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Adding and subtracting fractions with the same denominator

A pendulum bowling player knocked down two-ninths of the skittles with the first swing, three-ninths with the second swing, and four-ninths with the third swing.

\latex{\frac{2}{9}+\frac{3}{9}+\frac{4}{9}=\frac{9}{9}=1}

Example 1

Andrew’s birthday cake was cut into \latex{ 16 } equal slices. Andrew ate one, Eni ate three and Sam ate five slices. What fraction of the cake did the children eat?

Solution

Divide the cake into \latex{ 16 } equal slices, thus one slice is \latex{\frac{1}{16}} part of the cake. Use different colours to indicate the slices eaten by the children.

nine-sixteenths

one-sixteenths

three-sixteenths

five-sixteenths

\latex{\frac{1+3+5}{16}=\frac{9}{16}}

\latex{\frac{1}{16}}

\latex{\frac{3}{16}}

\latex{\frac{5}{16}}

The children ate \latex{\frac{9}{16}} of the cake.

\latex{\frac{1}{16}+\frac{3}{16}+\frac{5}{16}=\frac{1+3+5}{16}=\frac{9}{16}}

Fractions with the same denominator can be added by adding the numerators. The sum will be the numerator, while the denominator remains unchanged.

Example 2

The children ate \latex{\frac{9}{16}} of the cake.

a) What fraction of the cake is left?
b) Later, other guests arrived and ate \latex{ 4 } more slices.
What fraction of the cake was left after the new guests ate?

Solution

Use drawings to show the subtraction.

\latex{\frac{16-9}{16}=\frac{7}{16}}

\latex{\frac{16}{16}}

\latex{\frac{9}{16}}

a)

\latex{\frac{7}{16}} of the cake was left.

\latex{\frac{7-4}{16}=\frac{3}{16}}

\latex{\frac{7}{16}}

\latex{\frac{4}{16}}

b)

\latex{\frac{3}{16}} of the cake was left after the guests ate.

\latex{\frac{7}{16}-\frac{4}{16}=\frac{7-4}{16}=\frac{3}{16}}

When subtracting fractions with the same denominator, the numerator of the subtrahend is subtracted from the numerator of the minuend. This will be the numerator of the difference, and the common denominator will become the denominator.

When adding or subtracting fractions with the same denominator, the calculations are done with the same parts. It is enough to do the calculations with the numerators; the denominator remains unchanged.

Exercises

{{exercise_number}}. Perform the operations.

a) \latex{\frac{3}{7}+\frac{2}{7}}

b) \latex{\frac{6}{7}-\frac{4}{7}}

c) \latex{\frac{5}{11}+\frac{3}{11}+\frac{1}{11}}

d) \latex{\frac{10}{11}-\frac{7}{11}}

e) \latex{\frac{8}{21}+\frac{5}{21}+\frac{6}{21}}

f) \latex{\frac{18}{21}-\frac{7}{21}}

g) \latex{\frac{7}{9}+\frac{13}{9}-\frac{20}{9}}

h) \latex{\frac{9}{10}-\frac{5}{10}+\frac{6}{10}}

{{exercise_number}}. Perform the operations.

a) \latex{1+\frac{2}{3}}

b) \latex{1+\frac{7}{9}}

c) \latex{1+\frac{11}{18}+\frac{3}{18}}

d) \latex{\frac{2}{25}+1+\frac{7}{25}}

e) \latex{\frac{8}{30}+\frac{13}{30}+1}

f) \latex{1-\frac{13}{18}}

g) \latex{1-\frac{2}{15}-\frac{7}{15}}

h) \latex{1+\frac{12}{17}-\frac{8}{17}}

{{exercise_number}}. Which one is greater? By how much?

a) \latex{\frac{11}{9}\;or\;\frac{2}{9}}

b) \latex{\frac{7}{13}\;or\;\frac{10}{13}}

c) \latex{\frac{11}{10}\;or\;1\frac{5}{10}}

d) \latex{2\frac {3}{10}\;or\;\frac{19}{10}}

{{exercise_number}}. How much do you have to add to \latex{\frac{3}{7}} to get

a) \latex{\frac{5}{7};}

b) \latex{1;}

c) \latex{\frac{3}{7};}

d) \latex{\frac{19}{7}?}

{{exercise_number}}.

a) How much less is \latex{\frac{3}{10}} than \latex{\frac{7}{10}?}

b) How much larger is \latex{ 1 } than \latex{\frac{3}{10}?}

c) How much larger is \latex{\frac{8}{5}} than \latex{1?}

d) How much less is \latex{1} than \latex{\frac{11}{8}?}

{{exercise_number}}. What numbers should replace the symbols?

a)

b)

c)

\latex{ \frac{3}{8} }

\latex{ \frac{10}{8} }

\latex{ \frac{5}{8} }

\latex{ \frac{5}{7} }

\latex{ \frac{12}{7} }

\latex{ \frac{9}{7} }

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

\latex{ \frac{18}{9} }

\latex{ 1 }

\latex{ + }

\latex{ - }

{{exercise_number}}.

a) The sum of two fractions is \latex{\frac{13}{15}}. One of the fractions is \latex{\frac{4}{15}}. What is the other?
b) Which number is \latex{\frac{7}{8}} greater than \latex{\frac{5}{8}} ?
c) The difference of two numbers is \latex{\frac{5}{9}}. The subtrahend is \latex{\frac{3}{9}}. What is the minuend?
d) The difference of two numbers is \latex{\frac{4}{13}}. The minuend is \latex{\frac{11}{13}}. What is the subtrahend?

{{exercise_number}}. Write the following mixed numbers in fraction form.

a) \latex{4\frac{5}{6}}

b) \latex{7\frac{2}{3}}

c) \latex{10\frac{3}{5}}

d) \latex{8\frac{2}{9}}

\latex{15\frac{1}{4}}

\latex{23\frac{5}{7}}

{{exercise_number}}. Write the following fractions in mixed number form.

a) \latex{\frac{17}{3}}

b) \latex{\frac{10}{4}}

c) \latex{\frac{35}{8}}

d) \latex{\frac{13}{5}}

e) \latex{\frac{29}{7}}

f) \latex{\frac{400}{11}}

{{exercise_number}}. Which two fractions are equal?

a) \latex{2\frac{2}{3};1\frac{5}{6}}

b) \latex{12\frac{7}{8};1\frac{7}{6}}

c) \latex{3\frac{1}{2};\frac{63}{18}}

d) \latex{5\frac{1}{3};\frac{32}{6}}

{{exercise_number}}. How many \latex{ kilometres } did we walk during a \latex{ 3 }\latex{ -hour } tour if in the first \latex{ hour }, we walked

\latex{4\frac{3}{20}} \latex{ km }, in the second, \latex{3\frac{7}{20}} \latex{ km }, while in the third \latex{ hour, } \latex{1\frac{17}{20}} \latex{ km ?}

{{exercise_number}}. Perform the additions.

a) \latex{3+\frac{2}{7}+1}

b) \latex{2\frac{3}{4}+1\frac{1}{4}}

c) \latex{3\frac{4}{7}+ 2\frac{6}{7}}

d) \latex{2\frac{3}{8}+3\frac{5}{8}+4\frac{7}{8}}

{{exercise_number}}. Perform the operations.

a) \latex{2\frac{3}{11}+3\frac{5}{11}-1\frac{10}{11}}

b) \latex{7\frac{3}{4}+2\frac{3}{4}-6\frac{1}{4}}

c) \latex{5\frac{9}{20}-3\frac{11}{20}+10\frac{1}{20}}

{{exercise_number}}. The image shows how much paint was left in the three \latex{ 1 }\latex{ -litre } buckets. How many \latex{ litres } of paint is still left? (→)

{{exercise_number}}. A carpet is \latex{2\frac{3}{5}} \latex{ m } wide and \latex{3\frac{3}{5}} \latex{ m } long. How many \latex{ metres } of hem have I sewed if I have finished half of the carpet?

{{exercise_number}}. Zack is shovelling snow from a \latex{ 12 } \latex{ m }-long pavement. How many \latex{ metres } more does he have to go if he has cleaned \latex{5\frac{1}{4}} \latex{ m }?

{{exercise_number}}. \latex{ 20 } \latex{ litres } of lemonade were made. The guests drank \latex{\frac{18}{5}} \latex{litres }. How many \latex{ litres } of lemonade was left?

{{exercise_number}}. Write two additional members of each series. If possible, convert the fractions to the mixed number form.

a) \latex{\frac{3}{5};\frac{6}{5};\frac{9}{5};\frac{12}{5}...}

b) \latex{1\frac{1}{3};2\frac{2}{3};4;5\frac{1}{3}...}

c) \latex{10\frac{3}{4};9;7\frac{1}{4};5\frac{2}{4}...\;}

Quiz

You pour \latex{\frac{3}{5}} \latex{litres} of water, then \latex{\frac{4}{5}} \latex{litres} into a container with a volume of \latex{ 1 } \latex{litre}. How many \latex{litres} of water are in the container?

Mathematics 5.

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