Verified Answer :
- Convert \(a = 0.\overline{6}\) to the fraction \(\frac{2}{3}\) and \(b = 0.75\) to the fraction \(\frac{3}{4}\).
- Calculate the product: \(\frac{2}{3} \cdot \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\).
- Calculate the quotient: \(\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \cdot \frac{4}{3} = \frac{8}{9}\).
- The final answers are \(\boxed{a \cdot b = \frac{1}{2}}\) and \(\boxed{\frac{a}{b} = \frac{8}{9}}\).
Explanation
- Converting Decimals to Fractions
The problem requires us to work with two numbers, \(a\) and \(b\), which are provided in decimal form. Specifically, \(a = 0.\overline{6}\) and \(b = 0.75\). To find their product \(a \cdot b\) and their quotient \(\frac{a}{b}\), it is most efficient to first convert these decimals into their equivalent fractional representations.
For the repeating decimal \(a = 0.\overline{6}\), we can express it as:
\[a = \frac{6}{9} = \frac{2}{3}\]
For the terminating decimal \(b = 0.75\), we can express it as:
\[b = \frac{75}{100} = \frac{3}{4}\]
With these fractions, we can proceed to compute the requested values.
- Calculating the Product \(a \cdot b\)
Now that we have \(a = \frac{2}{3}\) and \(b = \frac{3}{4}\), we calculate their product \(a \cdot b\):
\[a \cdot b = \left(\frac{2}{3}\right) \cdot \left(\frac{3}{4}\right)\]
Multiplying the numerators and denominators:
\[a \cdot b = \frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12}\]
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 6:
\[a \cdot b = \frac{1}{2}\]
- Calculating the Quotient \(\frac{a}{b}\)
Next, we calculate the quotient \(\frac{a}{b}\):
\[\frac{a}{b} = \frac{\frac{2}{3}}{\frac{3}{4}}\]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[\frac{a}{b} = \frac{2}{3} \cdot \frac{4}{3}\]
Performing the multiplication:
\[\frac{a}{b} = \frac{2 \cdot 4}{3 \cdot 3} = \frac{8}{9}\]
The fraction \(\frac{8}{9}\) is already in its simplest form.
- Final Answer and Summary
We have determined the product and the quotient of the two numbers. The calculations yield the results \(a \cdot b = \frac{1}{2}\) and \(\frac{a}{b} = \frac{8}{9}\). Thus, we have \(\boxed{a \cdot b = \frac{1}{2}}\) and \(\boxed{\frac{a}{b} = \frac{8}{9}}\).
Examples
In real-world applications, converting decimals to fractions is a fundamental skill used in fields like finance and engineering. For instance, when dealing with measurements in inches where parts are often expressed as fractions (like 3/4 of an inch) or recurring interest rates (like 6.66...% which is 2/3 of 10%), being able to switch between representations allows for precise calculations. Using fractions avoids rounding errors and keeps data accurate throughout multi-step operations.