Math has always been a subject that sparks curiosity and intrigue, with its complex problems and surprising solutions. One such problem that may seem straightforward at first but can lead to a fascinating outcome is the calculation of 3 1/2 x 2/3. This expression involves the multiplication of a mixed number by a fraction, which can be simplified using basic arithmetic operations. In this article, we will delve into the step-by-step process of solving this problem and explore the interesting result that emerges from it.
Understanding Mixed Numbers and Fractions
To tackle the given problem, it’s essential to have a solid grasp of mixed numbers and fractions. A mixed number is a combination of a whole number and a fraction, such as 3 1⁄2, which represents three whole units and one half unit. On the other hand, a fraction is a way of expressing a part of a whole, like 2⁄3, which denotes two equal parts out of three. The multiplication of a mixed number by a fraction requires converting the mixed number into an improper fraction, which can then be multiplied by the given fraction.
Converting Mixed Numbers to Improper Fractions
The first step in solving the problem is to convert the mixed number 3 1⁄2 into an improper fraction. This can be done by multiplying the whole number part (3) by the denominator (2) and then adding the numerator (1). The result is the new numerator, while the denominator remains the same. Therefore, 3 1⁄2 can be converted into an improper fraction as follows: (3 x 2) + 1 = 7, so 3 1⁄2 = 7⁄2.
| Mixed Number | Improper Fraction |
|---|---|
| 3 1/2 | 7/2 |
Multiplying Fractions
Now that we have converted the mixed number into an improper fraction (7⁄2), we can proceed to multiply it by the given fraction (2⁄3). When multiplying fractions, we simply multiply the numerators together to get the new numerator and the denominators together to get the new denominator. Therefore, the multiplication of 7⁄2 by 2⁄3 can be calculated as follows: (7 x 2) / (2 x 3) = 14⁄6.
Simplifying the Result
The resulting fraction, 14⁄6, can be simplified further by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. This simplification yields a final result of 7⁄3. To express this improper fraction as a mixed number, we divide the numerator (7) by the denominator (3), giving us 2 with a remainder of 1. Therefore, the mixed number equivalent of 7⁄3 is 2 1⁄3.
Key Points
- The mixed number 3 1/2 can be converted into an improper fraction as 7/2.
- The multiplication of 7/2 by 2/3 yields 14/6, which can be simplified to 7/3.
- The improper fraction 7/3 is equivalent to the mixed number 2 1/3.
- Understanding the conversion of mixed numbers to improper fractions and the multiplication of fractions is crucial for solving this problem.
- The step-by-step process involves converting the mixed number, multiplying the fractions, and simplifying the result to obtain the final answer.
Conclusion and Implications
In conclusion, the calculation of 3 1⁄2 x 2⁄3 may seem daunting at first, but by breaking it down into manageable steps, we can arrive at a surprising and fascinating answer. The conversion of mixed numbers to improper fractions and the multiplication of fractions are essential skills that can be applied to a wide range of mathematical problems. By mastering these concepts, we can unlock the secrets of math and develop a deeper appreciation for the beauty and complexity of numbers.
What is the first step in solving the problem 3 1⁄2 x 2⁄3?
+The first step is to convert the mixed number 3 1⁄2 into an improper fraction, which is 7⁄2.
How do you multiply fractions?
+When multiplying fractions, you multiply the numerators together to get the new numerator and the denominators together to get the new denominator.
What is the final answer to the problem 3 1⁄2 x 2⁄3?
+The final answer is 2 1⁄3, which is the mixed number equivalent of the improper fraction 7⁄3.