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Chapter 1: Problem 16

Calculate and express each result in its simplest form:\(\frac{2}{3} \times \frac{5}{6}\)

Short Answer

Expert verified
The simplified result is \(\frac{5}{9}\).

Step by step solution

01

Multiply the Numerators

Multiply the numerators of the fractions together. For the fractions \(\frac{2}{3}\) and \(\frac{5}{6}\), multiply \(2 \times 5 = 10\).
02

Multiply the Denominators

Multiply the denominators of the fractions together. For the fractions \(\frac{2}{3}\) and \(\frac{5}{6}\), multiply \(3 \times 6 = 18\).
03

Form the New Fraction

Combine the results from Steps 1 and 2 to form a new fraction. This gives us \(\frac{10}{18}\).
04

Simplify the Fraction

To simplify \(\frac{10}{18}\), find the greatest common divisor (GCD) of 10 and 18, which is 2. Divide both the numerator and the denominator by 2 to get \(\frac{5}{9}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerator
The numerator is the top part of a fraction. It represents how many parts we have out of a whole. Think of the numerator as the "counter" or the "quantity" in the fraction. In the exercise given, which started with \(\frac{2}{3}\) and \(\frac{5}{6}\), the numerators are 2 and 5, respectively. When multiplying fractions, we multiply the numerators together. This means you're essentially finding how many parts are in total when multiple parts are multiplied together. It's like counting all the apples if you say 2 baskets with 5 apples each.
Denominator
The denominator is the bottom part of a fraction. It indicates the total number of equal parts the whole is divided into. When you look at the fractions \(\frac{2}{3}\) and \(\frac{5}{6}\), the denominators are 3 and 6. They tell us how each part of the whole is divided. In this context, multiplying the denominators (3 and 6) helps us find out how many total parts are defined when combining two sets into a new whole. This is why when you multiply fractions, you get a new fraction like \(\frac{10}{18}\); the 18 shows how finely the new whole is divided.
Simplification
Simplifying a fraction makes it easier to understand and work with by reducing the numbers. To simplify a fraction like \(\frac{10}{18}\), we need to find numbers that divide evenly both into the numerator and the denominator. This process helps create "neater" fractions that are easier to read and interpret. Simplification removes the common factors from the numerator and denominator, resulting in a more straightforward form, like \(\frac{5}{9}\), which is easier to visualize and use in further calculations. Always strive to simplify fractions to improve readability and accuracy.
Greatest Common Divisor (GCD)
The Greatest Common Divisor, or GCD, is the largest number that divides evenly into both the numerator and denominator of a fraction. Finding the GCD is crucial for simplifying fractions. To simplify \(\frac{10}{18}\), we need to identify the GCD of 10 and 18, which is 2. Using the GCD, we divide both the top and bottom of the fraction by this number, resulting in a simplified form. The GCD is an important tool in mathematics because it helps us find the most reduced form of a fraction swiftly and efficiently, ensuring clarity in math problems.

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Most popular questions from this chapter

Evaluate:\(4 \div 2+7 \times 2\)

Simplify by factorization and cancellation:\(\frac{3}{18}\)

Evaluate:\(25+144^{1 / 2}\)

Calculate and express each result in its simplest form:\(\frac{1}{2} \times \frac{3}{4}\)

Factorize in prime numbers:810
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