Free Math Worksheets

- How can you subtract fractions with different denominators? - What is the smallest number that is a multiple of both \(27\) and \(6\)? - When you make an equivalent fraction, multiply the numerator and denominator by the same number.

1. Find a common denominator. The least common multiple of \(27\) and \(6\) is \(54\). 2. Rewrite the fractions: \(\frac{5}{27} = \frac{10}{54}\) and \(\frac{1}{6} = \frac{9}{54}\). 3. Subtract: \(\frac{10}{54} - \frac{9}{54} = \frac{1}{54}\).

\(\frac{1}{54}\)

- What is the smallest common multiple of \(6\) and \(4\)? - How can you rewrite a fraction without changing its value? - Once the denominators match, which parts of the fractions are subtracted?

1. Find the least common denominator of \(6\) and \(4\), which is \(12\). 2. Rewrite the fractions as equivalent fractions with denominator \(12\): \(\frac{5}{6}=\frac{10}{12}\) and \(\frac{1}{4}=\frac{3}{12}\). 3. Subtract the numerators: \(\frac{10}{12}-\frac{3}{12}=\frac{7}{12}\).

Use the least common denominator \(12\), rewrite the fractions as \(\frac{10}{12}\) and \(\frac{3}{12}\), and subtract. The result is \(\frac{7}{12}\).

- Find a common denominator before subtracting fractions with different denominators. - Rewrite whole numbers as fractions when needed. - Simplify each final answer.

1. For a), use denominator \(12\): \(\frac{3}{4}-\frac{1}{6}=\frac{9}{12}-\frac{2}{12}=\frac{7}{12}\). 2. For b), use denominator \(15\): \(\frac{7}{5}-\frac{2}{3}=\frac{21}{15}-\frac{10}{15}=\frac{11}{15}\). 3. For c), rewrite \(2\) as \(\frac{16}{8}\): \(\frac{16}{8}-\frac{5}{8}=\frac{11}{8}\).

a) \(\frac{7}{12}\) b) \(\frac{11}{15}\) c) \(\frac{11}{8}\)

- After rewriting the mixed numbers as improper fractions, find a common denominator. - In part b), what can you do when the first fractional part is smaller than the second? - Compare the number and type of steps in the two methods.

1. For a), rewrite the mixed numbers: \(5 \frac{1}{6}=\frac{31}{6}\) and \(2 \frac{3}{4}=\frac{11}{4}\). 2. Use denominator \(12\): \(\frac{31}{6}-\frac{11}{4}=\frac{62}{12}-\frac{33}{12}=\frac{29}{12}=2 \frac{5}{12}\). 3. For b), regroup \(5 \frac{1}{6}\) as \(4 \frac{7}{6}\). 4. Use denominator \(12\) for the fractional parts: \(4 \frac{14}{12}-2 \frac{9}{12}=2 \frac{5}{12}\). Both methods give the same result; the preferred method can depend on which steps you find easier to track accurately.

a) \(2 \frac{5}{12}\) b) \(2 \frac{5}{12}\) Both methods are correct. Which one is less error-prone depends on the student's reasoning and organization.

- Subtract \(\frac{1}{6}\) from both sides. - Find a common denominator for \(\frac{1}{5}\) and \(\frac{1}{6}\). - Match the resulting unit fraction to \(\frac{1}{x}\).

1. Isolate the unknown fraction: \(\frac{1}{x}=\frac{1}{5}-\frac{1}{6}\). 2. Subtract using denominator \(30\): \(\frac{1}{5}-\frac{1}{6}=\frac{6}{30}-\frac{5}{30}=\frac{1}{30}\). Therefore, \(x=30\). 3. Check: \(\frac{1}{6}+\frac{1}{30}=\frac{5}{30}+\frac{1}{30}=\frac{6}{30}=\frac{1}{5}\).

\(x=30\)

- Calculate each difference separately. - Then rewrite the two results in a form that makes them easy to compare.

1. Calculate \(A\): \(\frac{5}{2}-\frac{3}{4}=\frac{10}{4}-\frac{3}{4}=\frac{7}{4}\). 2. Calculate \(B\): \(\frac{11}{6}-\frac{1}{3}=\frac{11}{6}-\frac{2}{6}=\frac{9}{6}=\frac{3}{2}\). 3. Rewrite \(\frac{3}{2}\) as \(\frac{6}{4}\). Since \(\frac{7}{4}>\frac{6}{4}\), \(A>B\).

\(A>B\), because \(\frac{7}{4}>\frac{3}{2}\).

- Look closely at the order of the numbers in \(2-5\). - Is the first fractional part large enough to subtract \(\frac{5}{8}\) from it directly? - How can one whole be regrouped as eighths?

1. The error occurs in the last step, where the student changes \((8-3)+\frac{2-5}{8}\) into \(5 \frac{3}{8}\). 2. The fractional subtraction \(\frac{2}{8}-\frac{5}{8}\) cannot become positive \(\frac{3}{8}\). Because \(\frac{2}{8}<\frac{5}{8}\), one whole must be regrouped before subtracting the fractional parts. 3. Rewrite \(8 \frac{2}{8}\) as \(7 \frac{10}{8}\). Then \(7 \frac{10}{8}-3 \frac{5}{8}=4 \frac{5}{8}\).

The error is in the last step: the student treats \(\frac{2}{8}-\frac{5}{8}\) as positive \(\frac{3}{8}\). After regrouping one whole, the correct result is \(4 \frac{5}{8}\).

- Begin with a unit fraction that is slightly greater than the target fraction. - Use a common denominator to subtract. - For part b), substitute \(10\) for \(n\) before finding the missing unit fraction.

1. For a), compare \(\frac{1}{6}\) with the next larger unit fraction, \(\frac{1}{5}\). Since \(\frac{1}{5}-\frac{1}{6}=\frac{6}{30}-\frac{5}{30}=\frac{1}{30}\), \(\frac{1}{6}=\frac{1}{5}-\frac{1}{30}\). 2. For b), substitute \(n=10\): \(\frac{1}{10}=\frac{1}{9}-\frac{1}{x}\). 3. Compute \(\frac{1}{9}-\frac{1}{10}=\frac{10}{90}-\frac{9}{90}=\frac{1}{90}\), so \(x=90\). 4. Check: \(\frac{1}{9}-\frac{1}{90}=\frac{10}{90}-\frac{1}{90}=\frac{9}{90}=\frac{1}{10}\).

a) \(\frac{1}{6}=\frac{1}{5}-\frac{1}{30}\) b) \(x=90\)

- Estimate the reciprocal of \(\frac{3}{14}\) to find the first denominator. - Use a common denominator to subtract the fractions. - Check whether the remainder is a unit fraction.

1. Since \(14\div3\approx4.67\), the smallest integer denominator that gives a unit fraction less than \(\frac{3}{14}\) is \(5\). Thus, the first term is \(\frac{1}{5}\). 2. Subtract: \(\frac{3}{14}-\frac{1}{5}=\frac{15}{70}-\frac{14}{70}=\frac{1}{70}\). 3. Therefore, \(\frac{3}{14}=\frac{1}{5}+\frac{1}{70}\).

\(\frac{3}{14}=\frac{1}{5}+\frac{1}{70}\)

- At each stage, choose the greatest unit fraction that does not exceed the current amount. - Use common denominators for each subtraction. - Stop when the remainder is a unit fraction.

1. The greatest unit fraction less than \(\frac{4}{5}\) is \(\frac{1}{2}\). 2. Subtract: \(\frac{4}{5}-\frac{1}{2}=\frac{8}{10}-\frac{5}{10}=\frac{3}{10}\). 3. The greatest unit fraction less than \(\frac{3}{10}\) is \(\frac{1}{4}\), because \(\frac{1}{3}>\frac{3}{10}\). 4. Subtract: \(\frac{3}{10}-\frac{1}{4}=\frac{6}{20}-\frac{5}{20}=\frac{1}{20}\). 5. Therefore, \(\frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}\).

\(\frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}\)