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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Fractions (revision)
Operations with fractions (revision)
Negative fractions
Multiplying fractions by fractions
The reciprocals of numbers
Division by a fraction
Properties of operations
Rational numbers
Mixed exercises
Proportion
Percentages
Probability, statistics
Supplementary material
Multiplying fractions by fractions
Example 1
Joe is putting tiles on the kitchen wall. The tiles are rectangular, and their sides measure \latex{\frac{1}{5}} and \latex{\frac{1}{4}} \latex{ m } in length.
  1. How many \latex{square \,metres} does one tile cover?
  2. As decoration, Joe made coloured rectangles with sides measuring  \latex{\frac{2}{5}} and \latex{\frac{3}{4}\, m} in length. How many \latex{ square \;metres } is the coloured area?
Solution
  1. The tiles are rectangular. The area of a rectangle is calculated by multiplying the lengths of adjacent sides.
The area of one tile is:
\latex{\frac{1}{5}{\,m} \times\frac{1}{4}{\,m}= \frac{1}{5} \times\frac{1}{4}{\,m}^2}.
Based on the image, the area of \latex{ 20 } tiles is 
\latex{1 \,{m}^2}, thus
the area of \latex{ 1 } tile is \latex{\frac{1}{20}\,{m}^2}.
So, \latex{\frac{1}{5}\times{\frac{1}{4}={\frac{1\times 1}{5\times 4}=\frac{1}{20}}}}.
This means that one tile covers an area of \latex{\frac{1}{20}\,{m}^2}.
\latex{\frac{1}{5}\,m}
\latex{\frac{1}{4}\,m}
\latex{ 5 } rows
\latex{ 4 } columns
\latex{1\,m^2}
Based on the changes
of the product:
 
\latex{\frac{1}{5}\times 1= \frac{1}{5}}
\latex{\downarrow\div4\downarrow\div4}
\latex{\frac{1}{5}\times\frac{1}{4}= \frac{1}{5}\div4 =}
\latex{=\frac{1}{5\times 4}}
  1. The area of the coloured area is
\latex{\frac{2}{5}\,{m}\times \frac{3}{4}\,{m}= \frac{2}{5} \times \frac{3}{4}\,{m}^2}.
The area of \latex{ 1 } tile is \latex{\frac{1}{20}\,{m}^2},
so that of \latex{ 6 } tiles is \latex{\frac{6}{20}\,{m}^2}.
So \latex{\frac{2}{5}\times \frac{3}{4}= \frac{2\times3}{5\times4}= \frac{6}{20}}.
The area of the coloured tiles is \latex{\frac{6}{20}\,{m}^2}.
\latex{\frac{3}{4}\,m}
\latex{\frac{2}{5}\,m}
\latex{1\,m^2}
Based on the
changes of the
product:
 
\latex{\frac{2}{5}\times 1= \frac{2}{5}}
\latex{\downarrow \space \div4}
\latex{\frac{2}{5}\times\frac{1}{4}= \frac{2}{5\times 4}}
\latex{\downarrow \times3}
\latex{\frac{2}{5}\times \frac{3}{4}=\frac{2\times3}{5\times 4}}
The fraction \latex{\frac{6}{20}} can be simplified: \latex{\frac{6}{20}=\frac{3}{10}} .
Simplification can be done while calculating:
\latex{\frac{\overset{1}{\cancel{2}}}{5} \times \frac{3}{\underset{2}{\cancel{4}}} = \frac{1}{5}\times\frac{3}{2}=\frac{1\times3}{5\times2}=\frac{3}{10}} .
When multiplying fractions, the product of the numerators is divided by the product of the denominators. (If possible, simplify the fractions before multiplying.)
\latex{\frac{2}{3}\times\frac{5}{7} = \frac{2\times5}{3\times 7}}
After grouping, it can be demonstrated that when multiplying more than two fractions, the result is still the quotient of the product of the numerators and that of the denominators:
  1. \latex{\frac{2}{3}\times\frac{5}{7}\times\frac{8}{9}=\left(\frac{2}{3}\times\frac{5}{7}\right)\times\frac{8}{9}=\frac{2\times5}{3\times7}\times\frac{8}{9}=\frac{2\times5\times8}{3\times7\times9}=\frac{80}{189}}
If possible, simplify before multiplying.
  1. \latex{\frac{5}{9}\times \frac{3}{4}\times \frac{8}{5}=\frac {{\overset{1}{\bcancel{5}}\times \overset{1}{\cancel{3}}}\times\overset{2}{\cancel{8}} } {\underset{{3}}{\cancel{9}}\times{\underset{{1}}{\cancel{4}}\times{\underset{{1}}{\bcancel{5}}}}}=\frac{2}{3}}  or like this: \latex{\frac{\overset{1}{\cancel{5}} }{\underset{3}{\cancel{9}}}\times\frac{\overset{1}{\cancel{3}} }{\underset{1}{\bcancel{4}}}\times \frac{\overset{2}{\bcancel{8}} }{\underset{1}{\cancel{5}}}=\frac{2}{3}}
Make sure the simplification is clear.
  1. \latex{\frac{7}{9}\times\frac{6}{5}\times \frac{3}{14} =\frac{\overset{1}{\cancel{7}} }{\underset{3}{\bcancel{9}}}\times\frac{6}{5}\times \frac{\overset{1}{\bcancel{3}} }{\underset{2}{\cancel{14}}}=\frac{6}{3\times5\times2}=\frac{\overset{1}{\cancel{6}} }{{\underset{1}{\cancel{6}}}\times5} = \frac{1}{5}}
Is it true?
The product of two
fractions is less than
the smaller fraction.
673645
If there is a negative fraction among the factors, determine the sign of the
product first.
 
  1. \latex{\frac{4}{5}\times\left(-\frac{2}{3}\right)=\frac{4\times(-2)}{5\times3}=\frac{-8}{15}=-\frac{8}{15}}    or    \latex{\frac{4}{5}\times\left(-\frac{2}{3}\right)=-\frac{4\times2}{5\times3}=-\frac{8}{15}};
  1. \latex{\left(-\frac{\overset{1}{\cancel{3}}}{\underset{1}{\cancel{7}}} \right)\times \frac{\overset{3}{\cancel{21}}}{5}\times\left(-\frac{2}{\underset{1}{\cancel{3}}}\right)= \frac{1\times3\times2}{1\times5\times1}=\frac{6}{5}=1\frac{1}{5}}
Exercises
{{exercise_number}}. Perform the multiplications.
  1. \latex{\frac{5}{6}\times7}
  1. \latex{\frac{2}{21}\times4}
  1. \latex{6\times\frac{7}{11}}
  1. \latex{(-5)\times\frac{2}{3}}
  1. \latex{-6\times\left(-\frac{2}{5}\right)}
{{exercise_number}}. Simplify before multiplying.
  1. \latex{7\times\frac{5}{14}}
  1. \latex{\frac{5}{6}\times(-4)}
  1. \latex{\frac{7}{5}\times\frac{15}{21}}
  1. \latex{\left(-\frac{13}{27}\right)\times\frac{18}{26}}
  1. \latex{\left(-\frac{108}{375}\right)\times\frac{125}{72}}
{{exercise_number}}. Complete the multiplications to make the equalities true.
  1. \latex{\frac{4}{7}\times\frac{5}{9}=\frac{20}{\Box}}
  1. \latex{\frac{\Box}{9}\times\frac{4}{7}=\frac{8}{63}}
  1. \latex{\frac{6}{13}\times\frac{\Box}{5}=\frac{54}{65}}
  1. \latex{\frac{\Box}{8}\times\frac{12}{25}=\frac{3}{10}}
  1. \latex{\frac{14}{27}\times\frac{3}{\Box}=\frac{7}{18}}
  1. \latex{\frac{15}{\Box}\times\frac{13}{12}=\frac{5}{8}}
{{exercise_number}}. Compare the magnitude of the first factor and the product. What do you notice?
  1. \latex{8\times\frac{5}{4}}
    \latex{8\times\frac{4}{4}}
    \latex{8\times\frac{3}{4}}
    \latex{8\times\frac{2}{4}}
  1. \latex{\frac{2}{3}\times\frac{5}{2}}
    \latex{\frac{2}{3}\times\frac{3}{2}}
    \latex{\frac{2}{3}\times\frac{2}{2}}
    \latex{\frac{2}{3}\times\frac{1}{2}}
  1. \latex{\left(-\frac{8}{9}\right)\times\frac{11}{8}}
    \latex{\left(-\frac{8}{9}\right)\times\frac{9}{8}}
    \latex{\left(-\frac{8}{9}\right)\times\frac{8}{8}}
    \latex{\left(-\frac{8}{9}\right)\times\frac{3}{8}}
  1. \latex{\left(-\frac{8}{9}\right)\times\left(-\frac{11}{8}\right)}
    \latex{\left(-\frac{8}{9}\right)\times\left(-\frac{9}{8}\right)}
    \latex{\left(-\frac{8}{9}\right)\times\left(-\frac{8}{8}\right)}
    \latex{\left(-\frac{8}{9}\right)\times\left(-\frac{3}{8}\right)}
{{exercise_number}}. Perform the following multiplications. Simplify before multiplying when possible.
  1. \latex{\frac{2}{3}\times\frac{3}{4}\times\frac{7}{5}}
  1. \latex{\left(-\frac{5}{6}\right)\times\left(-\frac{9}{10}\right)\times(-2)}
  1. \latex{\frac{7}{5}\times\left(-\frac{3}{7}\right)\times\left(-\frac{5}{9}\right)}
  1. \latex{\frac{9}{5}\times\frac{5}{9}\times\left(-\frac{7}{9}\right)\times\frac{6}{7}}
{{exercise_number}}. Perform the multiplications in the simplest way possible.
  1. \latex{\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times\frac{5}{6}}
  1. \latex{\frac{4}{5}\times\frac{1}{2}\times\frac{2}{3}\times\frac{1}{2}\times\frac{3}{4}}
  1. \latex{\frac{45}{90}\times\frac{12}{6}\times\frac{1}{3}\times\frac{48}{16}}
{{exercise_number}}. Tori weaves carpets. During a workday, she can make \latex{\frac{3}{8}} of a carpet. Can she make \latex{ 8 } carpets in \latex{ 20 } days at the same pace?
{{exercise_number}}. Perform the multiplications by converting the mixed numbers to improper fractions.
  1. \latex{1\frac{3}{4}\times8}
  1. \latex{5\times5\frac{3}{10}}
  1. \latex{2\frac{1}{2}\times\frac{1}{9}}
  1. \latex{\frac{3}{8}\times1\frac{7}{9}}
  1. \latex{4\frac{4}{5}\times2\frac{2}{9}}
  1. \latex{2\frac{1}{7}\times7\frac{1}{2}}
{{exercise_number}}. Determine the rule and write down the next four terms of each sequence using multiplication.
  1. \latex{6; 3; \frac{3}{2} ...}
  1. \latex{3\frac{1}{2} ;1\frac{3}{4}; \frac{7}{8}; ...}
  1. \latex{\frac{1}{2} ;\frac{1}{4}; \frac{1}{8}; ...}
  1. \latex{\frac{3}{5} ;\frac{1}{5}; \frac{1}{15}; ...}
  1. \latex{1\frac{4}{8} ;\frac{1}{2}; \frac{1}{6}; ...}
  1. \latex{25; 10; 4; ...}
{{exercise_number}}. How many \latex{ litres } of water can a fish tank hold if its edges are \latex{ 50\,{ cm}}, \latex{\frac{7}{20}\, m} and \latex{\frac{2}{5}\, m} long? How many \latex{ square \;metres } of glass are needed to manufacture it? (Ignore the thickness of the glass.)
{{exercise_number}}. The lengths of a rectangle's sides are \latex{3\frac{1}{5}} and \latex{\frac{7}{20}\,m}. How many \latex{ square\, metres } is its area?
{{exercise_number}}. On a class trip, the students went hiking at \latex{ 9 } AM. How many \latex{ kilometres } did they walk until lunch if they covered a distance of \latex{4\frac{1}{2}\,km} \latex{ per\,hour} on average and had a lunch break at \latex{ 12:45 } PM? After lunch, they walked for another \latex{ hour } \latex{ and } \latex{ a } \latex{ half } at a speed of \latex{5\frac{1}{4}\,km} \latex{ per\,hour}. How many \latex{ kilometres } did they walk in total?
Quiz
The fractions marked by points \latex{ P } and \latex{ R } are multiplied. Does point \latex{ M, S, N, } or \latex{ T } mark their product?
\latex{ M }
\latex{ N }
\latex{ P }
\latex{ R }
\latex{ S }
\latex{ T }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
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