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) \(5\div13\)

- Count the number of whole bars shown in the model. - The number of hikers tells how many equal shares are made. - Read the quotient as a fraction with the amount being shared on top and the number of shares on the bottom.

1. The model shows \(3\) whole bars being shared among \(4\) hikers, so the division expression is \(3\div4\). 2. A quotient of whole numbers can be written as a fraction: \(3\div4=\frac{3}{4}\). 3. Each hiker receives \(\frac{3}{4}\) of one whole bar.

\(3\div4=\frac{3}{4}\). Each hiker receives \(\frac{3}{4}\) of one whole bar.

- 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) \(7\frac{1}{5}\)

- Read \(\frac{x}{b}\) as \(x\div b\). - In a mixed quotient, the whole-number part is the quotient and the fractional numerator is the remainder when the denominator is the divisor. - Rebuild the dividend from divisor, quotient, and remainder.

1. For a), \(\frac{x}{7}\) means \(x\div7\). The mixed number says the quotient is \(5\) with remainder \(3\), so \(x=5\times7+3=38\). 2. For b), \(\frac{x}{9}\) means \(x\div9\). The quotient is \(8\) with remainder \(4\), so \(x=8\times9+4=76\).

a) \(38\div7=5\) remainder \(3\), so \(x=38\). b) \(76\div9=8\) remainder \(4\), so \(x=76\).

- 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.

- 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.

- First count how many sevenths are in the whole-number part and the fractional part together. - Then read the fraction bar as a division symbol.

1. Convert the mixed number: \(4\frac{2}{7}=\frac{4\times7+2}{7}=\frac{30}{7}\). 2. A fraction represents division, so \(\frac{30}{7}=30\div7\).

\(4\frac{2}{7}=\frac{30}{7}=30\div7\).

- Put the total amount in the dividend position and the number of equal shares in the divisor position. - Read the quotient as a fraction, then use the remainder to form a mixed number.

1. Equal sharing gives \(11\div4\). 2. As a fraction, \(11\div4=\frac{11}{4}\). 3. Since \(11=2\times4+3\), \(\frac{11}{4}=2\frac{3}{4}\).

Each display gets \(11\div4=\frac{11}{4}=2\frac{3}{4}\) yards.

- Equal sharing tells you which quantity is the dividend and which is the divisor. - To locate the quotient between whole numbers, compare the dividend with nearby multiples of the divisor. - Interpret the two numbers in the fraction using the original sharing situation.

1. Equal sharing gives \(13\div6=\frac{13}{6}\) pound per sculpture. 2. Since \(2\times6=12<13<18=3\times6\), dividing all three quantities by \(6\) gives \(2<\frac{13}{6}<3\). 3. The numerator \(13\) records the total pounds of clay, and the denominator \(6\) records the number of equal sculptures receiving shares.

a) \(13\div6=\frac{13}{6}\) pound per sculpture. b) \(2<\frac{13}{6}<3\), because \(12<13<18\) and \(12=2\times6\), \(18=3\times6\). c) \(13\) is the total pounds shared, and \(6\) is the number of equal shares.

- Read a fraction bar as division from numerator to denominator. - Keep the order of the original division expression when writing the fraction. - Use the whole-number quotient and remainder to form a mixed number.

1. A fraction \(\frac{a}{b}\) represents \(a\div b\), so \(17\div5=\frac{17}{5}\). 2. The numerator is the dividend \(17\), and the denominator is the divisor \(5\). 3. Since \(17=3\times5+2\), \(\frac{17}{5}=3\frac{2}{5}\).

\(17\div5=\frac{17}{5}=3\frac{2}{5}\). The dividend \(17\) becomes the numerator and the divisor \(5\) becomes the denominator, so reversing them would represent a different quotient.

- A fraction bar can represent the original division exactly. - Think about what the remainder represents relative to the divisor. - The divisor determines the denominator of the leftover fractional part.

1. The division \(23\div6\) is represented exactly by \(\frac{23}{6}\). 2. The quotient \(3\) accounts for \(18\) of the \(23\), leaving remainder \(5\). 3. The remainder \(5\) is five parts of size \(\frac{1}{6}\), so it becomes \(\frac{5}{6}\). 4. Therefore \(\frac{23}{6}=3\frac{5}{6}\).

\(23\div6=\frac{23}{6}=3\frac{5}{6}\). The denominator \(6\) shows the size of the equal parts left after forming whole groups.

- A fraction bar can represent division exactly. - Use whole-number division to find the quotient and remainder. - The remainder is measured in parts whose denominator is the divisor.

1. The exact quotient is \(\frac{41}{6}\). 2. Whole-number division gives \(41=6\times6+5\). 3. The quotient \(6\) becomes the whole-number part, and the remainder \(5\) becomes \(\frac{5}{6}\). 4. Therefore \(\frac{41}{6}=6\frac{5}{6}\).

\(41\div6=\frac{41}{6}=6\frac{5}{6}\). In \(41=6\times6+5\), the whole-number quotient \(6\) becomes the whole part and the remainder \(5\) becomes \(\frac{5}{6}\).

- Read the fraction bar as a division bar: identify which number is being divided and which number gives the equal-share count. - Compare \(18\) with nearby multiples of \(7\) rather than converting the quotient to another number form. - Use those multiples to bound the exact fractional quotient.

1. The quotient \(18\div7\) is \(\frac{18}{7}\): \(18\) is the amount being divided, so it is the numerator, and \(7\) is the number of equal shares, so it is the denominator. 2. Since \(14=2\times7<18<3\times7=21\), dividing by \(7\) gives \(2<\frac{18}{7}<3\).

a) \(18\div7=\frac{18}{7}\); the dividend \(18\) is the numerator and the divisor \(7\) is the denominator. b) The quotient lies between \(2\) and \(3\).

- Count the whole loaves in the model and identify the number of equal shares. - Write the dividend over the divisor to express the quotient as a fraction. - Use the thirds in the picture to see how many complete wholes and leftover thirds each share represents.

1. The model shows \(5\) whole loaves shared among \(3\) tables, so the division is \(5\div3\). 2. As a fraction, \(5\div3=\frac{5}{3}\). 3. Five thirds make \(1\) whole and \(2\) thirds, so \(\frac{5}{3}=1\frac{2}{3}\).

\(5\div3=\frac{5}{3}=1\frac{2}{3}\).

- Combine the shaded fifths across both bars. - Express that total number of fifths as one fraction before choosing the division expression. - Compare the size of the pictured value with what the reversed fraction would represent.

1. One full bar is \(\frac{5}{5}\), and the second bar shows \(\frac{4}{5}\), so the modeled value is \(\frac{9}{5}=1\frac{4}{5}\). 2. A fraction \(\frac{9}{5}\) represents \(9\div5\), so that is the matching expression. 3. The \(9\) counts the total fifths represented, while \(5\) tells how many fifths make one whole; reversing them would give \(\frac{5}{9}\), a value less than one and unlike the model.

The model matches \(9\div5\). Its value is \(\frac{9}{5}=1\frac{4}{5}\). The total number of fifths becomes the dividend/numerator, and the number of fifths in one whole becomes the divisor/denominator.

- Count the whole bars and the equal parts in each whole directly from each model. - In an equal-sharing model, the number of parts per whole matches the number of recipients. - Compare the two quotient fractions after you have read both situations from the images.

1. Model a) has \(2\) wholes divided into \(3\) equal recipient-parts per whole. One person receives one part from each whole, so the share is \(2\div3=\frac{2}{3}\). 2. Model b) has \(4\) wholes divided into \(6\) equal recipient-parts per whole. One person receives one part from each whole, so the share is \(4\div6=\frac{4}{6}=\frac{2}{3}\). 3. Doubling both the total number of wholes and the number of equal recipients doubles numerator and denominator of the fraction, so the amount per person is unchanged.

a) \(2\div3=\frac{2}{3}\) b) \(4\div6=\frac{4}{6}=\frac{2}{3}\) c) The shares are equal because both the total and the number of recipients were multiplied by \(2\), giving equivalent fractions.

- Read the individual share from shaded parts over total equal parts in the model. - Recombine five identical shares to recover the whole amount that was divided. - In the final division expression, identify which number is the total and which number is the number of equal shares.

1. The model has \(3\) of \(5\) equal parts filled, so each child received \(\frac{3}{5}\) of a bar. 2. Five children each receiving \(\frac{3}{5}\) of a bar use \(5\times\frac{3}{5}=3\) whole bars in all. 3. Therefore the sharing situation is \(3\div5=\frac{3}{5}\). The total number of bars is the dividend and becomes the numerator; the number of equal shares is the divisor and becomes the denominator.

a) \(\frac{3}{5}\) of a bar b) \(3\) whole bars c) \(3\div5=\frac{3}{5}\). The dividend \(3\) is the numerator and the divisor \(5\) is the denominator.

- 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 \(\frac{6}{5}\)

- 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) \(67\)

- Read an improper fraction as numerator divided by denominator. - Find the whole-number quotient and remainder for each division. - Match those two pieces to the whole-number part and fractional numerator of the mixed number.

1. \(\frac{46}{7}\) means \(46\div7\). Since \(46=6\times7+4\), the quotient is \(6\) with remainder \(4\), so \(x=4\). 2. \(\frac{23}{3}\) means \(23\div3\). Since \(23=7\times3+2\), the quotient is \(7\) with remainder \(2\), so \(y=7\).

a) \(46\div7=6\) remainder \(4\), so \(x=4\). b) \(23\div3=7\) remainder \(2\), so \(y=7\).

- 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 \(8\).

- First find the fraction that complements \(\frac{5}{11}\) to one whole. - Read \(\frac{x}{11}\) as \(x\div11\). - Use divisor, quotient, and remainder to reconstruct the dividend.

1. The value is between \(9\) and \(10\). Its fractional part is \(1-\frac{5}{11}=\frac{6}{11}\), so \(\frac{x}{11}=9\frac{6}{11}\). 2. Interpreting this as division, \(x\div11\) has quotient \(9\) and remainder \(6\). 3. Therefore \(x=9\times11+6=105\).

a) \(9\frac{6}{11}\) b) The division has quotient \(9\) and remainder \(6\), so \(x=105\) and \(105\div11=9\) remainder \(6\).

- Write each equal share as total pizzas divided by number of students. - Express each quotient as a fraction and simplify. - Think about what happens to a share when both the total and the number of recipients are doubled.

1. In Group A, each student receives \(3\div4=\frac{3}{4}\) pizza. 2. In Group B, each student receives \(6\div8=\frac{6}{8}\) pizza. 3. Simplify \(\frac{6}{8}=\frac{3}{4}\). 4. Doubling both the number of pizzas and the number of students keeps the amount per student unchanged.

Yes. Each student receives \(\frac{3}{4}\) pizza because \(3\div4=\frac{3}{4}\) and \(6\div8=\frac{6}{8}=\frac{3}{4}\).

- Divide to find the whole miles in each interval and the remainder. - Think about how the leftover miles are shared among all \(5\) intervals. - Use that equal sharing to write the fractional part of the mixed number.

1. Equal intervals give \(43\div5\), which is \(\frac{43}{5}\). 2. Since \(43=8\times5+3\), each interval gets \(8\) whole miles, with \(3\) miles still to share equally among the \(5\) intervals. 3. Each interval receives \(\frac{3}{5}\) of one more mile, so \(\frac{43}{5}=8\frac{3}{5}\).

\(43\div5=\frac{43}{5}=8\frac{3}{5}\) miles per interval. The remainder \(3\) becomes the numerator because the \(3\) leftover miles are divided equally among \(5\) intervals.

- Translate each equal-share situation directly into total hours divided by number of sessions. - Compare the two exact fractional quotients without changing them to decimals or mixed numbers. - Relate the top and bottom numbers of each fraction back to the quantities in the sharing situation.

1. The first average is \(17\div3=\frac{17}{3}\) hour per session, and the second is \(22\div4=\frac{22}{4}=\frac{11}{2}\) hour per session. 2. Using sixths, \(\frac{17}{3}=\frac{34}{6}\) and \(\frac{11}{2}=\frac{33}{6}\), so the first club's average is greater by \(\frac{1}{6}\) hour per session. 3. In each fraction, the numerator is the total number of hours being divided and the denominator is the number of equal sessions.

a) First club: \(\frac{17}{3}\) hour per session. Second club: \(\frac{11}{2}\) hours per session. b) The first club has the greater average by \(\frac{1}{6}\) hour per session. c) The numerator records total hours; the denominator records the number of equal sessions.

- Read the fraction bar as division from numerator to denominator. - Use whole-number division to identify the mixed-number part and remainder. - In the context, make the numerator a total amount and the denominator the number of equal shares.

1. The fraction \(\frac{29}{4}\) represents \(29\div4\). 2. Since \(29=7\times4+1\), \(\frac{29}{4}=7\frac{1}{4}\). 3. One valid situation is sharing \(29\) yards of material equally among \(4\) displays; each display receives \(7\frac{1}{4}\) yards.

\(\frac{29}{4}=29\div4=7\frac{1}{4}\). One valid context is sharing \(29\) yards equally among \(4\) displays. A valid story must use \(29\) as a total, \(4\) as the number of equal shares, and \(7\frac{1}{4}\) as the amount in each share.

- Express the whole number \(4\) using the same denominator as the quotient fraction. - Compare how many ninths the quotient has with how many ninths make \(4\) wholes. - Connect the denominator back to the divisor in the original division expression.

1. The exact quotient is \(\frac{34}{9}\). 2. Since \(4=\frac{36}{9}\), the amount needed is \(\frac{36}{9}-\frac{34}{9}=\frac{2}{9}\). 3. The denominator remains \(9\) because the quotient is measured in ninths: reaching \(4\) requires \(36\) ninths, and the quotient already contains \(34\) ninths.

\(34\div9=\frac{34}{9}\), and \(\frac{2}{9}\) must be added to reach \(4\). The missing amount is in ninths because the divisor \(9\) sets the size of the fractional parts.

- Compare the dividend with the nearby multiple of \(8\) that would give exactly \(6\) groups. - Think about what a difference of \(1\) in the dividend becomes after division by \(8\). - Keep the reasoning in division and fraction form.

1. The fraction \(\frac{47}{8}\) represents \(47\div8\). 2. Six whole groups of \(8\) would require \(48\), which is \(1\) more than the dividend \(47\). 3. One unit short in the dividend corresponds to \(\frac{1}{8}\) of one group after division by \(8\). Therefore \(\frac{47}{8}\) is \(\frac{1}{8}\) below \(6\).

\(\frac{47}{8}=47\div8\), and the exact quotient is \(\frac{1}{8}\) below \(6\). The dividend is one unit short of \(48=6\times8\), so dividing that one-unit shortfall by \(8\) gives \(\frac{1}{8}\).

- 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) \(z=12\)

- Interpret each fraction bar as a division sign. - For a), translate the fractional part of the mixed quotient into a remainder relative to the divisor. - For b), the remainder is two thirds of the unknown divisor; use that relationship with the four whole divisor groups.

1. For a), \(x\div8=15\frac{3}{4}\). Three fourths of a divisor of \(8\) is \(6\), so the division has quotient \(15\) and remainder \(6\). Thus \(x=15\times8+6=126\). 2. For b), \(210\div y=4\frac{2}{3}\). Imagine dividing one copy of the unknown divisor into \(3\) equal thirds. Four whole divisor groups contain \(12\) thirds, and the extra \(\frac{2}{3}\) makes \(14\) equal thirds altogether. 3. Since those \(14\) thirds total \(210\), one third is \(210\div14=15\). The whole divisor is \(3\times15=45\), so \(y=45\). 4. Check: \(210\div45=4\) remainder \(30\), and \(\frac{30}{45}=\frac{2}{3}\).

a) \(x=126\): \(126\div8=15\) remainder \(6\), and \(\frac{6}{8}=\frac{3}{4}\). b) \(y=45\): \(210\div45=4\) remainder \(30\), and \(\frac{30}{45}=\frac{2}{3}\).