Free Math Worksheets

- Think about the roles of the numerator and denominator in division. - A fraction bar can be read as a division symbol. - Identify which number is divided and which number it is divided by.

1. The numerator is the dividend, so the dividend is \(5\). 2. The denominator is the divisor, so the divisor is \(13\). 3. Therefore, \(\frac{5}{13}=5\div13\).

a) \(5\) b) \(13\) c) \(5\div13\)

- Read the fraction bar as division. - Find a number that divides both the dividend and divisor. - A quotient is a whole number only when the division has no remainder. - Use the greatest common factor to simplify in one step.

1. A quotient \(a\div b\) can be written as \(\frac{a}{b}\). 2. Simplify each fraction by dividing the numerator and denominator by a common factor. 3. The results are \(\frac{12}{15}=\frac{4}{5}\), \(\frac{45}{20}=\frac{9}{4}\), \(\frac{132}{11}=12\), \(\frac{14}{42}=\frac{1}{3}\), and \(\frac{75}{100}=\frac{3}{4}\).

a) \(\frac{4}{5}\) b) \(\frac{9}{4}\) c) \(12\) d) \(\frac{1}{3}\) e) \(\frac{3}{4}\)

- Think about sharing \(3\) pizzas among \(4\) children versus sharing \(4\) pizzas among \(3\) children. - Write both situations as fractions. - Decide what happens when the numerator and denominator are switched.

1. When \(3\) is the dividend and \(4\) is the divisor, the quotient is \(\frac{3}{4}\). 2. When \(4\) is the dividend and \(3\) is the divisor, the quotient is \(\frac{4}{3}\). 3. The fraction \(\frac{3}{4}\) is less than \(1\), while \(\frac{4}{3}\) is greater than \(1\). The quotients are different, so Lucas is not correct.

a) \(\frac{3}{4}\) b) \(\frac{4}{3}\) c) No. \(\frac{3}{4}<1\), while \(\frac{4}{3}>1\), so changing the order changes the quotient.

- Identify the total amount being shared and the number of equal groups. - The amount being divided appears first in the division expression. - Connect the dividend and divisor to the numerator and denominator.

1. The \(7\) bagfuls are the amount being divided, so \(7\) is the dividend. 2. The \(10\) bins show how many equal groups are made, so \(10\) is the divisor. 3. The division expression is \(7\div10\), and each bin receives \(\frac{7}{10}\) of a bagful.

a) Dividend b) Divisor c) \(7\div10=\frac{7}{10}\) of a bagful

- Divide the numerator by the denominator. - Use the quotient as the whole-number part and the remainder as the new numerator. - Simplify the fractional part.

1. For \(\frac{26}{4}\), divide: \(26\div4=6\) remainder \(2\). Thus, \(\frac{26}{4}=6\frac{2}{4}=6\frac{1}{2}\). 2. For \(\frac{50}{12}\), divide: \(50\div12=4\) remainder \(2\). Thus, \(\frac{50}{12}=4\frac{2}{12}=4\frac{1}{6}\). 3. For \(\frac{108}{15}\), divide: \(108\div15=7\) remainder \(3\). Thus, \(\frac{108}{15}=7\frac{3}{15}=7\frac{1}{5}\).

a) \(6\frac{1}{2}\) b) \(4\frac{1}{6}\) c) \(7\frac{1}{5}\)

- Rewrite each mixed number as an improper fraction. - Multiply the whole number by the denominator. - Add the numerator of the fractional part.

1. Convert \(5\frac{3}{7}\) to an improper fraction: \(5\times7+3=38\). Therefore, \(x=38\). 2. Convert \(8\frac{4}{9}\) to an improper fraction: \(8\times9+4=76\). Therefore, \(x=76\).

a) \(x=38\) b) \(x=76\)

- For a mixed number, count how many fractional parts are in the whole numbers. - The denominator stays the same when you change forms. - For an improper fraction, divide the numerator by the denominator. - Use the remainder as the numerator of the fractional part.

1. For a), multiply the whole number by the denominator and add the numerator: \(8\times9+4=76\). Thus, \(8\frac{4}{9}=\frac{76}{9}\). 2. For b), \(11\times12+7=139\). Thus, \(11\frac{7}{12}=\frac{139}{12}\). 3. For c), divide \(53\div6\). The quotient is \(8\) with remainder \(5\), so \(\frac{53}{6}=8\frac{5}{6}\). 4. For d), divide \(127\div10\). The quotient is \(12\) with remainder \(7\), so \(\frac{127}{10}=12\frac{7}{10}\).

a) \(\frac{76}{9}\) b) \(\frac{139}{12}\) c) \(8\frac{5}{6}\) d) \(12\frac{7}{10}\)

- Write each sharing situation as a fraction. - Divide the numerator and denominator by a common factor. - When the numerator is greater than the denominator, divide to write a mixed number.

1. For a), each person receives \(\frac{12}{16}=\frac{3}{4}\) of a pizza. 2. For b), each pitcher receives \(\frac{14}{20}=\frac{7}{10}\) quart. 3. For c), each piece is \(\frac{21}{6}=\frac{7}{2}=3\frac{1}{2}\) feet long.

a) \(\frac{3}{4}\) of a pizza b) \(\frac{7}{10}\) quart c) \(\frac{7}{2}\) feet, or \(3\frac{1}{2}\) feet

- Divide each total distance by \(12\) miles per loop. - A fraction bar represents division. - Simplify before writing the result as a mixed number.

1. For the Miller family, divide the total distance by the loop length: \(\frac{30}{12}=\frac{5}{2}=2\frac{1}{2}\). 2. For the training group, \(\frac{50}{12}=\frac{25}{6}=4\frac{1}{6}\).

The Miller family completes \(2\frac{1}{2}\) loops. The training group completes \(4\frac{1}{6}\) loops.

- First look for a common factor of the numerator and denominator. - Divide the numerator by the denominator. - Use the remainder as the numerator of the fractional part.

1. Simplify \(\frac{22}{4}\) by \(2\): \(\frac{22}{4}=\frac{11}{2}=5\frac{1}{2}\). 2. Simplify \(\frac{45}{10}\) by \(5\): \(\frac{45}{10}=\frac{9}{2}=4\frac{1}{2}\). 3. The fraction \(\frac{19}{6}\) is already in simplest form. Since \(19\div6=3\) remainder \(1\), \(\frac{19}{6}=3\frac{1}{6}\). 4. Simplify \(\frac{32}{12}\) by \(4\): \(\frac{32}{12}=\frac{8}{3}=2\frac{2}{3}\).

a) \(5\frac{1}{2}\) b) \(4\frac{1}{2}\) c) \(3\frac{1}{6}\) d) \(2\frac{2}{3}\)

- Imagine cutting each pizza into \(3\) equal pieces. - Count how many one-third pieces each child receives. - A fraction bar represents division.

1. Two whole pizzas are divided among \(3\) children. 2. Write the sharing as \(2\div3\). 3. Since \(2\div3=\frac{2}{3}\), each child receives \(\frac{2}{3}\) of a pizza.

Each child receives \(\frac{2}{3}\) of a pizza.

- Write each quotient as a fraction and simplify it. - Record the simplified fractions so you can compare them. - Make sure every fraction is in simplest form before grouping.

1. Write and simplify each quotient: \(A=\frac{15}{10}=\frac{3}{2}\), \(B=\frac{24}{16}=\frac{3}{2}\), and \(C=\frac{6}{4}=\frac{3}{2}\). 2. Also, \(D=\frac{18}{15}=\frac{6}{5}\) and \(E=\frac{12}{10}=\frac{6}{5}\). 3. Therefore, \(A\), \(B\), and \(C\) form one group, while \(D\) and \(E\) form the other.

Group 1: \(A\), \(B\), and \(C\), each equal to \(\frac{3}{2}\) Group 2: \(D\) and \(E\), each equal to \(\frac{6}{5}\)

- Divide the numerator by the denominator using long division. - The quotient is the whole-number part. - The remainder becomes the numerator of the fractional part. - The denominator stays the same.

1. Divide \(475\div21\). The quotient is \(22\) with remainder \(13\), because \(21\times22=462\) and \(475-462=13\). Therefore, \(\frac{475}{21}=22\frac{13}{21}\). 2. Divide \(1000\div37\). The quotient is \(27\) with remainder \(1\), because \(37\times27=999\) and \(1000-999=1\). Therefore, \(\frac{1000}{37}=27\frac{1}{37}\).

a) \(22\frac{13}{21}\) b) \(27\frac{1}{37}\)

- Write each division expression as a fraction. - Divide the numerator and denominator by a common factor. - For an improper fraction, divide the numerator by the denominator. - A remainder of \(0\) gives a whole-number result.

1. Write \(210\div45\) as \(\frac{210}{45}\). Simplifying by \(15\) gives \(\frac{14}{3}=4\frac{2}{3}\). 2. Write \(444\div24\) as \(\frac{444}{24}\). Simplifying by \(12\) gives \(\frac{37}{2}=18\frac{1}{2}\). 3. Write \(1005\div15\) as \(\frac{1005}{15}\). Simplifying by \(15\) gives \(67\).

a) \(\frac{14}{3}=4\frac{2}{3}\) b) \(\frac{37}{2}=18\frac{1}{2}\) c) \(67\)

- Use the rule for converting a mixed number to an improper fraction. - Write an equation for the missing value. - Check by converting the completed mixed number.

1. For a), converting the mixed number gives \(6\times7+x=46\). Thus, \(42+x=46\), so \(x=4\). 2. For b), converting the mixed number gives \(3y+2=23\). Thus, \(3y=21\), so \(y=7\).

a) \(x=4\) b) \(y=7\)

- Think about how a mixed number and an improper fraction represent the same value. - Multiply the whole number by the denominator, then add the numerator. - Use division with a remainder to convert an improper fraction. - The denominator stays the same when the form changes.

1. For a), \(7\times9+2=65\), so the missing numerator is \(65\). 2. For b), \(53\div6=8\) remainder \(5\), so the missing numerator is \(5\). 3. For c), \(11\times4+3=47\). The denominator remains \(4\), so the missing denominator is \(4\).

a) \(65\) b) \(5\) c) \(4\)

- Write the amount per child as a fraction for each group. - Rewrite the fractions with a common denominator. - Find a common multiple of \(8\) and \(5\).

1. In Group A, each child receives \(\frac{5}{8}\) of a pizza. 2. In Group B, each child receives \(\frac{3}{5}\) of a pizza. 3. Use denominator \(40\): \(\frac{5}{8}=\frac{25}{40}\) and \(\frac{3}{5}=\frac{24}{40}\). 4. Since \(\frac{25}{40}>\frac{24}{40}\), each child in Group A receives more pizza.

Group A, because \(\frac{5}{8}>\frac{3}{5}\).

- Divide the numerator by the denominator. - Use the remainder as the numerator of the fractional part. - Find the fraction that completes the fractional part to \(1\).

1. For a), \(19\div3=6\) remainder \(1\), so \(\frac{19}{3}=6\frac{1}{3}\). The next whole number is \(7\), and \(1-\frac{1}{3}=\frac{2}{3}\). 2. For b), \(55\div8=6\) remainder \(7\), so \(\frac{55}{8}=6\frac{7}{8}\). The next whole number is \(7\), and \(1-\frac{7}{8}=\frac{1}{8}\). 3. For c), \(113\div15=7\) remainder \(8\), so \(\frac{113}{15}=7\frac{8}{15}\). The next whole number is \(8\), and \(1-\frac{8}{15}=\frac{7}{15}\).

a) \(6\frac{1}{3}\); add \(\frac{2}{3}\) to reach \(7\). b) \(6\frac{7}{8}\); add \(\frac{1}{8}\) to reach \(7\). c) \(7\frac{8}{15}\); add \(\frac{7}{15}\) to reach \(8\).

- A number whose next whole number is \(10\) lies between \(9\) and \(10\). - Find the fraction that complements \(\frac{5}{11}\) to \(1\). - Convert the mixed number by multiplying the whole number by the denominator and adding the numerator.

1. Since the next whole number is \(10\), the whole-number part is \(9\). 2. The fractional part is \(1-\frac{5}{11}=\frac{6}{11}\), so the mixed number is \(9\frac{6}{11}\). 3. Convert to an improper fraction: \(9\times11+6=105\). Therefore, \(9\frac{6}{11}=\frac{105}{11}\).

The mixed number is \(9\frac{6}{11}\), and the improper fraction is \(\frac{105}{11}\).

- Write each quotient as a fraction. - Decide whether to scale up or simplify the fraction. - Check by substituting each missing number into the original equation.

1. For a), \(\frac{x}{12}=\frac{3}{4}\). Multiply the numerator and denominator of \(\frac{3}{4}\) by \(3\) to get \(\frac{9}{12}\), so \(x=9\). 2. For b), \(\frac{56}{y}=\frac{7}{8}\). Since \(56=7\times8\), the denominator must be \(8\times8=64\), so \(y=64\). 3. For c), \(\frac{144}{60}=\frac{z}{5}\). Divide the numerator and denominator by \(12\): \(\frac{144}{60}=\frac{12}{5}\), so \(z=12\).

a) \(x=9\) b) \(y=64\) c) \(z=12\)

- Convert each mixed number to an improper fraction. - Equivalent fractions multiply the numerator and denominator by the same factor. - Find the scale factor between the known numerators.

1. Convert \(15\frac{3}{4}\) to an improper fraction: \(15\times4+3=63\), so \(15\frac{3}{4}=\frac{63}{4}\). To make the denominator \(8\), multiply the numerator and denominator by \(2\): \(\frac{63}{4}=\frac{126}{8}\). Therefore, \(x=126\). 2. Convert \(4\frac{2}{3}\) to an improper fraction: \(4\times3+2=14\), so \(4\frac{2}{3}=\frac{14}{3}\). Since \(210=14\times15\), multiply the denominator by the same factor: \(y=3\times15=45\).

a) \(x=126\) b) \(y=45\)