Compare rational numbers

Learning Objectives Compare any two rational numbers using the least common denominator (LCD) method. Efficiently compare two rational numbers using the cross-multiplication property. By the end of of this lesson, students will be able to convert rational numbers to decimals to determine their relative size. Justify the comparison of two rational numbers, forming a basis for simple proofs. Apply comparison techniques to rational expressions involving variables. Select the most appropriate comparison method based on the numbers involved. Which is a better performance: scoring 17 out of 21 on a test, or 15 out of 19? 🤔 Let's find out how to answer this definitively! While comparing fractions might seem like a review, mastering this skill with speed and accuracy is cruci...

TermDefinitionExample Rational NumberAny number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.-7/3, 5 (which is 5/1), 0.25 (which is 1/4) NumeratorThe top number in a fraction, which represents the number of parts being considered.In the fraction 8/11, the numerator is 8. DenominatorThe bottom number in a fraction, which represents the total number of equal parts in the whole.In the fraction 8/11, the denominator is 11. Least Common Multiple (LCM)The smallest positive integer that is a multiple of two or more integers. When used for denominators, it is called the Least Common Denominator (LCD).The LCM of 12 and 18 is 36. Number LineA line on which numbers are marked at intervals, used to illustrate simple numerical operations. Numbers to the right a...

Common Denominator Method To compare \( \frac{a}{b} \) and \( \frac{c}{d} \), find a common denominator, BD. The comparison is then between \( \frac{a \cdot d}{b \cdot d} \) and \( \frac{c \cdot b}{d \cdot b} \). This method transforms the fractions into equivalent forms that are easy to compare directly by looking at their numerators. It is reliable but can involve large numbers if the denominators are large. Cross-Multiplication Property For two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) with positive denominators (b, d > 0): If \( a \cdot d > c \cdot b \), then \( \frac{a}{b} > \frac{c}{d} \). If \( a \cdot d < c \cdot b \), then \( \frac{a}{b} < \frac{c}{d} \). This is often the fastest method. It's a shortcut derived from the common denominato...