Problem 37 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\frac{2}{\frac{2}{3}}-\frac{\frac{2}{3}}{2}\)

Step-by-Step Solution

Answer

The result of the expression is \( \frac{8}{3} \).

Step 1: Simplify the first term

First, we need to simplify the expression \( \frac{2}{\frac{2}{3}} \). This is a division problem between fractions, which can be resolved by multiplying by the reciprocal. Thus, we multiply \( 2 \) by \( \frac{3}{2} \) to get \( 2 \times \frac{3}{2} = 3 \).

Step 2: Simplify the second term

Next, simplify \( \frac{\frac{2}{3}}{2} \). Again, this is a division of fractions, so we multiply \( \frac{2}{3} \) by the reciprocal of 2, which is \( \frac{1}{2} \). Therefore, \( \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} \).

Step 3: Subtract the two simplified terms

Now, we subtract the simplified results from Step 1 and Step 2: \( 3 - \frac{1}{3} \). To perform this operation, convert 3 into a fraction: \( 3 = \frac{9}{3} \). The subtraction becomes \( \frac{9}{3} - \frac{1}{3} \).

Step 4: Perform the subtraction

Subtract the fractions: \( \frac{9}{3} - \frac{1}{3} = \frac{8}{3} \).

Step 5: Conclusion

The expression simplifies to \( \frac{8}{3} \).

Key Concepts

Reciprocals

Reciprocals are incredibly handy in mathematics, especially when it comes to dividing fractions. The reciprocal of a number is simply 1 divided by that number. If you have a fraction, like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). When you multiply a number by its reciprocal, the result is always 1.

• Example: The reciprocal of 2 is \( \frac{1}{2} \), because \( 2 \times \frac{1}{2} = 1 \).
• Another Example: The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \), since \( \frac{2}{3} \times \frac{3}{2} = 1 \).

Reciprocals are particularly useful when you need to perform division between fractions. Rather than dividing directly, you multiply by the reciprocal. This is seen in our problem, where division \( \frac{2}{\frac{2}{3}} \) is simplified by multiplying: \( 2 \times \frac{3}{2} \). This transforms the complex process of division into straightforward multiplication.

Fraction Subtraction

Subtracting fractions can initially seem daunting, but understanding the process makes it simpler. To subtract fractions, it’s crucial that they share a common denominator. If they don't, you will have to find it. This allows you to effectively compare and operate on both fractions.

• In this task, after simplifying \( 3 - \frac{1}{3} \), we convert 3 into a fraction with a common denominator: \( 3 = \frac{9}{3} \).
• This transforms the subtraction into \( \frac{9}{3} - \frac{1}{3} \).

Once the denominators are the same, subtract the numerators, keeping the denominator intact. This gives us \( \frac{8}{3} \) as a result. Remember, simplifying such results by ensuring the lowest terms will help in presenting the final answer clearly.

Multiplication of Fractions

When multiplying fractions, the process is straightforward and involves multiplying the numerators together and the denominators together. The result is a new fraction.

• For example, when multiplying \( 2 \times \frac{3}{2} \), you multiply the numerators: \( 2 \times 3 = 6 \), and the denominators: \( 1 \times 2 = 2 \).
• This gives you \( \frac{6}{2} \) which can be simplified to 3.

Another critical step is simplifying the fraction post-multiplication. Always check if the fraction can be reduced to its simplest form. During the process of \( \frac{\frac{2}{3}}{2} \), we multiply \( \frac{2}{3} \) by \( \frac{1}{2} \), resulting in \( \frac{1}{3} \). This exemplifies the importance of understanding both multiplication rules and simplification in fraction operations.