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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
Operations with fractions (revision)
Adding and subtracting fractions
Example 1
Dan left school at \latex{ 1} PM. It took him \latex{ 20\;minutes } to get home, then he rested for a quarter of an \latex{ hour } and spent one and a half \latex{ hours } doing his homework. After that, he played football with his friends for  \latex{1 \frac{3}{4}} \latex{ hours }. His parents arrived home half an \latex{ hour } after he returned from the football field. At what time did Dan's parents come home?
Solution
Convert the data to fractions, then add them. \latex{ 20 } \latex{ minutes } = \latex{\frac{1}{3}} \latex{ hour };
quarter of an \latex{ hour }= \latex{\frac{1}{4}} \latex{ hour }; one and a half \latex{ hours } = \latex{\frac{3}{2}} \latex{ hours }; \latex{1\frac{3}{4}} \latex{ hours }= \latex{\frac{7}{4}} \latex{ hours };
half an \latex{ hour } = \latex{\frac{1}{2}} \latex{ hour }.
\latex{\frac{1}{3}+\frac{1}{4}+\frac{3}{2}+\frac{7}{4}+\frac{1}{2}=\frac{4}{12}+\frac{3}{12}+\frac{18}{12}+\frac{21}{12}+\frac{6}{12}=\frac{52}{12}}
To add fractions, find a common denominator.
Written as a mixed number, then simplified: \latex{\frac{52}{12}=4\frac{4}{12}=4\frac{1}{3}}
\latex{13+4\frac{1}{3}=17\frac{1}{3}};   \latex{17\frac{1}{3}=17} \latex{ hours } \latex{20} \latex{ minutes. }
Check:
\latex{ 20 \;minutes} + \latex{\frac{1}{4}} \latex{ hour } + \latex{\frac{3}{2}} \latex{ hours} + \latex{1\frac{3}{4}} \latex{ hours} + \latex{\frac{1}{2}} \latex{ hour } =
=\latex{ 20 \;min } + \latex{ 15 \;min } + \latex{ 90 \;min } + \latex{ 105 \;min } + \latex{ 30 \;min } = \latex{ 260 \;min } = \latex{ 4 \;hours \;20 \;minutes }.
Dan’s parents arrived home at \latex{17} \latex{ hours } \latex{20} \latex{ minutes } (\latex{5}:\latex{20} PM).
Example 2
To make a Hawaiian cocktail, you need \latex{\frac{3}{20}} \latex{ l } of orange juice, \latex{\frac{3}{50}} \latex{ l } of lemon juice, \latex{\frac{4}{50}} \latex{l} of pineapple juice and \latex{\frac{6}{50}} \latex{l} of ginger juice. 
Moreover, you need to add sparkling water to make the volume of your cocktail \latex{ 1 \;litre }.
How many \latex{ litres } of sparkling water do you need to add to your Hawaiian cocktail to have \latex{ 1 \;litre } of beverage?
Look it up.
Where is Hawaii? What country does it belong to?
Solution
You can calculate the amount of sparkling water needed by subtracting the sum of the other ingredients from \latex{ 1\; litre }.
 
\latex{1-\left(\frac{3}{20}+\frac{3}{50}+\frac{4}{50}+\frac{6}{50} \right)=1-\left(\frac{3}{20}+\frac{13}{50} \right)=1-\left(\frac{15}{100}+\frac{26}{100} \right)=}
\latex{=1-\frac{41}{100}=\frac{100}{100}-\frac{41}{100}=\frac{59}{100}}
The amount of sparkling water needed is \latex{\frac{59}{100}} \latex{l}.
When adding and subtracting fractions, the lowest common multiple of the denominators should be used as the common denominator.
Example:
\latex{\frac{7}{16}+\frac{1}{12}=\frac{7\times 3+1\times 4}{48} =\frac{25}{48}}
Expanding:

\latex{\frac{8}{16}-\frac{3}{12}=\frac{8\times 3-3\times 4}{48}=\frac{12}{48}=\frac{1}{4}}
Simplifying:

\latex{\frac{8}{16}-\frac{3}{12}=\frac{1}{2}-\frac{1}{4}=\frac{2}{4}-\frac{1}{4}=\frac{1}{4}}
\latex{ 16 }
\latex{ 8 }
\latex{ 4 }
\latex{ 2 }
\latex{ 1 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 2 }
\latex{ 12 }
\latex{ 6 }
\latex{ 3 }
\latex{ 1 }
\latex{ 3 }
\latex{ 2 }
\latex{ 2 }
\latex{16=2\times 2\times 2\times 2}
\latex{12=}
\latex{2\times2\times3}
LCM of \latex{ 16 } and \latex{ 12 } \latex{=2\times 2\times 2\times 2\times 3=48 }
When adding or subtracting fractions with different denominators, find the common denominator by expanding or simplifying them, then perform the operation.
Multiplying fractions by integers
Example 3
A sixth grader spends \latex{\frac{5}{24}} of a \latex{ day } at school every \latex{ day }. How many \latex{ days } does a sixth grader spend at school during one \latex{ week }?
Solution
Add the time spent at school each \latex{ day }:
 
\latex{\frac{5}{24} +\frac{5}{24}+ \frac{5}{24}+\frac{5}{24}+\frac{5}{24}=\frac{25}{24}=1\frac{1}{24}}
 
An addition with the same addends can be expressed as a product:
 
\latex{\frac{5}{24} +\frac{5}{24}+ \frac{5}{24}+\frac{5}{24}+\frac{5}{24}=5\times \frac{5}{24}=\frac{5\times 5}{24}=\frac{25}{24}=1\frac{1}{24}}
 
A sixth grader spends \latex{25} \latex{ hours }, that is, \latex{1\frac{1}{24}} \latex{ days } at school each \latex{ week }.
Since the factors are interchangeable, you can write it as:
\latex{5\times \frac{5}{24}= \frac{5}{24}\times 5}
When multiplying a fraction by an integer, multiply the numerator by the integer and leave the denominator unchanged.
Example:
\latex{8\times \frac{5}{24}=\frac{8\times 5}{24}=\frac{40}{24}=\frac{5}{3}}
If the denominator of the fraction is divisible by the integer, then it can be multiplied by dividing the denominator by the integer and leaving the numerator unchanged.
Example:
\latex{8\times \frac{5}{24}=\frac{5}{24\div 8}=\frac{5}{3} }
In other words:
If the denominator and the multiplier have a common factor, then the terms can be simplified before performing the multiplication.
Example:
\latex{\overset{1}{\cancel{8}}\times \frac{5}{\underset{3}{\cancel{24}} }=\frac{5}{3}}
Before performing the multiplication, it is always worth contemplating every possibility to simplify the terms.
Example:
\latex{24\times \frac{\overset{7}{\cancel{14}} }{\underset{16}{\cancel{32}} }=\overset{3}{\cancel{24}}\times \frac{7}{\underset{2}{\cancel{16}} }=\frac{21}{2}=10\frac{1}{2}}
Dividing fractions by integers
Example 4
\latex{\frac{3}{8}} \latex{ kg } of butter is added to the dough. Identical buns are formed using the dough. How many kilograms of butter are in each bun if
  1. \latex{ 5 };
  2. \latex{ 3 } buns are made?
Solution
  1. The dough containing \latex{\frac{3}{8}} \latex{ kg } of butter is divided into \latex{ 5} buns; therefore, one bun will contain one-fifth of the total amount, that is, \latex{\frac{3}{8} \div 5(kg)} of butter.
\latex{\frac{3}{8} \div 5=\frac{3}{8\times 5}=\frac{3}{40}}
One bun contains \latex{\frac{3}{40}\;kg} of butter.
  1. The dough containing  \latex{\frac{3}{8}\;kg} of butter is divided into \latex{ 3 } buns:
\latex{\frac{3}{8} \div 3=\frac{3\div 3}{8}=\frac{1}{8}}
A bun will contain \latex{\frac{1}{8}\;kg} of butter.
You can always divide a fraction by an integer by multiplying its denominator by the integer and leaving its numerator unchanged.
Example:
\latex{\frac{10}{7} \div 2=\frac{10}{2\times 7}=\frac{10}{14}=\frac{5}{7}}
If the numerator of a fraction is divisible by the integer, you can divide the numerator by it and leave the denominator unchanged.
Example:
\latex{\frac{10}{7} \div 2=\frac{10\div 2}{7}=\frac{5}{7} }
When a mixed number is divided by an integer, convert the mixed number to an improper fraction and perform the division.
Example:
\latex{1\frac{3}{7}\div 2=\frac{10}{7}\div 2=\frac{5}{7} }
Exercises
{{exercise_number}}. Perform the additions. Find the easiest solutions.
  1. \latex{\frac{21}{32}+\frac{5}{32}}
  1. \latex{\frac{25}{48}+\frac{15}{48}}
  1. \latex{\frac{5}{14}+\frac{13}{14}}
  1. \latex{\frac{7}{56}+\frac{40}{64}}
  1. \latex{\frac{22}{55}+\frac{14}{35}}
  1. \latex{\frac{4}{8}+\frac{1}{2}+\frac{32}{8} }
{{exercise_number}}. What is the sum?
  1. \latex{\frac{5}{21}+\frac{2}{3}}
  1. \latex{\frac{3}{7}+\frac{5}{12}}
  1. \latex{\frac{13}{6}+\frac{4}{5}+\frac{7}{3}}
  1. \latex{3\frac{1}{2}+\frac{5}{18}}
  1. \latex{3\frac{1}{2}+\frac{5}{8}}
  1. \latex{3\frac{1}{2}+\frac{8}{5}}
  1. \latex{\frac{4}{28}+\frac{5}{7}+\frac{10}{14}}
  1. \latex{\frac{2}{5}+3\frac{12}{20}}
  1. \latex{3\frac{2}{3}+2\frac{1}{5}+6\frac{5}{6}}
{{exercise_number}}. What number is indicated by the \latex{\triangle} ?
  1. \latex{4\frac{1}{5}+\triangle =8\frac{3}{5}}
  1. \latex{\triangle +1\frac{3}{8}=7\frac{1}{8}}
  1. \latex{3\frac{1}{3}-\triangle =2\frac{5}{6}}
  1. \latex{7\frac{1}{7}-\triangle =2\frac{3}{7}}
  1. \latex{3\frac{1}{5}-\left(1\frac{1}{6}+\triangle \right) =1}
  1. \latex{3+\triangle =\frac{59}{17}}
{{exercise_number}}. Perform the following operations using the smallest common denominator.
  1. \latex{\frac{13}{15}+\frac{5}{18}}
  1. \latex{\frac{11}{15}-\frac{3}{18}}
  1. \latex{2\frac{1}{15}-1\frac{5}{18}}
  1. \latex{\frac{54}{162}+\frac{31}{48}}
  1. \latex{\frac{54}{162}-\frac{31}{48}}
  1. \latex{\frac{54}{162}+3\frac{5}{45}}
{{exercise_number}}. 
  1. Which number is \latex{\frac{5}{6}} greater than \latex{3\frac{1}{3}?} 
  2. How much greater is \latex{\frac{7}{12}} than \latex{\frac{1}{5}?} 
  3. Which number is \latex{\frac{4}{3}} greater than \latex{\frac{7}{15}?} 
{{exercise_number}}. Determine the rule and write down the next two terms of the sequences. Then, calculate the sum of the five terms.
  1. \latex{\frac{1}{2};\frac{1}{4};\frac{1}{8};...; }
  1. \latex{\frac{1}{4};\frac{1}{16};\frac{1}{64};...; }
  1. \latex{\frac{1}{10};\frac{1}{100};\frac{1}{1000};...; }
  1. \latex{\frac{9}{1000000};\frac{9}{1000000};\frac{9}{1000000};...; }
{{exercise_number}}. Joe baked a cake. The flour and the bowl together weighed \latex{1\frac{1}{4}} \latex{ kg }. The bowl was \latex{\frac{2}{5}} \latex{ kg }. How much did the flour weigh in \latex{ kilograms?}
{{exercise_number}}. Zoe and Nora went fishing. How many \latex{ kilograms } of fish did they catch in total if they caught two \latex{\frac{4}{5}} \latex{ kg } carps and one \latex{\frac{17}{20}} \latex{ kg } catfish?
{{exercise_number}}. Martha walks \latex{\frac{1}{10}} of an \latex{ hour } to the bus stop, waits \latex{\frac{1}{20}} of an \latex{ hour } for the bus, travels \latex{\frac{1}{6}} of an \latex{ hour } on the bus, and it takes her \latex{\frac{1}{30}} of an \latex{ hour } to arrive at the school after getting off. How many \latex{ hours } does Martha spend travelling to school during a 5-\latex{ day } \latex{ week? }
{{exercise_number}}. What is the perimeter of a square if the length of its sides is \latex{2\frac{2}{5}} \latex{ m? }
{{exercise_number}}. One side of a rectangle is \latex{\frac{7}{20}} \latex{ m } long, and its perimeter is \latex{\frac{51}{40}} \latex{ m }. How long is the other side of the rectangle in \latex{ metres? } How long are the sides of the rectangles in \latex{ centimetres? }
{{exercise_number}}. What is the result of the series of operations?
  1. \latex{\left(7\frac{1}{2}-\frac{3}{4} \right)+\left(5\frac{3}{8}+\frac{1}{4} \right)=}
  1. \latex{(+7)+\left(\frac{5}{8} \right) -\left(+3\frac{3}{8} \right) +17=}
  1. \latex{\frac{7}{32}+\frac{12}{128}-\frac{5}{64}+2\frac{7}{8}+\frac{1}{2}-2\frac{1}{4}=}
  1. \latex{\frac{11}{16}+\left[3\frac{1}{2}-\left(\frac{1}{2}-\frac{14}{7} \right) \right]-\left(\frac{3}{4}-\frac{7}{16} \right)=}
{{exercise_number}}. Calculate the products. Try to calculate in the simplest possible way.
  1. \latex{5\times \frac{1}{9}}
  1. \latex{3\times \frac{2}{5}}
  1. \latex{5\times \frac{4}{6}}
  1. \latex{6\times \frac{5}{12}}
  1. \latex{16\times \frac{7}{24}}
  1. \latex{12\times \frac{5}{12}}
  1. \latex{4\times \frac{15}{5}}
  1. \latex{14\times \frac{13}{7}}
  1. \latex{3\times \frac{9}{27}}
  1. \latex{20\times \frac{4}{5}}
{{exercise_number}}. Substitute the symbols with numbers that make the equalities true.
  1. \latex{\square \times \frac{5}{8}=\frac{15}{8}=1\frac{\triangle }{8}}
  1. \latex{\square \times \frac{5}{8}=\frac{\bigcirc}{2}}
  1. \latex{\triangle \times \frac{5}{8}=1\frac{1}{4}}
{{exercise_number}}. Multiply the following numbers by six. Simplify the products if possible.
  1. \latex{\frac{3}{4}}
  1. \latex{\frac{5}{6}}
  1. \latex{\frac{7}{11}}
  1. \latex{1\frac{1}{2}}
  1. \latex{\frac{5}{3}}
  1. \latex{2\frac{11}{12}}
{{exercise_number}}. Determine the rule and continue each sequence with three additional terms.
  1. \latex{\frac{1}{2};1\frac{1}{2};4\frac{1}{2};...;}
  1. \latex{\frac{5}{128};\frac{5}{64};\frac{5}{32};...;}
  1. \latex{1\frac{1}{3};2\frac{2}{3};5\frac{1}{3};...}
{{exercise_number}}. The height of the steps in a long staircase is \latex{\frac{4}{25}\;m}. How high can you get after climbing \latex{ 7; 28 } and \latex{ 100 } steps? What is the height of each step in \latex{ centimetres? }
{{exercise_number}}. Calculate using the simplest method possible.
  1. \latex{\frac{4}{5}\div 2}
  1. \latex{\frac{36}{37}\div 6}
  1. \latex{2\frac{1}{5}\div 11}
  1. \latex{4\frac{2}{7} \div 5}
  1. \latex{\frac{12}{6}\div 4}
  1. \latex{\frac{7}{18}\div 2}
  1. \latex{\frac{3}{40}\div 6}
  1. \latex{\frac{10}{19}\div 25}
  1. \latex{\frac{16}{17}\div 20}
  1. \latex{\frac{9}{11}\div 27}
{{exercise_number}}. Which number is replaced by the \latex{\square} ?
  1. \latex{\frac{2}{3}\div \square =\frac{2}{9}}
  1. \latex{\frac{\square }{7}\div 5=\frac{3}{35}}
  1. \latex{\frac{5 }{\square }\div 6=\frac{5}{42}}
  1. \latex{\frac{5 }{45 }\div \square =\frac{1}{18}}
  1. \latex{\frac{40 }{88 }\div 11 =\frac{\square }{121}}
  1. \latex{\frac{96 }{72 }\div 5 =\frac{4 }{\square }}
{{exercise_number}}. 
  1. What is the dividend if the divisor is \latex{ 30 } and the quotient is \latex{\frac{54}{7}?}  
  2. Twelve times which number is \latex{\frac{72}{5}?} 
  3. If a number is multiplied by \latex{ 5 } and then divided by \latex{ 2 }, the result is  \latex{1\frac{3}{7}}. What is the original number?
{{exercise_number}}. Jane had to write the correct relation symbols between the following amounts. Correct her mistakes.
  1. \latex{3\frac{1}{2}} \latex{ m } \latex{=350} \latex{ cm }
  1. \latex{1\frac{1}{3}} \latex{ hours }  \latex{\gt 80} \latex{ min } 
  1. \latex{\frac{5}{6}} \latex{ days }  \latex{\gt 22} \latex{ hours }
  1. \latex{\frac{7}{8}} \latex{ km } \latex{=840} \latex{ m }
  1. \latex{2\frac{3}{4}} \latex{ kg } \latex{=2750} \latex{ g }
  1. one and a half \latex{ days } \latex{=2\frac{1}{2}} \latex{ days }
Quiz
Convert the fraction \latex{\frac{1}{9}} to decimal form, then write it as the sum of fractions whose numerator is \latex{ 1 }. What do you notice?
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