Answer :
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]
### 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 \( \frac{10}{3} \).
