Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. $\frac{6}{2}÷\frac{3}{4}=$ ______

Short answer

4

Step-by-step solution

Rewrite Division as Multiplication

To perform division with fractions, we rewrite the division problem as a multiplication problem by multiplying by the reciprocal of the divisor. The problem is initially $\frac{6}{2}÷\frac{3}{4}$. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. Therefore, the new expression is: $\frac{6}{2}\times \frac{4}{3}$.

Simplify Each Fraction

Before multiplying, simplify each fraction if possible. $\frac{6}{2}$ simplifies to $3$ because 6 divided by 2 is 3. Now, the multiplication problem is: $3\times \frac{4}{3}$.

Multiply the Fractions

Multiply the numerators and the denominators separately. The expression $3\times \frac{4}{3}$ gives: $$\frac{3\times 4}{1\times 3}=\frac{12}{3}$$.

Reduce to Lowest Terms

To reduce $\frac{12}{3}$ to its lowest terms, divide both the numerator and the denominator by their greatest common divisor, which is 3. $\frac{12}{3}=4$. Thus, the simplified expression is 4.

Key concepts

Reciprocal

Understanding the reciprocal is crucial when dealing with fraction division. The reciprocal of a number is simply what you get when you flip the numerator and the denominator. For example, the fraction $\frac{3}{4}$ has the reciprocal $\frac{4}{3}$. This flipped version plays a significant role in transforming division problems into multiplication problems. When dividing fractions, take the reciprocal of the divisor, and you are ready to multiply. This makes things simpler and is a key step in the process.

Simplifying Fractions

Before diving into complex calculations, it's often easier to simplify fractions. Simplifying involves reducing fractions to their simplest form so that calculations become straightforward. For instance, if we have $\frac{6}{2}$, we can simplify it to 3 because both 6 and 2 are divisible by 2. Simplification is about division: find the greatest number that divides both the numerator and the denominator. This step makes the latter processes, like multiplication, more manageable and efficient.

Multiplying Fractions

Once the fractions are simplified (if needed) and the reciprocal is found, the next step is to multiply them. Multiplication of fractions is relatively straightforward. Multiply the numerators together to get the new numerator, and the denominators together to get the new denominator. In the case of $3\times \frac{4}{3}$, multiply 3 by 4 to get 12, and 1 by 3 to get 3. Thus, the multiplication yields $\frac{12}{3}$. This sequence ensures that fraction multiplication results in a correct fraction representation.

Reducing to Lowest Terms

Reducing a fraction to its lowest terms is about simplifying it so that it cannot be reduced further. This means the greatest common divisor (GCD) of the numerator and the denominator is 1. For the fraction $\frac{12}{3}$, divide both numbers by their GCD, which is 3. This results in the final value of 4. Always check your final fraction to determine if it can be reduced any further; if not, you have reached the lowest terms.