Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{1}{9} \times \frac{7}{3}=\) ______

Short answer

\(\frac{7}{27}\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

The given fractions are already in proper form so no conversion is needed in this case. The fractions are \( \frac{1}{9} \) and \( \frac{7}{3} \).

Multiply the Numerators

Multiply the numerators of both fractions. For \( \frac{1}{9} \) and \( \frac{7}{3} \), the numerators are 1 and 7. Therefore, \( 1 \times 7 = 7 \).

Multiply the Denominators

Multiply the denominators of both fractions. The denominators are 9 and 3. Therefore, \( 9 \times 3 = 27 \).

Form the Product Fraction

Combine the results from the previous two steps to form the product fraction: \( \frac{7}{27} \).

Reduce to Lowest Terms

Check if the fraction \( \frac{7}{27} \) can be simplified. The greatest common divisor of 7 and 27 is 1, so the fraction is already in its simplest form.

Key concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used when the value is greater than one, making them easier for us to interpret in practical situations. For instance, if you have a pizza and someone eats all but one slice, you might consider how many whole pizzas were eaten and how much of another is left over. Mixed numbers can be converted into improper fractions for mathematical operations like multiplication or division. This involves multiplying the whole number by the denominator of the fraction, adding the result to the numerator of the fraction, and writing this value over the original denominator. However, in our exercise, the numbers provided were already in a suitable fraction form, so no conversion was necessary. - Example: To convert 3 1/2, multiply 3 (whole number) by 2 (denominator), resulting in 6. Add the numerator, 1, to get 7. The improper fraction would be 7/2.

Reduce to Lowest Terms

Reducing to lowest terms, also known as simplifying a fraction, involves rewriting the fraction in the simplest form where the numerator and denominator have no common divisors except 1. This makes fractions easier to understand and use in further calculations. To reduce a fraction to its lowest terms, you divide both the numerator and the denominator by their greatest common divisor (GCD). If there's no common factor other than 1, then the fraction is already in its simplest form.

• It keeps calculations simpler.
• Helps clearly compare fractions.
• Ensures any arithmetic done with fractions uses the most straightforward numbers.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Knowing the GCD helps in reducing fractions to their simplest terms.To find the GCD, list the factors of both the numerator and the denominator. Identify the greatest factor that both numbers share. In the solution to the given exercise, the fraction \( \frac{7}{27} \) already appears in its simplest form. The GCD of 7 and 27 is 1, meaning no further reduction is possible. - Finding the GCD can involve inspecting numbers.- Alternatively, methods like prime factorization or the Euclidean algorithm can also be used.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means the fraction represents a value equal to or greater than one. Improper fractions are common when discussing or converting mixed numbers, especially in calculations involving multiplication or division. To transform a mixed number into an improper fraction, you multiply the whole number by the fraction's denominator, add the numeral to the original numerator, and place it over the original denominator. Using improper fractions can simplify multiplication, as the multiplication of numerators and denominators becomes a straightforward process without needing to worry about keeping track of whole numbers separately.

For example, converting 4 2/3 into an improper fraction involves multiplying the whole number 4 by the denominator 3, adding the numerator 2, which equals 14. Thus, 14/3 is the improper fraction equivalent.