cadaVR anatomy by Mozaik Education

Dividing by a fraction

Example 1

How many people can share \latex{ 3 } pizzas if each person is given

\latex{ 1 };

\latex{\frac{1}{2}};

\latex{\frac{1}{5}} pizza?

Solution 1

The number of resulting slices can be seen in the image above. The same can be calculated using division:

a) \latex{3 \div 1 = 3}; b) \latex{3 \div \frac{1}{2}=6}; c) \latex{3 \div \frac{1}{5} = 15}.

Solution 2

Based on the properties of the quotient, it can be confirmed that if the dividend is unchanged and the factor is reduced to its half or one fifth part, then the quotient will be multiplied by two or five, respectively.

b) \latex{\quad3 \div 1 = 3}

\latex{\div2}

\latex{\times 2}

\latex{\downarrow}

\latex{3 \div \frac{1}{2} = 3 \times 2}

c) \latex{\quad3 \div 1 = 3}

\latex{\quad3 \div \frac{1}{5} = 3 \times 5}

\latex{\downarrow}

\latex{\div5}

\latex{\times 5}

If the pizzas are left whole, \latex{ 3 } people can share, if they are cut in half, then \latex{ 6 }, and if they are cut in five, \latex{ 15 } people can share.

Division by \latex{\frac{1}{2}} means multiplication by \latex{2}, while division by \latex{\frac{1}{5}} means multiplication by \latex{5}.

Example 2

We poured four \latex{ litres } of peach juice into

\latex{1}-\latex{ litre } bottles;
\latex{\frac{1}{3}}-\latex{ litre } glasses;
\latex{\frac{2}{3}}-\latex{ litre } jugs.

How many bottles, glasses and jugs were filled?

Solution

bottles

glasses

jugs

Based on the figure:

Based on the changes of the quotient:

number of bottles: \latex{4 \div 1 = 4}.

number of glasses: \latex{4 \div \frac{1}{3} = 4\times 3 = 12}.

number of jugs: \latex{4 \div \frac{2}{3}= 6}.

\latex{4 \div 1 = 4}

\latex{4 \div \frac{1}{3} = 4 \times 3}

\latex{4 \div \frac{2}{3} = 4 \times \frac{3}{2}}.

\latex{\downarrow}

\latex{\div 3}

\latex{\times3}

\latex{\div 2}

\latex{\times 2}

Division by \latex{\frac{2}{3}} means multiplication by \latex{\frac{3}{2}}. The reciprocal of \latex{\frac{2}{3}} is \latex{\frac{3}{2}}.

Example 3

Perform the following divisions.

\latex{3 \div \frac{1}{5}}

\latex{3 \div \frac{4}{5}}

\latex{\frac{3}{7} \div \frac{4}{5}}

Solution

In Example 1, you have seen that \latex{3 \div \frac{1}{5}=3\times 5= 15}.
Calculate on the basis of the changes of the quotient.

\latex{3 \div \frac{1}{5}=3 \times5}

\latex{3 \div \frac{4}{5}= \left(3 \times5\right)\div4=\frac{3 \times 5}{4}=3\times\frac{5}{4}=\frac{15}{4}}

\latex{3 \div {\color{009ed5}{\frac{4}{5}}}=3 \times {\color{009ed5}{\frac{5}{4}}}}

\latex{\downarrow}

\latex{\times4}

\latex{\div 4}

the factor is
multiplied by \latex{ 4 }

the quotient is divided by \latex{ 4 }

Calculate on the basis of exercise b) and the changes of the quotient.

\latex{3 \div \frac{4}{5}=3 \times\frac{5}{4}}

\latex{\frac{3}{7} \div \frac{4}{5}= \frac{3}{7}\times \frac{5}{4}=\frac{15}{28}}

\latex{\frac{3}{7} \div {\color{009ed5}{\frac{4}{5}}}=\frac{3}{7} \times {\color{009ed5}{\frac{5}{4}}}}

\latex{\downarrow}

\latex{\div7}

\latex{\div 7}

the dividend is
divided by \latex{ 7 }

the quotient is divided by \latex{ 7 }

Dividing by a fraction can be accomplished by multiplying the dividend by the reciprocal of the factor.

When dividing by a fraction, simplification can be performed after the division has been transformed into a multiplication by the reciprocal of the factor.

\latex{\frac{12}{5}\div\frac{15}{4}=\frac{12}{5}\times\frac{4}{15}=\frac{\overset{4}{\bcancel{12}}}{5}\times\frac{4}{\underset{5}{\bcancel{15}}}=\frac{16}{25}}

If there is a mixed fraction in the division, convert it to an improper fraction first.

\latex{2\frac{1}{4}\div1\frac{3}{5}=\frac{9}{4}\div\frac{8}{5}=\frac{9}{4}\times\frac{5}{8}=\frac{45}{32}=1\frac{13}{32}}

If there is a negative fraction in the division, the sign of the quotient is determined the same way as in the case of integers.

\latex{\frac{4}{5}\div\left(-\frac{2}{3}\right)=\frac{4}{5}\times\left(-\frac{3}{2}\right)=-\left(\frac{\overset{2}{\bcancel{4}}}{5} \times \frac{3}{\underset{1}{\bcancel{2}}}\right)=-\frac{6}{5}=-1\frac{1}{5}}

\latex{\left(-\frac{7}{8}\right)\div\left(-\frac{3}{4}\right) = \left(-\frac{7}{8}\right)\times\left(-\frac{4}{3}\right)=\frac{7}{\underset{2}{\bcancel{8}}}\times \frac{\overset{1}{\bcancel{4}}}{3}=\frac{7}{6} = 1\frac{1}{6}}

Exercises

{{exercise_number}}. Perform the following divisions, then check your answers.

\latex{14 \div \frac{7}{12}}

\latex{14 \div \frac{12}{7}}

\latex{14 \div 1\frac{5}{7}}

\latex{\frac{28}{2} \div \frac{7}{12}}

\latex{\left(-5\right)\div \frac{3}{4}}

\latex{7\div \left(-\frac{2}{5}\right)}

\latex{\left(-\frac{5}{7}\right) \div \left(-\frac{5}{7}\right)}

\latex{\frac{5}{13}\div \frac{13}{5}}

{{exercise_number}}. Calculate the quotients, then check your answers.

\latex{\frac{5}{7} \div 1\frac{2}{3}}

\latex{\frac{6}{11} \div 4\frac{2}{7}}

\latex{\frac{30}{49} \div 4\frac{2}{7}}

\latex{\left(-\frac{7}{10}\right) \div 2\frac{4}{5}}

\latex{4\frac{1}{4}\div 8\frac{1}{2}}

\latex{\left(-9\frac{1}{2}\right)\div \left(-2\frac{1}{9}\right)}

\latex{\left(-7\frac{4}{5}\right) \div 1\frac{3}{10}}

\latex{5\frac{1}{8}\div \left(-8\frac{1}{5}\right)}

{{exercise_number}}. Add the quotients of the divisions written on the wagons to find out the number each train carries.

\latex{\frac{2}{5}\div\frac{2}{3}}

\latex{\frac{4}{9}\div4}

\latex{\frac{3}{2}\div\frac{1}{6}}

\latex{\frac{3}{2}\div\frac{3}{11}}

\latex{2\div\frac{5}{2}}

\latex{\frac{1}{2}\div \frac{5}{3}}

\latex{\frac{23}{50}\div\frac{2}{25}}

\latex{\frac{3}{46}\div\frac{5}{23}}

{{exercise_number}}. Use the following numbers to write as many divisions as you can. Make sure that the dividend, the factor and even the quotient is chosen from the group.

\latex{\frac{9}{8}}; \latex{\quad-4}; \latex{\quad\frac{5}{2}}; \latex{\quad 4}; \latex{\quad\frac{6}{5}}; \latex{\quad-\frac{5}{8}}; \latex{\quad\frac{3}{2}}

{{exercise_number}}. Vince and Carl have been cycling for an \latex{ hour } and a half and are now at \latex{\frac{3}{4}} of their trip. What length of their trip did they complete in the first \latex{ hour? } If they continue at the same speed, how much time will they need to arrive at their destination?

{{exercise_number}}. What numbers should replace the symbols to make the equalities true?

\latex{\frac{2}{3} \div \triangle = \frac{4}{3}}

\latex{\left(-\frac{3}{5}\right) \div \triangle =\frac{9}{10}}

\latex{\frac{5}{8} \div \triangle = \frac{3}{14}}

\latex{\left(-1\frac{4}{5}\right) \div \triangle = \frac{5}{6}}

\latex{\frac{3}{4}\div \triangle=2\frac{1}{2}}

\latex{\frac{21}{25} \div \triangle = 2\frac{9}{20}}

{{exercise_number}}.

What is the quotient of \latex{\left(-6\frac{1}{4}\right)} divided by \latex{4\frac{1}{6}}?
By what number does \latex{4\frac{1}{6}} need to be multiplied to get \latex{\left(-6\frac{1}{4}\right)}?
By what number does \latex{\left(-6\frac{1}{4}\right)} need to be multiplied to get \latex{4\frac{1}{6}}?
If multiplied by \latex{4\frac{1}{6}} what number will result in \latex{\left(-6\frac{1}{4}\right)}?

{{exercise_number}}. The area of a rectangle is \latex{25\,m^2}. Calculate the perimeter of the rectangle if one of its sides is \latex{8\frac{1}{2}\,m} long.

{{exercise_number}}. Perform the following divisions. In each case, determine whether the dividend or the quotient is greater.

\latex{\frac{2}{3} \div \frac{4}{5}}; \latex{\quad\frac{7}{8} \div \frac{1}{2}}; \latex{\quad\frac{2}{9} \div \frac{2}{3}}

\latex{\frac{2}{3} \div \frac{3}{2}}; \latex{\quad1\frac{1}{3} \div \frac{4}{3}}; \latex{\quad\frac{9}{4} \div \frac{5}{4}}

Quiz

A number was first multiplied by \latex{\frac{3}{4}}, then divided by \latex{\frac{3}{5}}. Which of the following operations is equivalent to those stated here?

division by \latex{\frac{4}{3}}

division by \latex{\frac{9}{20}}

multiplication by \latex{\frac{9}{20}}

multiplication by \latex{\frac{5}{4}}

division by \latex{\frac{5}{4}}

Mathematics 6.

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