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Adding and subtracting fractions with different denominators
Example 1
Two types of flowers were planted on the main square of the city. Tulips covered \latex{\frac{2}{{5}}} of the area, while begonias covered \latex{\frac{1}{{2}}}.
a) What fraction of the available area was planted with flowers?
b) What fraction lacks flowers?
Solution (a)
\latex{\frac{2}{5} +\frac{1}{2}=\; ?}
Only fractions with the same denominator can be added, so find a common denominator. In this case, you need to expand them.
Since \latex{\frac{2}{5} =\frac{4}{10}} and \latex{\frac{1}{2} =\frac{5}{10}}, thus \latex{\frac{2}{5} +\frac{1}{2}=\frac{4}{10} +\frac{5}{10}=\frac{9}{10} }.
\latex{\frac{\text{9}}{\text{10}}} of the available area was planted with flowers.
Since \latex{\frac{2}{5} =\frac{4}{10}} and \latex{\frac{1}{2} =\frac{5}{10}}, thus \latex{\frac{2}{5} +\frac{1}{2}=\frac{4}{10} +\frac{5}{10}=\frac{9}{10} }.
\latex{\frac{\text{9}}{\text{10}}} of the available area was planted with flowers.
Visualising the addition:
\latex{\frac{2}{5}}
\latex{+}
\latex{\frac{1}{2}}
\latex{=}
\latex{\frac{4}{10}}
\latex{+}
\latex{\frac{5}{10}}
\latex{=}
\latex{\frac{9}{10}}
Solution (b)
The drawing shows that one-tenth of the area can still be planted with flowers. Calculations also prove this.
Since \latex{1=\frac{10}{10}}, thus \latex{1-\frac{9}{10}=\frac{10}{10}-\frac{9}{10}=\frac{10 - 9}{10}=\frac{1}{10}}.
Example 2
What is the result of \latex{\frac{8}{10} - \frac{3}{5} ?}
Solution 1
Note that \latex{\frac{8}{10} = \frac{4}{5} }, so you can use simplification to find a common denominator.
Subtraction: \latex{\frac{8}{10} - \frac{3}{5} = \frac{4}{5} - \frac{3}{5} = \frac{1}{5} }.
Solution 2
Of course, you can also use expansion to find a common denominator.
Since \latex{\frac{3}{5} =\frac{6}{10} }, thus \latex{\frac{8}{10} -\frac{3}{5} = \frac{8}{10} -\frac{6}{10} = \frac{8-6}{10}=\frac{2}{10}}.
The result can be simplified: \latex{\frac{2}{10} =\frac{1}{5} }.
expansion
simplification
\latex{\frac{6}{10}}
\latex{\frac{8}{10}}
\latex{\frac{3}{5}}
\latex{\frac{4}{5}}
\latex{\frac{1}{5}}
When adding or subtracting fractions with different denominators, find a common denominator first, then perform the calculations.
The addition of mixed numbers is similar to adding fractions with different denominators.
Add the wholes first, then the fractions, then take the sum of these.
\latex{3\frac{1}{2} + 1\frac{2}{3} = \left(3+\frac{1}{2}\right) + \left(1+\frac{2}{3}\right) = 3+1+\frac{1}{2}+\frac{2}{3} = 4+\frac{3}{6}+\frac{4}{6}=4\frac{7}{6} = 5\frac{1}{6} }
You can also convert mixed numbers to fraction form and then perform the addition.
\latex{3\frac{1}{2} + 1\frac{2}{3} = \frac{7}{2} +\frac{5}{3} = \frac{21}{6}+\frac{10}{6}=\frac{31}{6} = 5\frac{1}{6} }
Exercises
{{exercise_number}}. Perform the additions.
a)\latex{\; \frac{1}{2} +\frac{2}{3}}
b)\latex{\; \frac{2}{5} +\frac{3}{4}}
c)\latex{\; \frac{2}{3} +\frac{3}{4}}
d)\latex{\; \frac{4}{5} +\frac{1}{2}}
e)\latex{\; \frac{3}{4} +\frac{4}{7}}
f)\latex{\; \frac{4}{5} +\frac{3}{8}}
g)\latex{\; \frac{5}{7} +\frac{1}{3}}
h)\latex{\; \frac{2}{3} +\frac{3}{4}}
{{exercise_number}}. Perform the subtractions.
a)\latex{\; \frac{1}{2} -\frac{1}{4}}
b)\latex{\; \frac{2}{3} -\frac{1}{6}}
c)\latex{\; \frac{3}{4} -\frac{5}{8}}
d)\latex{\; \frac{2}{3} -\frac{5}{9}}
e)\latex{\; \frac{3}{4} -\frac{7}{12}}
f)\latex{\; \frac{4}{5} -\frac{11}{20}}
g)\latex{\; \frac{5}{8} -\frac{7}{40}}
h)\latex{\; \frac{5}{6} -\frac{9}{24}}
{{exercise_number}}. Perform the operations.
a)\latex{\; \frac{1}{4} +\frac{1}{6}}
b)\latex{\; \frac{5}{6} -\frac{3}{4}}
c)\latex{\; \frac{2}{9} +\frac{1}{6}}
d)\latex{\; \frac{7}{10} +\frac{8}{15}}
e)\latex{\; \frac{5}{6} -\frac{3}{8}}
f)\latex{\; \frac{7}{12} +\frac{5}{8}}
g)\latex{\; \frac{7}{9} -\frac{5}{12}}
h)\latex{\; \frac{7}{20} -\frac{7}{30}}
{{exercise_number}}. Perform the operations. (Use simplification to find the common denominator.)
a)\latex{\; \frac{1}{2} +\frac{2}{4}}
b)\latex{\; \frac{2}{3} -\frac{3}{9}}
c)\latex{\; \frac{3}{4} -\frac{9}{12}}
d)\latex{\; \frac{3}{5} +\frac{8}{20}}
e)\latex{\; \frac{5}{6} -\frac{12}{18}}
f)\latex{\; \frac{5}{9} +\frac{6}{27}}
g)\latex{\; \frac{8}{12} +\frac{3}{9}}
h)\latex{\; \frac{15}{20} -\frac{3}{12}}
{{exercise_number}}. How many \latex{ litres } of beverage will you have if you mix \latex{\frac{3}{4}} \latex{ litres } of water and \latex{\frac{1}{2}} \latex{ litre } of syrup?
{{exercise_number}}. I loaded two playlists on my MP\latex{ 3 } player today. One is \latex{\frac{11}{12}} \latex{ hours } long, while the other is
\latex{\frac{19}{20}} \latex{ hours } long. How many \latex{ hours } can I listen to music?
{{exercise_number}}. Fred spends half of his summer vacation at his grandparents' house, \latex{\frac{2}{7}} of it at a summer camp and the rest of the time, he will be at home. What fraction of his summer vacation will Fred spend at home?
{{exercise_number}}. Perform the operations in your head.
a)\latex{\; \frac{1}{2} +\frac{1}{4}}
b)\latex{\; \frac{1}{5} +\frac{1}{2}}
c)\latex{\; \frac{1}{4} +\frac{1}{8}}
d)\latex{\; \frac{1}{2} +\frac{3}{6}}
e)\latex{\; \frac{1}{2} -\frac{1}{4}}
f)\latex{\; \frac{1}{3} -\frac{1}{6}}
g)\latex{\; \frac{7}{8} -\frac{1}{2}}
h)\latex{\; \frac{15}{16} -\frac{5}{8}}
{{exercise_number}}. Perform the operations.
a)\latex{\; \frac{5}{12} -\frac{2}{5}}
b)\latex{\; \frac{6}{7} +\frac{5}{6}}
c)\latex{\; \frac{9}{10} -\frac{3}{4}}
d)\latex{\; \frac{13}{16} -\frac{7}{12}}
e)\latex{\; \frac{3}{20} +\frac{2}{30}}
f)\latex{\; \frac{5}{36} +\frac{7}{24}}
g)\latex{\; \frac{5}{8} -\frac{8}{32}}
h)\latex{\; \frac{3}{18} +\frac{4}{24}}
{{exercise_number}}. A spider starts to make a web. It completes \latex{\frac{3}{8}} of the web in the first \latex{ hour } and \latex{\frac{5}{12}} in the second. How much of the web is still missing?
{{exercise_number}}. I read a book in four \latex{ days }. I read \latex{\frac{3}{20} } of it on the first \latex{ day }, \latex{\frac{2}{5} } on the second, and \latex{\frac{1}{6} } on the third. How much did I read on the fourth \latex{ day ?} On which \latex{ day } did I read the most?
{{exercise_number}}. \latex{\frac{2}{7} } of a nursery is planted with roses, \latex{\frac{1}{6} } with gerberas, and \latex{\frac{5}{14} } with lilies, while hyacinths are grown in the rest of the nursery. What fraction of the nursery contains hyacinths?
{{exercise_number}}. During a \latex{ 5 }-\latex{ day } training camp, a swimmer swam \latex{\frac{1}{2} } \latex{ km } on the first \latex{ day. } On the rest of the \latex{ days }, he swam \latex{\frac{1}{5} } \latex{ km } more than the \latex{ day } before. How many \latex{ kilometres } did the swimmer swim in \latex{ 5 } \latex{ days ?}
{{exercise_number}}. Perform the operations.
a)\latex{\; 1\frac{1}{4} +\frac{1}{2}}
b)\latex{\; 2\frac{2}{3} +3\frac{1}{4}}
c)\latex{\; 1\frac{3}{8} +2\frac{5}{6}}
d)\latex{\; 2\frac{5}{8} +5\frac{7}{12}}
e)\latex{\; 4\frac{5}{6} -3\frac{4}{5}}
f)\latex{\; 4\frac{1}{6} -3\frac{5}{8}}
g)\latex{\; 1\frac{5}{24} -2\frac{7}{12}}
h)\latex{\; 2\frac{3}{12} -1\frac{3}{4}}
{{exercise_number}}. What fraction should replace the ◯ to make the equalities true?
a)\latex{\; ◯ +\frac{1}{3}=\frac{5}{6}}
b)\latex{\; ◯ -\frac{1}{4}=\frac{1}{3}}
c)\latex{\; \frac{7}{10} - ◯=\frac{1}{5}}
d)\latex{\; \frac{8}{9} +(1-◯)=1\frac{5}{6}}
e)\latex{\; 1\frac{1}{8}+◯=1\frac{3}{4}}
f)\latex{\; \frac{17}{18} - \left(\frac{1}{2}+◯\right)=\frac{1}{3}}
{{exercise_number}}. Perform the operations.
a)\latex{\; 1+\frac{2}{3} -\frac{1}{2} }
b)\latex{\; 5-\frac{7}{4} -\frac{4}{3} }
c)\latex{\; \frac{3}{8} +1-\frac{4}{3} }
d)\latex{\; 1+\frac{7}{36} -\frac{4}{9} }
e) \latex{ \left(\frac{3}{7}+\frac{3}{4}\right)-\left(\frac{1}{2}-\frac{5}{14}\right) }
f) \latex{\left(3\frac{1}{2}-1\frac{5}{18}\right)-\left(\frac{5}{9}+\frac{3}{4}\right) }
g) \latex{ 4\frac{2}{5}-\left(1\frac{2}{7}-\frac{3}{35}\right) }
{{exercise_number}}. What is the rule? Write the next three members of each series.
a)\latex{\; \frac{1}{2};\; 1\frac{1}{4};\; 2; ...}
b)\latex{\; 3\frac{2}{5};\; 3\frac{4}{5};\; 4\frac{1}{5}; ...}
c)\latex{\; 4\frac{2}{3};\; 4\frac{1}{6};\; 3\frac{2}{3}; ... }
{{exercise_number}}. Fill in the numbers so that each is the sum of the two numbers below it.
\latex{ 1\frac{2}{3} }
\latex{ \frac{3}{4} }
\latex{ \frac{7}{12} }
\latex{ 4\frac{1}{5} }
\latex{ 2\frac{1}{3} }
\latex{ \frac{1}{2} }
\latex{ 1\frac{2}{3} }
\latex{ \frac{1}{2} }
\latex{ \frac{1}{6} }
\latex{ 3 }
\latex{ 1\frac{1}{2} }
\latex{ \frac{4}{5} }
{{exercise_number}}. Perform the operations.
a)\latex{\; \frac{1}{6} +\frac{2}{3}+\frac{5}{4} }
b)\latex{\;\frac{3}{8}+\frac{4}{3}-\frac{5}{6}}
c)\latex{\;\frac{1}{2}+\frac{4}{9}-\frac{7}{18}}
d)\latex{\;\frac{9}{7}-\frac{7}{9}+\frac{1}{3}}
{{exercise_number}}. Perform the additions in the simplest way possible.
a)\latex{\; \frac{3}{10} +\frac{2}{3}+\frac{7}{10} }
b)\latex{\;\frac{5}{12}+\frac{1}{3}+\frac{3}{12}}
c)\latex{\;\frac{3}{8}+\frac{2}{7}+\frac{1}{8}+\frac{5}{7}}
{{exercise_number}}. Fill in the magic squares, so that the sum of the numbers is the same in all the rows, columns and diagonals.
a)
\latex{ \frac{4}{15} }
\latex{ \frac{1}{3} }
\latex{ \frac{2}{5} }
\latex{ \frac{8}{15} }
b)
\latex{ \frac{3}{4} }
\latex{ \frac{3}{4} }
\latex{ \frac{7}{12} }
\latex{ \frac{1}{2} }
\latex{ \frac{1}{4} }
c)
\latex{ \frac{4}{3} }
\latex{ \frac{3}{2} }
\latex{ \frac{5}{3} }
\latex{ \frac{7}{6} }
d)
\latex{ \frac{7}{6} }
\latex{ \frac{8}{3} }
\latex{1 \frac{11}{12} }
\latex{5 \frac{2}{3} }
{{exercise_number}}. How long is the route from the car park to
a) the observation tower through the castle;
b) the minigolf course, passing by the fish pond;
c) the castle, passing by the minigolf course?
d) Think of a route which takes you back to the car park. How many \latex{ kilometres } do you have to walk?
b) the minigolf course, passing by the fish pond;
c) the castle, passing by the minigolf course?
d) Think of a route which takes you back to the car park. How many \latex{ kilometres } do you have to walk?
\latex{4} \latex{ km }
\latex{2\frac{5}{8}} \latex{ km }
\latex{3\frac{7}{10}} \latex{ km }
\latex{\frac{3}{4}} \latex{ km }
\latex{5\frac{1}{10}} \latex{ km }
\latex{3\frac{4}{5}} \latex{ km }
\latex{2\frac{1}{2}} \latex{ km }
\latex{1\frac{9}{10}} \latex{ km }
Quiz
A slacker spends his \latex{day} in the following way: he sleeps \latex{\frac{5}{12} } of the \latex{day}, spends \latex{\frac{1}{4} } of the \latex{day} eating, drinking and getting dressed, \latex{\frac{1}{8} } of the \latex{day} sitting in the armchair and yawning, and \latex{\frac{3}{16} } of the \latex{day} walking around. He goes to work the rest of the \latex{day}. What fraction of the \latex{day} does the slacker spend working? How many \latex{hours} is that?
