Solving Rational Number And Measurement Problems Calculating Rational Multipliers, Cloth Length Per Trouser, And Fabric Pieces
When dealing with rational numbers, a common question arises: what number do we need to multiply a given rational number by to get a desired result? This is precisely the type of problem we'll tackle in this section. Let’s delve into the question of by what rational number should -8/39 be multiplied to obtain 5/26?
To solve this, we need to understand the fundamental concept of inverse operations. Multiplication and division are inverse operations, meaning one can undo the other. Therefore, to find the required rational number, we need to divide the target number (5/26) by the given number (-8/39). This can be represented mathematically as:
(5/26) ÷ (-8/39)
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of -8/39 is -39/8. So, our equation becomes:
(5/26) * (-39/8)
Now, we can multiply the numerators and the denominators separately:
(5 * -39) / (26 * 8)
This gives us:
-195 / 208
To simplify this fraction, we need to find the greatest common divisor (GCD) of 195 and 208. The GCD is 13. Dividing both the numerator and the denominator by 13, we get:
(-195 ÷ 13) / (208 ÷ 13)
Which simplifies to:
-15 / 16
Therefore, the rational number -8/39 should be multiplied by -15/16 to obtain 5/26. This exercise highlights the importance of understanding rational number operations, particularly division and simplification. Mastering these concepts allows for the efficient solving of similar problems. In conclusion, when faced with a problem asking for a multiplier to achieve a specific result with rational numbers, remember to use division and simplification techniques to arrive at the correct answer. The ability to manipulate fractions and identify common factors is crucial for success in this area of mathematics. Furthermore, practicing with different examples will solidify your understanding and build confidence in handling rational number problems. Rational numbers are a fundamental part of mathematics, and a solid grasp of their properties and operations is essential for tackling more advanced concepts.
In practical scenarios, we often encounter situations where we need to calculate material requirements for a specific number of items. Consider the question: If 24 pairs of trousers of equal size can be prepared with 54 m of cloth, what length of cloth is required for each pair of trousers? This is a classic problem involving division and understanding unit rates.
To solve this, we need to divide the total length of cloth by the number of trouser pairs. This will give us the length of cloth required for a single pair of trousers. Mathematically, this can be represented as:
Total cloth length / Number of trouser pairs
In this case, we have 54 meters of cloth and 24 pairs of trousers. So, the equation becomes:
54 m / 24 pairs
Performing the division, we get:
- 25 m/pair
This means that each pair of trousers requires 2.25 meters of cloth. This answer is a rational number represented in decimal form. We can also express this as a fraction. To convert 2.25 to a fraction, we can write it as 225/100. Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 25, we get:
(225 ÷ 25) / (100 ÷ 25)
Which simplifies to:
9/4
So, each pair of trousers requires 9/4 meters of cloth. This problem demonstrates how division is used to find unit rates. Unit rates are essential in various real-world applications, such as calculating the cost per item, the speed of a vehicle, or the amount of ingredients needed per serving. Understanding how to calculate and interpret unit rates is a valuable skill.
Furthermore, this problem highlights the relationship between decimals and fractions. Being able to convert between these two forms of rational numbers is crucial for flexibility in problem-solving. In summary, to determine the amount of cloth required for each pair of trousers, we divided the total cloth length by the number of pairs. This gave us a unit rate of 2.25 meters per pair, which can also be expressed as the fraction 9/4. This type of calculation is fundamental in many practical situations involving resource allocation and material requirements. Mastering these concepts not only aids in solving mathematical problems but also equips individuals with practical skills for everyday life. Understanding the relationship between quantities and how to calculate unit rates is a key aspect of mathematical literacy.
Another common problem in mathematics involves determining how many smaller units can be obtained from a larger one. This often arises in scenarios such as cutting pieces from a length of material. Let’s consider the question: How many pieces, each of length 3 3/4 m, can be cut from a length of cloth?
To solve this, we need to know the total length of the cloth. Since the question doesn't provide the total length, we'll assume a total length for demonstration purposes. Let's assume we have a total cloth length of 45 meters. The next step is to divide the total length of the cloth by the length of each piece. This will give us the number of pieces that can be cut.
However, we need to address a small correction in the original question. The total length of the cloth is required to determine the number of pieces. Let’s assume the total length of the cloth is 45 meters. The length of each piece is given as 3 3/4 meters. We need to divide the total length by the length of each piece:
Total length / Length of each piece
In this case, we have 45 meters as the total length and 3 3/4 meters as the length of each piece. First, we need to convert the mixed fraction 3 3/4 to an improper fraction. To do this, we multiply the whole number (3) by the denominator (4) and add the numerator (3):
(3 * 4) + 3 = 12 + 3 = 15
So, 3 3/4 is equal to 15/4. Now, our equation becomes:
45 ÷ (15/4)
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 15/4 is 4/15. So, our equation becomes:
45 * (4/15)
Now, we can multiply:
(45 * 4) / 15
This gives us:
180 / 15
Dividing 180 by 15, we get:
12
Therefore, 12 pieces, each of length 3 3/4 meters, can be cut from a cloth of 45 meters. This problem emphasizes the importance of working with fractions and mixed numbers. Converting mixed numbers to improper fractions is a crucial step in performing division. Additionally, understanding the concept of reciprocals is essential for dividing by fractions. In summary, to find the number of pieces that can be cut from a given length, we divide the total length by the length of each piece. This often involves converting mixed numbers to improper fractions and multiplying by the reciprocal. These skills are fundamental in various practical applications, from tailoring and construction to cooking and crafting. Mastering fraction operations is a key component of mathematical proficiency and enables individuals to solve a wide range of real-world problems. Furthermore, consistent practice with these types of problems will enhance your ability to visualize and solve them efficiently.
In conclusion, this exploration has covered three distinct but related mathematical problems. These examples showcase the practical application of rational numbers, fractions, and division in everyday scenarios. By mastering these fundamental concepts, individuals can confidently tackle a wide array of mathematical challenges and apply these skills in real-world situations.
