Question

Calculate (a) $\frac{7}{9}÷\frac{2}{3}$ (b) $\frac{8}{15}÷\frac{4}{5}$ (c) $\frac{9}{10}÷\frac{9}{20}$ (d) $\frac{6}{11}÷\frac{7}{12}$ (e) $\frac{12}{13}÷\frac{6}{7}$

Short answer

Question: Find the result of each given division of fractions: a) $\frac{7}{9}÷\frac{2}{3}$ b) $\frac{8}{15}÷\frac{4}{5}$ c) $\frac{9}{10}÷\frac{9}{20}$ d) $\frac{6}{11}÷\frac{7}{12}$ e) $\frac{12}{13}÷\frac{6}{7}$ Answer: a) $\frac{7}{6}$ b) $\frac{2}{3}$ c) $2$ d) $\frac{72}{77}$ e) $\frac{14}{13}$

Step-by-step solution

(a) Find the reciprocal of the divisor

The divisor is $\frac{2}{3}$. To find the reciprocal, we just switch the numerator and the denominator: $\frac{3}{2}$

(a) Perform the multiplication

Since dividing by the fraction is the same as multiplying by its reciprocal, we can rewrite the problem: $$\frac{7}{9}÷\frac{2}{3}=\frac{7}{9}\cdot \frac{3}{2}$$ Now, we perform the multiplication: $$\frac{7\cdot 3}{9\cdot 2}=\frac{21}{18}$$

(a) Simplify the result

To simplify the fraction, we can divide numerator and denominator by their greatest common divisor, which is 3: $$\frac{21÷3}{18÷3}=\frac{7}{6}$$

(b) Find the reciprocal of the divisor

The divisor is $\frac{4}{5}$. The reciprocal is: $\frac{5}{4}$

(b) Perform the multiplication

Rewrite the problem using the reciprocal of the divisor and multiply: $$\frac{8}{15}÷\frac{4}{5}=\frac{8}{15}\cdot \frac{5}{4}=\frac{8\cdot 5}{15\cdot 4}=\frac{40}{60}$$

(b) Simplify the result

Dividing numerator and denominator by the greatest common divisor, which is 20, we get: $$\frac{40÷20}{60÷20}=\frac{2}{3}$$

(c) Find the reciprocal of the divisor

The divisor is $\frac{9}{20}$. The reciprocal is: $\frac{20}{9}$

(c) Perform the multiplication

Rewrite the problem and multiply: $$\frac{9}{10}÷\frac{9}{20}=\frac{9}{10}\cdot \frac{20}{9}=\frac{9\cdot 20}{10\cdot 9}=\frac{180}{90}$$

(c) Simplify the result

Dividing numerator and denominator by the greatest common divisor, which is 90, we get: $$\frac{180÷90}{90÷90}=\frac{2}{1}=2$$

(d) Find the reciprocal of the divisor

The divisor is $\frac{7}{12}$. The reciprocal is: $\frac{12}{7}$

(d) Perform the multiplication

Rewrite the problem and multiply: $$\frac{6}{11}÷\frac{7}{12}=\frac{6}{11}\cdot \frac{12}{7}=\frac{6\cdot 12}{11\cdot 7}=\frac{72}{77}$$

(d) Simplify the result

The result is already in its simplest form, so the final answer is: $$\frac{72}{77}$$

(e) Find the reciprocal of the divisor

The divisor is $\frac{6}{7}$. The reciprocal is: $\frac{7}{6}$

(e) Perform the multiplication

Rewrite the problem and multiply: $$\frac{12}{13}÷\frac{6}{7}=\frac{12}{13}\cdot \frac{7}{6}=\frac{12\cdot 7}{13\cdot 6}=\frac{84}{78}$$

(e) Simplify the result

Dividing numerator and denominator by the greatest common divisor, which is 6, we get: $$\frac{84÷6}{78÷6}=\frac{14}{13}$$

Key concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is essential in the world of mathematics, especially when dealing with division of fractions. The reciprocal of a given fraction is simply another fraction where the numerator and the denominator are switched. For instance, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. It's important to note that the reciprocal of a fraction is essentially what you multiply the original fraction by to get 1.

When dividing by a fraction, you can think of it as multiplying by its reciprocal. This is why, during the division of fractions, we flip the second fraction (the divisor) and proceed with multiplication. The process can be summarized as follows: If you want to divide by a fraction, you multiply by its reciprocal.

Simplifying Fractions

Simplifying a fraction, also known as reducing a fraction, is the process of making the fraction as simple as possible. This is done by identifying a number by which both the numerator (top number) and the denominator (bottom number) can be divided evenly. This number is known as the greatest common divisor (GCD).

Once the GCD is found, both the numerator and the denominator are divided by this number to obtain the simplified fraction. For example, $\frac{21}{18}$ can be simplified by dividing both numbers by their GCD, which is 3, resulting in the simplified fraction $\frac{7}{6}$. Ensuring a fraction is in its simplest form is crucial for clarity and can often make further calculations easier.

Multiplication of Fractions

Multiplying fractions is straightforward: you simply multiply the numerators together and the denominators together. The formula is $\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}$.

Let's consider an example. If you have $\frac{7}{9}$ and $\frac{3}{2}$ and you want to multiply them, you would calculate $\frac{7\cdot 3}{9\cdot 2}=\frac{21}{18}$. Then, the resulting fraction could often be simplified. Multiplication of fractions is used in various mathematical and real-world applications, making it an important concept to master. When multiplying fractions, unlike adding or subtracting them, you don't need to find a common denominator, which can sometimes simplify the process.