, the product of the three fractions is $\frac{25}{39}$. Now, we need to find the difference between 1 and this product:

$$1 - \frac{25}{39}$$

To subtract a fraction from a whole number, we need to express the whole number as a fraction with the same denominator as the fraction we are subtracting. In this case, we can write 1 as $\frac{39}{39}$:

$$1 - \frac{25}{39} = \frac{39}{39} - \frac{25}{39}$$

Now we can subtract the numerators:

$$\frac{39}{39} - \frac{25}{39} = \frac{39 - 25}{39} = \frac{14}{39}$$

Therefore, the difference between 1 and the product of the fractions is $\frac{14}{39}$. This comprehensive solution breaks down each step, emphasizing the importance of simplification and the conversion of mixed numbers to improper fractions. It not only provides the answer but also reinforces the underlying principles of fraction arithmetic.

In this section, we delve into the concept of multiplicative inverses, also known as reciprocals. The multiplicative inverse of a number is the value that, when multiplied by the original number, results in a product of 1. Finding the multiplicative inverse is a fundamental operation in mathematics, particularly in algebra and number theory. It is essential for solving equations, simplifying expressions, and understanding the structure of number systems. This segment will guide you through the process of identifying and calculating multiplicative inverses, reinforcing the understanding of reciprocal relationships in mathematics.

To find the multiplicative inverse of a number, we simply flip the fraction, meaning we swap the numerator and the denominator. For example, the multiplicative inverse of $\frac{2}{3}$ is $\frac{3}{2}$, because $\frac{2}{3} \times \frac{3}{2} = 1$. For whole numbers, we can think of them as fractions with a denominator of 1. For example, the multiplicative inverse of 5 (which can be written as $\frac{5}{1}$) is $\frac{1}{5}$. Let’s consider some examples to illustrate this concept further:

1. Multiplicative Inverse of a Fraction: To find the multiplicative inverse of a fraction, you simply swap the numerator and the denominator. For example, the multiplicative inverse of $\frac{a}{b}$ (where a and b are non-zero) is $\frac{b}{a}$. This is because when you multiply these two fractions together, the result is 1:

$$\frac{a}{b} \times \frac{b}{a} = 1$$

2. Multiplicative Inverse of a Whole Number: Whole numbers can be considered fractions with a denominator of 1. For instance, the number 7 can be written as $\frac{7}{1}$. To find its multiplicative inverse, we flip the fraction to get $\frac{1}{7}$. Thus, the multiplicative inverse of 7 is $\frac{1}{7}$, since:

$$7 \times \frac{1}{7} = 1$$

3. Multiplicative Inverse of a Mixed Number: Before finding the multiplicative inverse of a mixed number, it must first be converted into an improper fraction. For example, let’s consider the mixed number $2 \frac{3}{4}$. To convert it into an improper fraction, we multiply the whole number part (2) by the denominator (4) and add the numerator (3). The result is placed over the original denominator:

$$2 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$$

Now that we have the improper fraction $\frac{11}{4}$, we can find its multiplicative inverse by swapping the numerator and the denominator, which gives us $\frac{4}{11}$. Therefore, the multiplicative inverse of $2 \frac{3}{4}$ is $\frac{4}{11}$.

4. Multiplicative Inverse of a Negative Number: The multiplicative inverse of a negative number is also negative. For example, let’s find the multiplicative inverse of -3. We can write -3 as $\frac{-3}{1}$. Flipping the fraction gives us $\frac{1}{-3}$, which is the same as $-\frac{1}{3}$. To verify, we multiply the number and its inverse:

$$-3 \times \left(-\frac{1}{3}\right) = 1$$

5. Multiplicative Inverse of 1: The multiplicative inverse of 1 is 1 itself, because 1 multiplied by 1 equals 1.

6. Multiplicative Inverse of -1: Similarly, the multiplicative inverse of -1 is -1, as -1 multiplied by -1 equals 1.

Important Note: The number 0 does not have a multiplicative inverse. This is because any number multiplied by 0 is 0, not 1. The reciprocal of 0 is undefined.

Understanding multiplicative inverses is not just an abstract mathematical concept; it has practical applications in various areas of mathematics and real-world problem-solving. For example, in algebra, multiplicative inverses are used to solve equations involving fractions. In cryptography, they play a crucial role in encryption and decryption algorithms. Furthermore, in engineering and physics, understanding reciprocals is essential for dealing with rates, ratios, and proportions.

In conclusion, the multiplicative inverse is a fundamental concept in mathematics that simplifies many calculations and provides a deeper understanding of number relationships. Mastering this concept is essential for success in higher-level mathematics and its applications in various fields. The ability to quickly identify and calculate multiplicative inverses is a valuable skill that enhances mathematical fluency and problem-solving capabilities. By consistently practicing with different types of numbers—fractions, whole numbers, mixed numbers, and negative numbers—one can develop a strong intuition for multiplicative inverses and their role in mathematical operations.