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0-4. Find the quotients:

a) [tex]\frac{15}{19} \div \frac{5}{3} = \frac{15}{19} \times \frac{3}{5} = \frac{9}{19}[/tex]

b) [tex]2 \div \frac{3}{5} =[/tex]


Sagot :

Let's solve these problems step-by-step:

### Part (a)
Given: [tex]\(\frac{15}{19} \div \frac{5}{3}\)[/tex]

To divide fractions, we multiply by the reciprocal of the divisor. So, we need to multiply [tex]\(\frac{15}{19}\)[/tex] by the reciprocal of [tex]\(\frac{5}{3}\)[/tex], which is [tex]\(\frac{3}{5}\)[/tex].

Thus,

[tex]\[ \frac{15}{19} \div \frac{5}{3} = \frac{15}{19} \times \frac{3}{5} \][/tex]

When multiplying fractions, we multiply the numerators together and the denominators together:

[tex]\[ \frac{15 \times 3}{19 \times 5} = \frac{45}{95} \][/tex]

Now, simplify [tex]\(\frac{45}{95}\)[/tex]:

1. Find the greatest common divisor (GCD) of 45 and 95, which is 5.
2. Divide the numerator and the denominator by 5:

[tex]\[ \frac{45 \div 5}{95 \div 5} = \frac{9}{19} \][/tex]

So, the result is:

[tex]\[ \frac{15}{19} \div \frac{5}{3} = \frac{9}{19} \][/tex]

#### Which equals approximately:

[tex]\[ 0.47368421052631576 \][/tex]

### Part (b)

Given: [tex]\(2 \div \frac{3}{5}\)[/tex]

To divide by a fraction, we multiply by its reciprocal. The reciprocal of [tex]\(\frac{3}{5}\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].

Thus,

[tex]\[ 2 \div \frac{3}{5} = 2 \times \frac{5}{3} \][/tex]

Multiply 2 by [tex]\(\frac{5}{3}\)[/tex]:

[tex]\[ 2 \times \frac{5}{3} = \frac{2 \times 5}{3} = \frac{10}{3} \][/tex]

So, converting to a decimal:

[tex]\(\frac{10}{3}\)[/tex] equals approximately:

[tex]\[ 3.3333333333333335 \][/tex]

### Final Quotients:
a) [tex]\(\frac{15}{19} \div \frac{5}{3} = 0.47368421052631576\)[/tex]

b) [tex]\(2 \div \frac{3}{5} = 3.3333333333333335\)[/tex]

Thus, these are the detailed solutions:

[tex]\[ \begin{aligned} \text{(a)} \quad & \frac{15}{19} \div \frac{5}{3} \approx 0.47368421052631576 \\ \text{(b)} \quad & 2 \div \frac{3}{5} \approx 3.3333333333333335 \end{aligned} \][/tex]

Answer:To find the quotient \( \frac{2}{\frac{3}{5}} \), we interpret it as the multiplication of 2 by the reciprocal of \( \frac{3}{5} \).

First, find the reciprocal of \( \frac{3}{5} \):

\[

\text{Reciprocal of } \frac{3}{5} = \frac{5}{3}

\]

Now, multiply 2 by the reciprocal \( \frac{5}{3} \):

\[

2 \cdot \frac{5}{3} = \frac{2 \cdot 5}{3} = \frac{10}{3}

\]

Therefore, the quotient \( \frac{2}{\frac{3}{5}} \) simplifies to \( \frac{10}{3} \).

So, the answer for part b) is \( \boxed{\frac{10}{3}} \).

Step-by-step explanation:Sure, let's break down the steps to find the quotient \( \frac{2}{\frac{3}{5}} \).

We can rewrite the expression \( \frac{2}{\frac{3}{5}} \) as multiplication by the reciprocal of \( \frac{3}{5} \):

1. **Identify the expression:**

  \[

  \frac{2}{\frac{3}{5}}

  \]

2. **Find the reciprocal of \( \frac{3}{5} \):**

  \[

  \text{Reciprocal of } \frac{3}{5} = \frac{5}{3}

  \]

3. **Rewrite the division as multiplication:**

  \[

  \frac{2}{\frac{3}{5}} = 2 \cdot \frac{5}{3}

  \]

4. **Perform the multiplication:**

  \[

  2 \cdot \frac{5}{3} = \frac{2 \cdot 5}{3} = \frac{10}{3}

  \]

Therefore, \( \frac{2}{\frac{3}{5}} = \frac{10}{3} \).

The step-by-step process confirms that the quotient \( \frac{2}{\frac{3}{5}} \) simplifies to \( \frac{10}{3} \).