Replace each pair of fractions by equivalent fractions having the smallest common denominator. Show your work. a. 3 10 and 7 15 b. -1 6 and 3 8 c. 6 5 and 9 2
The correct answer and explanation is:
To replace each pair of fractions with equivalent fractions having the smallest common denominator, we first find the least common denominator (LCD) for each pair, then rewrite the fractions using that denominator.
a. 310\frac{3}{10} and 715\frac{7}{15}
Step 1: Find the LCD of 10 and 15
- 10 = 2 × 5
- 15 = 3 × 5
- LCD = 2 × 3 × 5 = 30
Step 2: Convert each fraction to have denominator 30 310=3×310×3=930\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} 715=7×215×2=1430\frac{7}{15} = \frac{7 \times 2}{15 \times 2} = \frac{14}{30}
Answer: 930\frac{9}{30} and 1430\frac{14}{30}
b. −16-\frac{1}{6} and 38\frac{3}{8}
Step 1: Find the LCD of 6 and 8
- 6 = 2 × 3
- 8 = 2 × 2 × 2
- LCD = 2 × 2 × 2 × 3 = 24
Step 2: Convert each fraction to have denominator 24 −16=−1×46×4=−424-\frac{1}{6} = -\frac{1 \times 4}{6 \times 4} = -\frac{4}{24} 38=3×38×3=924\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}
Answer: −424-\frac{4}{24} and 924\frac{9}{24}
c. 65\frac{6}{5} and 92\frac{9}{2}
Step 1: Find the LCD of 5 and 2
- LCD = 10
Step 2: Convert each fraction to have denominator 10 65=6×25×2=1210\frac{6}{5} = \frac{6 \times 2}{5 \times 2} = \frac{12}{10} 92=9×52×5=4510\frac{9}{2} = \frac{9 \times 5}{2 \times 5} = \frac{45}{10}
Answer: 1210\frac{12}{10} and 4510\frac{45}{10}
Explanation:
To compare or operate with fractions, it is often necessary to express them with the same denominator. This helps in addition, subtraction, or comparison. The smallest common denominator, also called the least common denominator, is the smallest number that both denominators divide evenly into.
We find the LCD by determining the least common multiple (LCM) of the denominators. Once we find the LCD, we adjust the numerator and denominator of each fraction by multiplying both by the same number so the denominator becomes the LCD. This process keeps the value of the fraction unchanged while making calculations easier and more consistent.