Math worksheet for finding the least common denominator of fraction pairs.
Worksheet titled "Least Common Denominators" with nine fraction pairs and blank spaces to find the LCD.
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Step-by-step solution for: Least Common Denominator - Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Least Common Denominator - Worksheets
To solve the problem of finding the least common denominator (LCD) for each pair of fractions, we need to determine the smallest number that is a multiple of both denominators in each pair. This involves finding the least common multiple (LCM) of the denominators.
Let's solve each part step by step:
---
- Denominators: 6 and 3
- Prime factorization:
- \( 6 = 2 \times 3 \)
- \( 3 = 3 \)
- LCM: Take the highest power of each prime factor.
- \( 2^1 \) (from 6)
- \( 3^1 \) (from both 6 and 3)
- LCM = \( 2 \times 3 = 6 \)
LCD = 6
---
- Denominators: 5 and 3
- Prime factorization:
- \( 5 = 5 \)
- \( 3 = 3 \)
- LCM: Since 5 and 3 are coprime (no common factors other than 1), their LCM is simply their product.
- LCM = \( 5 \times 3 = 15 \)
LCD = 15
---
- Denominators: 4 and 6
- Prime factorization:
- \( 4 = 2^2 \)
- \( 6 = 2 \times 3 \)
- LCM: Take the highest power of each prime factor.
- \( 2^2 \) (from 4)
- \( 3^1 \) (from 6)
- LCM = \( 2^2 \times 3 = 4 \times 3 = 12 \)
LCD = 12
---
- Denominators: 2 and 4
- Prime factorization:
- \( 2 = 2 \)
- \( 4 = 2^2 \)
- LCM: Take the highest power of each prime factor.
- \( 2^2 \) (from 4)
- LCM = \( 4 \)
LCD = 4
---
- Denominators: 8 and 9
- Prime factorization:
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- LCM: Since 8 and 9 have no common prime factors, their LCM is their product.
- LCM = \( 8 \times 9 = 72 \)
LCD = 72
---
- Denominators: 10 and 5
- Prime factorization:
- \( 10 = 2 \times 5 \)
- \( 5 = 5 \)
- LCM: Take the highest power of each prime factor.
- \( 2^1 \) (from 10)
- \( 5^1 \) (from both 10 and 5)
- LCM = \( 2 \times 5 = 10 \)
LCD = 10
---
- Denominators: 6 and 9
- Prime factorization:
- \( 6 = 2 \times 3 \)
- \( 9 = 3^2 \)
- LCM: Take the highest power of each prime factor.
- \( 2^1 \) (from 6)
- \( 3^2 \) (from 9)
- LCM = \( 2 \times 3^2 = 2 \times 9 = 18 \)
LCD = 18
---
- Denominators: 8 and 10
- Prime factorization:
- \( 8 = 2^3 \)
- \( 10 = 2 \times 5 \)
- LCM: Take the highest power of each prime factor.
- \( 2^3 \) (from 8)
- \( 5^1 \) (from 10)
- LCM = \( 2^3 \times 5 = 8 \times 5 = 40 \)
LCD = 40
---
- Denominators: 3 and 8
- Prime factorization:
- \( 3 = 3 \)
- \( 8 = 2^3 \)
- LCM: Since 3 and 8 have no common prime factors, their LCM is their product.
- LCM = \( 3 \times 8 = 24 \)
LCD = 24
---
\[
\boxed{
\begin{array}{lll}
\text{a. } & \text{LCD = 6} & \\
\text{b. } & \text{LCD = 15} & \\
\text{c. } & \text{LCD = 12} & \\
\text{d. } & \text{LCD = 4} & \\
\text{e. } & \text{LCD = 72} & \\
\text{f. } & \text{LCD = 10} & \\
\text{g. } & \text{LCD = 18} & \\
\text{h. } & \text{LCD = 40} & \\
\text{i. } & \text{LCD = 24} & \\
\end{array}
}
\]
Let's solve each part step by step:
---
a. \( \frac{1}{6} \) and \( \frac{2}{3} \)
- Denominators: 6 and 3
- Prime factorization:
- \( 6 = 2 \times 3 \)
- \( 3 = 3 \)
- LCM: Take the highest power of each prime factor.
- \( 2^1 \) (from 6)
- \( 3^1 \) (from both 6 and 3)
- LCM = \( 2 \times 3 = 6 \)
LCD = 6
---
b. \( \frac{1}{5} \) and \( \frac{2}{3} \)
- Denominators: 5 and 3
- Prime factorization:
- \( 5 = 5 \)
- \( 3 = 3 \)
- LCM: Since 5 and 3 are coprime (no common factors other than 1), their LCM is simply their product.
- LCM = \( 5 \times 3 = 15 \)
LCD = 15
---
c. \( \frac{1}{4} \) and \( \frac{5}{6} \)
- Denominators: 4 and 6
- Prime factorization:
- \( 4 = 2^2 \)
- \( 6 = 2 \times 3 \)
- LCM: Take the highest power of each prime factor.
- \( 2^2 \) (from 4)
- \( 3^1 \) (from 6)
- LCM = \( 2^2 \times 3 = 4 \times 3 = 12 \)
LCD = 12
---
d. \( \frac{1}{2} \) and \( \frac{1}{4} \)
- Denominators: 2 and 4
- Prime factorization:
- \( 2 = 2 \)
- \( 4 = 2^2 \)
- LCM: Take the highest power of each prime factor.
- \( 2^2 \) (from 4)
- LCM = \( 4 \)
LCD = 4
---
e. \( \frac{5}{8} \) and \( \frac{4}{9} \)
- Denominators: 8 and 9
- Prime factorization:
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- LCM: Since 8 and 9 have no common prime factors, their LCM is their product.
- LCM = \( 8 \times 9 = 72 \)
LCD = 72
---
f. \( \frac{3}{10} \) and \( \frac{1}{5} \)
- Denominators: 10 and 5
- Prime factorization:
- \( 10 = 2 \times 5 \)
- \( 5 = 5 \)
- LCM: Take the highest power of each prime factor.
- \( 2^1 \) (from 10)
- \( 5^1 \) (from both 10 and 5)
- LCM = \( 2 \times 5 = 10 \)
LCD = 10
---
g. \( \frac{1}{6} \) and \( \frac{2}{9} \)
- Denominators: 6 and 9
- Prime factorization:
- \( 6 = 2 \times 3 \)
- \( 9 = 3^2 \)
- LCM: Take the highest power of each prime factor.
- \( 2^1 \) (from 6)
- \( 3^2 \) (from 9)
- LCM = \( 2 \times 3^2 = 2 \times 9 = 18 \)
LCD = 18
---
h. \( \frac{7}{8} \) and \( \frac{4}{10} \)
- Denominators: 8 and 10
- Prime factorization:
- \( 8 = 2^3 \)
- \( 10 = 2 \times 5 \)
- LCM: Take the highest power of each prime factor.
- \( 2^3 \) (from 8)
- \( 5^1 \) (from 10)
- LCM = \( 2^3 \times 5 = 8 \times 5 = 40 \)
LCD = 40
---
i. \( \frac{2}{3} \) and \( \frac{1}{8} \)
- Denominators: 3 and 8
- Prime factorization:
- \( 3 = 3 \)
- \( 8 = 2^3 \)
- LCM: Since 3 and 8 have no common prime factors, their LCM is their product.
- LCM = \( 3 \times 8 = 24 \)
LCD = 24
---
Final Answer:
\[
\boxed{
\begin{array}{lll}
\text{a. } & \text{LCD = 6} & \\
\text{b. } & \text{LCD = 15} & \\
\text{c. } & \text{LCD = 12} & \\
\text{d. } & \text{LCD = 4} & \\
\text{e. } & \text{LCD = 72} & \\
\text{f. } & \text{LCD = 10} & \\
\text{g. } & \text{LCD = 18} & \\
\text{h. } & \text{LCD = 40} & \\
\text{i. } & \text{LCD = 24} & \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of common denominator worksheet.
