krit.club logo

Number - Simplifying, Comparing, and Ordering Fractions

Grade 6IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. This is also known as a fraction in its lowest terms.

•

Equivalent fractions are fractions that represent the same value. They are created by multiplying or dividing the numerator and the denominator by the same non-zero number: ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}.

•

To compare two fractions with different denominators, you must first find a common denominator, typically the Least Common Multiple (LCM) of the denominators.

•

Once fractions have a common denominator, the fraction with the larger numerator is the larger fraction: if b=db = d, then ab>cd\frac{a}{b} > \frac{c}{d} if a>ca > c.

•

Ordering multiple fractions involves converting all fractions in the set to have a common denominator and then arranging them based on their numerators in ascending (smallest to largest) or descending (largest to smallest) order.

📐Formulae

Simplifying: ab=a÷HCF(a,b)b÷HCF(a,b)\text{Simplifying: } \frac{a}{b} = \frac{a \div \text{HCF}(a, b)}{b \div \text{HCF}(a, b)}

Cross-Multiplication Comparison: ab>cd if a×d>b×c\text{Cross-Multiplication Comparison: } \frac{a}{b} > \frac{c}{d} \text{ if } a \times d > b \times c

Finding Common Denominator: LCM(b,d)\text{Finding Common Denominator: } \text{LCM}(b, d)

💡Examples

Problem 1:

Simplify the fraction 2460\frac{24}{60} to its lowest terms.

Solution:

2460=24÷1260÷12=25\frac{24}{60} = \frac{24 \div 12}{60 \div 12} = \frac{2}{5}

Explanation:

To simplify, find the Highest Common Factor (HCF) of 24 and 60. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The HCF is 12. Dividing both terms by 12 gives 25\frac{2}{5}.

Problem 2:

Compare 34\frac{3}{4} and 57\frac{5}{7} using a common denominator.

Solution:

34=3×74×7=2128\frac{3}{4} = \frac{3 \times 7}{4 \times 7} = \frac{21}{28} 57=5×47×4=2028\frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28} Since 2128>2028, then 34>57\text{Since } \frac{21}{28} > \frac{20}{28}, \text{ then } \frac{3}{4} > \frac{5}{7}

Explanation:

The LCM of 4 and 7 is 28. Convert both fractions to equivalent fractions with a denominator of 28. Comparing the numerators (21 and 20) shows that 34\frac{3}{4} is greater.

Problem 3:

Order the following fractions from least to greatest: 12\frac{1}{2}, 23\frac{2}{3}, and 56\frac{5}{6}.

Solution:

LCM of 2, 3, and 6 is 6.\text{LCM of 2, 3, and 6 is 6.} 12=1×32×3=36\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} 23=2×23×2=46\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} 56=56\frac{5}{6} = \frac{5}{6} Order: 36<46<56  ⟹  12,23,56\text{Order: } \frac{3}{6} < \frac{4}{6} < \frac{5}{6} \implies \frac{1}{2}, \frac{2}{3}, \frac{5}{6}

Explanation:

First, find the LCM of the denominators 2, 3, and 6, which is 6. Convert each fraction to an equivalent fraction with 6 as the denominator. Then, compare the numerators to determine the order.