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Multiplying fractions by natural numbers

Example 1

The students complete \latex{\frac{4}{17}} of the hike in one \latex{ hour }.

What fraction of the entire route do they complete in \latex{ 3 } \latex{ hours } if they walk at a uniform speed?

Solution

Make a sketch.

\latex{\frac{4}{17}}

distance travelled
in \latex{ 1 }^st \latex{ hour }

\latex{0}

\latex{\frac{4}{17}}

\latex{\frac{17}{17}}

distance travelled
in \latex{ 2 }^nd \latex{ hour }

distance travelled
in \latex{ 3 }^rd \latex{ hour }

distance travelled in \latex{ 3 } \latex{ hours }

\latex{3\times\frac{4}{17}}

\latex{=}

\latex{\frac{4}{17}}

\latex{+}

\latex{\frac{4}{17}}

\latex{+}

\latex{\frac{4}{17}}

\latex{=}

\latex{\frac{4+4+4}{17}}

\latex{=}

\latex{\frac{3\times4}{17}}

\latex{=}

\latex{\frac{12}{17}}

The students complete \latex{\frac{12}{17}} of the hike in \latex{ 3 } \latex{ hours }.

When multiplying a fraction by a natural number, multiply the numerator by the natural number and leave the denominator unchanged.

Example 2

Perform the following multiplications.

a) \latex{5\times\frac{3}{10}}; b) \latex{\frac{2}{15}\times3}.

Solution 1

a) \latex{5 \times \frac{3}{10} = \frac{5 \times 3}{10} = \frac{15}{10}=\frac{3}{2}};

b) \latex{\frac{2}{15} \times 3 = \frac{2 \times 3}{15} = \frac{6}{15}=\frac{2}{5}}.

Solution 2

Note that when dividing the denominator by the given natural number, you get the same result:

\latex{5 \times \frac{3}{10}=\frac{3}{10\div5}=\frac{3}{2}}

unchanged

\latex{\div5}

\latex{\frac{2}{15}\times 3=\frac{2}{15\div3}=\frac{2}{5}}

unchanged

\latex{\div3}

A fraction can also be multiplied by a natural number by dividing the denominator by the given natural number and leaving the numerator unchanged. This method can only be used if the natural number is a factor of the denominator.

Example 3

Perform the following multiplications.

a) \latex{\frac{7}{24} \times 9}; b) \latex{10 \times \frac{8}{15}}.

Solution

If the denominator of the fraction and the natural number have a common factor, then they can be simplified by the common factor.

a) \latex{\frac{7}{\cancel{24}} \times \cancel{9}=\frac{7}{8} \times 3 = \frac{21}{8} = 2\frac{5}{8}};

\latex{\textcolor{049fe0}{8}}

\latex{\textcolor{049fe0}{3}}

b) \latex{\cancel{10} \times \frac{8}{\cancel{15}} = 2 \times \frac{8}{3} = \frac{16}{3} = 5\frac{1}{3}}.

\latex{\textcolor{049fe0}{3}}

\latex{\textcolor{049fe0}{2}}

Example 4

Multiply \latex{2\frac{1}{5}} by \latex{3}.

Solution

Method 1: \latex{2\frac{1}{5}\times3= \left( 2 + \frac{1}{5}\right) \times 3 = 2 \times 3 + \frac{1}{5} \times 3 = 6 + \frac{3}{5} = 6 \frac{3}{5}}.

Method 2: \latex{2\frac{1}{5}\times3= \left( 2 + \frac{1}{5} \right) \times 3 = \left( \frac{10}{5} + \frac{1}{5} \right) \times 3 = \frac{11}{5} \times 3 = \frac{33}{5} = 6 \frac{3}{5}}.

Mixed numbers can be multiplied by natural numbers in two ways:

– you can multiply the whole number and the fraction by the natural number and then add the products;

– or you can convert the mixed number to fraction form and multiply the fraction by the natural number.

Exercises

{{exercise_number}}. Perform the multiplications. Simplify the result if possible.

a) \latex{\frac{7}{5} \times 4}

b) \latex{\frac{3}{8} \times 7}

c) \latex{\frac{6}{5} \times 4}

d) \latex{\frac{10}{3} \times 2}

e) \latex{3 \times \frac{9}{7}}

f) \latex{4 \times \frac{7}{8}}

g) \latex{\frac{11}{15}\times5}

h) \latex{\frac{4}{10} \times 5}

i) \latex{\frac{7}{72} \times 9}

j) \latex{\frac{3}{4} \times 8}

k) \latex{10 \times \frac{7}{5}}

l) \latex{9 \times \frac{2}{3}}

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

a) \latex{\frac{4}{5} \times a = \frac{28}{5}}

b) \latex{\frac{7}{6}\times b = \frac{35}{6}}

c) \latex{c \times \frac{3}{10} = \frac{3}{2}}

d) \latex{d \times \frac{7}{9} = \frac{7}{3}}

e) \latex{\frac{e}{11} \times 3=\frac{18}{11}}

f) \latex{\frac{f}{6} \times 3=\frac{5}{2}}

g) \latex{g \times \frac{5}{8} = \frac{5}{2}}

h) \latex{9 \times \frac{h}{12} = 3}

{{exercise_number}}. Mum buys \latex{\frac{3}{4}} \latex{ kg } of bread every \latex{ day }. How many \latex{ kilograms } does she buy in \latex{ 6 } \latex{ days ?}

{{exercise_number}}. Each member of a family of \latex{ 4 } drinks \latex{ 2 } and a half \latex{ dl } of milk a \latex{ day }. How many \latex{ decilitres } of milk do they drink in \latex{ 3 } \latex{ days ?}

{{exercise_number}}. A snail moves \latex{\frac{2}{7}} \latex{ m } forwards every \latex{ 10 } \latex{ minutes }. How far does it get in \latex{ 50 } \latex{ minutes ?}

{{exercise_number}}. When painting a housing complex, the painters use \latex{4\frac{2}{3}} buckets of paint a \latex{ day }. How many buckets of paint were used in May if they worked \latex{ 21 } \latex{ days ?}

{{exercise_number}}. On average, Val studies \latex{\frac{5}{6}} \latex{ hours } a \latex{ day }. How many \latex{ hours } does she study in \latex{ 30 } \latex{ days? }

{{exercise_number}}. Perform the multiplications. What do you notice?

a) \latex{\frac{6}{5} \times 5}

b) \latex{\frac{2}{7}\times 7}

c) \latex{\frac{6}{11} \times 11}

d) \latex{\frac{9}{4} \times 4}

e) \latex{2\frac{5}{5} \times 6}

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

a) \latex{\frac{4}{9} \times a = 4}

b) \latex{\frac{10}{7}\times b = 10}

c) \latex{\frac{8}{13} \times c = 8}

d) \latex{\frac{d}{5} \times 5 = 9}

e) \latex{\frac{e}{7} \times 0 = 0}

{{exercise_number}}. Which one is greater?

a) Fifteen times \latex{\frac{7}{5}} or twelve times \latex{\frac{5}{4}}?

b) Five times \latex{\frac{6}{13}} or six times \latex{\frac{5}{13}}?

c) Three times \latex{\frac{15}{21}} or two times \latex{\frac{15}{14}}?

{{exercise_number}}. What is the rule? Write down the next three members of each sequence.

a) \latex{\frac{3}{5}}; \latex{\frac{6}{5}}; \latex{\frac{12}{5}}; ...

b) \latex{\frac{2}{21}}; \latex{\frac{2}{7}}; \latex{\frac{6}{7}}; ...

c) \latex{\frac{3}{10}}; \latex{\frac{3}{5}}; \latex{\frac{6}{5}}; ...

c) \latex{\frac{13}{9}}; \latex{\frac{13}{3}}; \latex{13}; ...

{{exercise_number}}. Perform the multiplications.

a) \latex{1\frac{1}{2}\times 2}

b) \latex{4\frac{1}{5}\times 5}

c) \latex{2\frac{5}{9}\times 7}

d) \latex{10\frac{2}{3}\times 6}

e) \latex{7\frac{3}{10}\times 6}

f) \latex{5\frac{6}{11}\times 3}

{{exercise_number}}. Convert the following units of measurement. Simplify the result if possible.

a) \latex{\frac{1}{5}} \latex{ m = } ...... \latex{ dm }

b) \latex{\frac{3}{2}} \latex{ m = } ...... \latex{ cm }

c) \latex{\frac{6}{10}} \latex{ dkg = } ...... \latex{ g }

d) \latex{\frac{49}{100}} \latex{ kg = } ...... \latex{ dkg }

e) \latex{\frac{1}{25}} \latex{ km = } ...... \latex{ m }

f) \latex{\frac{5}{8}} \latex{ kg = } ...... \latex{ g }

Quiz

\latex{\frac{5}{9}} part of every side of the cube has been painted. What fraction of the surface of the cube is painted?

Mathematics 5.

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