Question

What is the value of \(\frac{\frac{1}{4} \times \frac{6}{11}}{\frac{3}{7}} ?\)

Short answer

The value of \(\frac{\frac{1}{4} \times \frac{6}{11}}{\frac{3}{7}}\) is \(\frac{7}{22}\).

Step-by-step solution

- Simplify the numerator

The numerator of the fraction is \(\frac{1}{4} \times \frac{6}{11}\). Multiply these two fractions by multiplying their numerators and denominators: \[ \frac{1 \times 6}{4 \times 11} = \frac{6}{44} \]

- Simplify the fraction \(\frac{6}{44}\)

Simplify \(\frac{6}{44}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \[ \frac{6 \text{÷} 2}{44 \text{÷} 2} = \frac{3}{22} \]

- Set up the division of fractions

The original expression now looks like this: \[ \frac{\frac{3}{22}}{\frac{3}{7}} \]

- Divide by multiplying by the reciprocal

To divide by a fraction, multiply by its reciprocal. So divide \(\frac{3}{22}\) by \( \frac{3}{7} \) by multiplying \(\frac{3}{22}\) by \(\frac{7}{3}\): \[ \frac{3}{22} \times \frac{7}{3} \]

- Simplify the multiplication

Multiply the fractions by multiplying the numerators and denominators: \[ \frac{3 \times 7}{22 \times 3} = \frac{21}{66} \]. Next, simplify \(\frac{21}{66}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \[ \frac{21 \text{÷} 3}{66 \text{÷} 3} = \frac{7}{22} \]

Key concepts

simplifying fractions

Simplifying fractions means making the fraction as simple as possible. This often involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number.
In the example from the exercise, the fraction \(\frac{6}{44}\) was simplified by identifying that both 6 and 44 can be divided by their GCD, which is 2. So, \(\frac{6}{44} = \frac{6 \text{ ÷ } 2}{44 \text{ ÷ } 2} = \frac{3}{22}\).
Simplifying fractions makes them easier to work with and understand. It’s an essential skill in fraction operations like addition, subtraction, multiplication, and division.

dividing by a fraction

Dividing by a fraction involves multiplying by the reciprocal of the fraction. The reciprocal of a fraction is created by swapping its numerator and denominator.
For example, in the given exercise, we needed to divide \(\frac{3}{22}\) by \(\frac{3}{7}\). To do this, we multiply \(\frac{3}{22}\) by the reciprocal of \(\frac{3}{7}\), which is \(\frac{7}{3}\). This converts the division problem into multiplication: \(\frac{3}{22} \times \frac{7}{3}\).
This method works because multiplying by a reciprocal effectively cancels out the original fraction, leaving a simple multiplication operation.

multiplying fractions

Multiplying fractions is a straightforward process that involves multiplying the numerators together and the denominators together.
For instance, in the exercise, to multiply \(\frac{1}{4}\) by \(\frac{6}{11}\), you multiply the numerators \(\frac{1 \times 6} \) to get 6 and the denominators \(\frac{4 \times 11}\) to get 44, resulting in \(\frac{6}{44}\).
Then, the fraction \(\frac{6}{44}\) can be simplified to \(\frac{3}{22}\) by dividing both the numerator and the denominator by their GCD, which is 2. This step is essential because it keeps the fraction as simple and as easy to understand as possible.