Find The Product Of The Following Fractions And Reduce Them To Their Lowest Terms, If Possible: (i) 5/8 * 3/4 (ii) 16/35 * 21/48 (iii) 76/200 * 2/19 (iv) 20 * 10 3/4

In this comprehensive guide, we will delve into the core concept of multiplying fractions and simplifying the results to their lowest terms. This fundamental skill is crucial in various mathematical contexts and real-world applications. We'll explore step-by-step solutions to several examples, providing you with a solid understanding of the process. Before diving into the examples, let's establish the basic rule for multiplying fractions: To multiply fractions, simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Once you've performed the multiplication, it's essential to reduce the resulting fraction to its simplest form, if possible. This involves finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.

Understanding how to multiply fractions and reduce them to their lowest terms is a fundamental skill in mathematics. Fractions represent parts of a whole, and multiplying them allows us to determine a fraction of another fraction. For instance, if you have half a pizza and you want to give a third of that half to a friend, you're essentially multiplying 1/2 by 1/3. The result, 1/6, tells you that your friend will receive one-sixth of the whole pizza. This concept extends beyond simple examples like pizza sharing. In more advanced mathematics, multiplying fractions is essential for solving equations, working with ratios and proportions, and understanding probability. Moreover, in real-world applications, you might use fraction multiplication when scaling recipes, calculating discounts, or determining measurements in construction or engineering projects. Mastering this skill provides a solid foundation for tackling more complex mathematical problems and confidently applying mathematical concepts in practical situations. Therefore, a thorough understanding of fraction multiplication and simplification is not just an academic exercise; it's a valuable tool for problem-solving in both theoretical and practical contexts.

(i) Multiplying rac{5}{8} by rac{3}{4}

Let's begin with our first example: rac{5}{8} \times rac{3}{4}. To multiply these fractions, we multiply the numerators (5 and 3) and the denominators (8 and 4). This gives us:

{rac{5}{8} \times rac{3}{4} = \frac{5 \times 3}{8 \times 4} = \frac{15}{32}}

Now, we need to check if the fraction rac{15}{32} can be reduced to its lowest terms. To do this, we look for the greatest common factor (GCF) of 15 and 32. The factors of 15 are 1, 3, 5, and 15. The factors of 32 are 1, 2, 4, 8, 16, and 32. The only common factor is 1, which means that 15 and 32 are relatively prime, and the fraction is already in its simplest form. Therefore, the product of rac{5}{8} and rac{3}{4} is rac{15}{32}.

Understanding how to determine if a fraction is in its simplest form is crucial for mastering fraction manipulation. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. This means that you cannot divide both the numerator and the denominator by any whole number greater than 1 and still get whole numbers as the result. To check if a fraction is in its simplest form, you need to identify the factors of both the numerator and the denominator. Factors are the numbers that divide evenly into a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because 12 can be divided by each of these numbers without leaving a remainder. Once you have identified the factors of both the numerator and the denominator, you need to find the greatest common factor (GCF). The GCF is the largest factor that the numerator and the denominator share. If the GCF is 1, then the fraction is already in its simplest form, because there is no number other than 1 that can divide both the numerator and the denominator evenly. If the GCF is greater than 1, then the fraction can be simplified further by dividing both the numerator and the denominator by the GCF. This process is called reducing the fraction to its lowest terms. Recognizing when a fraction is in its simplest form is a fundamental skill that helps in simplifying calculations and comparing fractions more easily.

(ii) Multiplying rac{16}{35} by rac{21}{48}

Next, let's consider the example: rac{16}{35} \times rac{21}{48}. Multiplying the numerators and denominators, we get:

{rac{16}{35} \times rac{21}{48} = \frac{16 \times 21}{35 \times 48} = \frac{336}{1680}}

Now, we need to simplify rac{336}{1680}. Finding the GCF of 336 and 1680 directly might be challenging, so we can simplify in stages. We notice that both numbers are even, so we can divide both by 2:

${\frac{336}{1680} = \frac{336 \div 2}{1680 \div 2} = \frac{168}{840}}$

Both numbers are still even, so we divide by 2 again:

${\frac{168}{840} = \frac{168 \div 2}{840 \div 2} = \frac{84}{420}}$

Again, both numbers are even:

${\frac{84}{420} = \frac{84 \div 2}{420 \div 2} = \frac{42}{210}}$

Both are still even:

${\frac{42}{210} = \frac{42 \div 2}{210 \div 2} = \frac{21}{105}}$

Now, both numbers are divisible by 3:

${\frac{21}{105} = \frac{21 \div 3}{105 \div 3} = \frac{7}{35}}$

Finally, both numbers are divisible by 7:

${\frac{7}{35} = \frac{7 \div 7}{35 \div 7} = \frac{1}{5}}$

Therefore, the product of rac{16}{35} and rac{21}{48}, reduced to its lowest terms, is rac{1}{5}.

Simplifying fractions in stages is a valuable technique, particularly when dealing with larger numbers where finding the greatest common factor (GCF) directly can be cumbersome. This method involves dividing both the numerator and the denominator by a common factor, then repeating the process until the fraction is in its simplest form. The key advantage of this approach is that it breaks down a potentially complex simplification into a series of smaller, more manageable steps. For instance, when you have a fraction like 336/1680, instead of trying to determine the GCF of 336 and 1680 all at once, you can start by noticing that both numbers are even and thus divisible by 2. Dividing both numerator and denominator by 2 gives you 168/840, which is a simpler fraction. You can then continue this process, dividing by common factors like 2, 3, 5, or 7, until you arrive at a fraction where the numerator and denominator have no common factors other than 1. This iterative process not only makes simplification easier but also reduces the risk of errors. By simplifying in stages, you ensure that each step is straightforward, making it more likely that you will arrive at the correct simplified fraction. This method is particularly useful in mental math or when working without a calculator, as it allows you to handle complex fractions with greater confidence and accuracy.

(iii) Multiplying rac{76}{200} by rac{2}{19}

Let's move on to the next example: rac{76}{200} \times rac{2}{19}. Multiplying the numerators and denominators gives us:

{rac{76}{200} \times rac{2}{19} = \frac{76 \times 2}{200 \times 19} = \frac{152}{3800}}

Now, let's simplify rac{152}{3800}. We can start by dividing both numbers by 2:

${\frac{152}{3800} = \frac{152 \div 2}{3800 \div 2} = \frac{76}{1900}}$

Divide by 2 again:

${\frac{76}{1900} = \frac{76 \div 2}{1900 \div 2} = \frac{38}{950}}$

Divide by 2 again:

${\frac{38}{950} = \frac{38 \div 2}{950 \div 2} = \frac{19}{475}}$

Now, we notice that 19 is a prime number. Let's see if 475 is divisible by 19. Dividing 475 by 19, we get 25. So, we can simplify further:

${\frac{19}{475} = \frac{19 \div 19}{475 \div 19} = \frac{1}{25}}$

Therefore, the product of rac{76}{200} and rac{2}{19}, reduced to its lowest terms, is rac{1}{25}.

Recognizing prime numbers and using them to simplify fractions can significantly streamline the simplification process. A prime number is a whole number greater than 1 that has only two distinct positive divisors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, and so on. When simplifying fractions, if you notice that either the numerator or the denominator is a prime number, it immediately narrows down the possibilities for common factors. For instance, in the fraction 19/475, recognizing that 19 is a prime number means that the only possible common factors between the numerator and the denominator are 1 and 19. This simplifies the task of finding the greatest common factor (GCF) because you only need to check if the denominator is divisible by the prime number. If the denominator is divisible by the prime number, as in the case of 475 being divisible by 19, then you can simplify the fraction by dividing both the numerator and the denominator by that prime number. This technique is particularly useful because prime numbers cannot be further factored, making them powerful tools in reducing fractions to their simplest form. By identifying prime numbers early in the simplification process, you can often avoid unnecessary steps and arrive at the simplest form more quickly and efficiently.

(iv) Multiplying 20 by 10 rac{3}{4}

Now, let's tackle the example: 20 \times 10 \frac{3}{4}. First, we need to convert the mixed number 10 \frac{3}{4} into an improper fraction. To do this, we multiply the whole number (10) by the denominator (4) and add the numerator (3). This gives us (10 \times 4) + 3 = 43. So, 10 \frac{3}{4} is equal to \frac{43}{4}. Now we can rewrite the problem as:

${20 \times 10 \frac{3}{4} = 20 \times \frac{43}{4}}$

We can express 20 as a fraction by writing it as \frac{20}{1}. Now we multiply the fractions:

${\frac{20}{1} \times \frac{43}{4} = \frac{20 \times 43}{1 \times 4} = \frac{860}{4}}$

Now, we simplify rac{860}{4}. We can divide both numbers by 4:

${\frac{860}{4} = \frac{860 \div 4}{4 \div 4} = \frac{215}{1} = 215}$

Therefore, the product of 20 and 10 \frac{3}{4} is 215.

Converting mixed numbers to improper fractions is a crucial step in performing arithmetic operations, particularly multiplication and division, with fractions. A mixed number is a number consisting of a whole number and a proper fraction, such as 10 3/4. To convert a mixed number to an improper fraction, you follow a systematic process that ensures accurate calculations. First, you multiply the whole number part by the denominator of the fractional part. This step calculates the number of fractional units contained in the whole number. For example, in the mixed number 10 3/4, you multiply 10 by 4, which equals 40. This indicates that the whole number 10 contains 40 quarters (since the denominator is 4). Next, you add the numerator of the fractional part to the result obtained in the first step. In this case, you add 3 to 40, which equals 43. This new number represents the total number of fractional units in the mixed number. Finally, you write this sum as the numerator of the improper fraction, keeping the same denominator as the original fractional part. Therefore, the improper fraction equivalent of 10 3/4 is 43/4. This conversion is essential because it transforms the mixed number into a single fraction, making it easier to perform multiplication, division, and other arithmetic operations. By converting mixed numbers to improper fractions, you eliminate the complexity of dealing with both whole numbers and fractions simultaneously, leading to more straightforward and accurate calculations.

In conclusion, mastering fraction multiplication and simplification involves understanding the basic rules, practicing step-by-step simplification, and employing techniques like identifying prime numbers and simplifying in stages. These skills are fundamental to success in mathematics and have wide-ranging applications in various fields.