cadaVR anatomy by Mozaik Education

Operations with rational numbers

Revision

Game • Calculate with rational numbers

Make the following cards.

Rational numbers:

\latex{\frac{2}{3}}

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

\latex{\frac{3}{4}}

\latex{\frac{5}{6}}

\latex{0}

\latex{12}

\latex{3.5}

\latex{1.2}

signs:

\latex{\times}

\latex{+}

\latex{-}

\latex{,}

\latex{;}

operations:

\latex{\times}

\latex{+}

\latex{-}

\latex{\div }

\latex{, }

\latex{.}

Flip the cards and shuffle them. Then, start drawing the cards as the shapes indicate it.

Write down the resulting operation and calculate the result. Put the cards back and draw a new series of operations. The winner is the player who gets the largest number when adding the results of four operations.

Multiplying sums and differences

Example 1

Perform the following operations.

\latex{(-2)\times \left(1-\frac{1}{3} \right) }

\latex{-\left(\frac{3}{4}+\frac{1}{2} \right) }

\latex{\left(-\frac{2}{3} \right) \times \left(1+\frac{1}{4}-\frac{1}{2} \right)}

Solution

There are two ways to perform these operations.

First, perform the operations between brackets.

\latex{(-2)\times \left(1-\frac{1}{3} \right)=(-2)\times \left(\frac{3}{3}-\frac{1}{3} \right)=(-2)\times \frac{2}{3}=-\frac{4}{3}.}

You already know that multiplying sums and differences can be performed by multiplying each term separately and then adding or subtracting the products. This is called removing the brackets.

\latex{(-2)\times \left(1-\frac{1}{3} \right)=(-2)\times 1-(-2)\times \frac{1}{3}=-2-\left(-\frac{2}{3} \right)=\\= -2+\frac{2}{3} =-\frac{6}{3}+\frac{2}{3}=-\frac{4}{3} .}

Performing the operations between brackets first:

\latex{-\left(\frac{3}{4}+\frac{1}{2} \right)=-\left(\frac{3}{4}+\frac{2}{4}\right)=-\left(\frac{5}{4} \right)=-\frac{5}{4}.}

Removing the brackets first:

\latex{-\left(\frac{3}{4}+\frac{1}{2} \right)=(-1)\times \frac{3}{4}+(-1)\times \frac{1}{2}=-\frac{3}{4}-\frac{1}{2}=-\frac{3}{4}-\frac{2}{4}=-\frac{5}{4}.}

Performing the operations between brackets first:

\latex{\left(-\frac{2}{3} \right) \times \left(1+\frac{1}{4}-\frac{1}{2} \right)=\left(-\frac{2}{3} \right)\times \left(\frac{4}{4}+\frac{1}{4}-\frac{2}{4} \right)=\left(-\frac{\bcancel{2 }^{1} }{\cancel{3}^{1} } \right)\times\frac{\cancel{3}^{1} }{\bcancel{4}^{2} }=-\frac{1}{2}.}

Removing the brackets first:

\latex{\left(-\frac{2}{3} \right)\times \left(1+\frac{1}{4}-\frac{1}{2} \right)=\left(-\frac{2}{3} \right)\times 1+\left(-\frac{\cancel{2}^{1} }{3} \right)\times \frac{1}{\cancel{4}^{2} }-\left(-\frac{\cancel{2}^{1} }{3} \right)\times \frac{1}{\cancel{2}^{1} }=}

\latex{=-\frac{2}{3}+\left(-\frac{1}{6} \right)-\left(-\frac{1}{3} \right)=-\frac{4}{6}-\frac{1}{6}+\frac{2}{6}=-\frac{\cancel{3}^{1} }{\cancel{6}^{2} }=-\frac{1}{2}.}

Example 2

Perform the following operations.

\latex{\frac{2}{3}-\left(\frac{2}{3}-\frac{1}{5} \right);}

\latex{2-12\times \left(\frac{5}{6}-\frac{1}{3} \right).}

Solution

Performing the operations between brackets first:

\latex{\frac{2}{3}-\left(\frac{2\times 5}{3\times 5}-\frac{1\times 3}{5\times 3} \right)=\frac{2}{3}-\left(\frac{10}{15}-\frac{3}{15} \right)=\frac{2}{3}-\frac{7}{15}=}

\latex{=\frac{2\times 5}{3\times 5}-\frac{7}{15}=\frac{10}{15}-\frac{7}{15}=\frac{3}{15}=\frac{1}{5}.}

Removing brackets first:

\latex{\frac{2}{3}- \left(\frac{2}{3}-\frac{1}{5} \right)=\frac{2}{3}-\frac{2}{3}+\frac{1}{5}=\frac{1}{5}.}

This solution required less calculation. Its worth considering which method to choose when solving an exercise.

Performing the operations between brackets first:

\latex{2-12\times \left(\frac{5}{6}-\frac{1}{3} \right)=2-12\times \left(\frac{5}{6}-\frac{1\times 2}{3\times 2} \right)=}
\latex{=2-12\times \left(\frac{5}{6}-\frac{2}{6} \right)=2-\cancel{12}^{2}\times \frac{3}{\cancel{6}}=2-6=-4.}

Removing the brackets first:

\latex{2-\cancel{12}^{2}\times \frac{5}{\cancel{6}}+(-\cancel{12}^{4} ) \times \left(-\frac{1}{\cancel{3}} \right)=2-10+4=-4.}

Example 3

Perform the following operations.

\latex{1-\frac{1}{3}\times \frac{2}{5}-0,1;}

\latex{\left(1-\frac{1}{3} \right)\times \frac{2}{5}-0,1;}

\latex{1-\frac{1}{3}\times \left(\frac{2}{5}-0,1 \right);}

\latex{\left(1-\frac{1}{3} \right)\times \left(\frac{2}{5}-0,1 \right).}

Solution

The difference between these operations is that, due to the brackets, the order of operations is different.

It is better to use the fraction form because of \latex{\frac{1}{3}.}

\latex{1-\frac{1}{3}\times \frac{2}{5}-\frac{1}{10}=1-\frac{2}{15}-\frac{1}{10}=\frac{30-4-3}{30}=\frac{23}{30};}

\latex{\left(1-\frac{1}{3} \right)\times \frac{2}{5}-\frac{1}{10}=\frac{2}{3}\times \frac{2}{5}-\frac{1}{10}=\frac{4}{15}-\frac{1}{10}=\frac{8-3}{30}=\frac{5}{30};}

\latex{1-\frac{1}{3}\times \left(\frac{2}{5}-\frac{1}{10} \right)=1-\frac{1}{3}\times \frac{4-1}{10}=1-\frac{1}{3}\times \frac{3}{10}=1-\frac{1}{10}=\frac{9}{10}=\frac{27}{30};}

\latex{\left(1-\frac{1}{3} \right)\times \left(\frac{2}{5}-\frac{1}{10} \right)=\frac{2}{3}\times \frac{3}{10}=\frac{6}{30}.}

Example 4

Which number is greater: \latex{A=\frac{\frac{1}{2} }{3},} or \latex{B=\frac{1}{\frac{2}{3} }?}

Solution

In the case of compound fractions, the order of operations is determined by the main fraction bar. The main fraction bar is always written at the same height as the equals sign.

\latex{A=\frac{\frac{1}{2} }{3}=\frac{1}{2}\div 3=\frac{1}{2\times 3}=\frac{1}{6};} \latex{B=\frac{1}{\frac{2}{3} }=1\div \frac{2}{3}=1\times \frac{3}{2}=\frac{3}{2}.}

Since \latex{\frac{3}{2}=\frac{9}{6},} \latex{A\lt B.}

Note that depending on where the main fraction bar is placed, the position of the brackets may change:

\latex{A=\frac{\frac{1}{2} }{3}=(1\div 2)\div 3,} while \latex{B=\frac{1}{\frac{2}{3} }=1\div (2\div 3) .}

Example 5

Simplify the compound fraction \latex{1+\frac{4}{1-\frac{1}{3} }.}

Solution

The numerator of the fraction is \latex{ 4 }, and the denominator is \latex{1-\frac{1}{3}=\frac{2}{3}.}

The fraction can be simplified in the following way:

\latex{1+\frac{4}{1-\frac{1}{3} }=1+\frac{4}{\frac{2}{3} }=1+4\div \frac{2}{3}=1+4\times \frac{3}{2}=1+2\times 3=1+6=7.}

Exercises

{{exercise_number}}. Perform the following operations.

\latex{1+2\times \left(\frac{1}{2}-4 \right) }

\latex{4-3\times (1-2\times 3)}

\latex{3,5-(4\times 3-9)\times \frac{1}{2}}

\latex{1-3\times \left(\frac{1}{3}+\frac{5}{6} \right) }

\latex{\frac{1}{2}+10\times \left(0,2\frac{1}{4} \right)}

\latex{-3\times \left(\frac{1}{9}+\frac{1}{27} \right)+\frac{1}{3}}

{{exercise_number}}. Perform the following operations.

\latex{\frac{2}{3}+\left(\frac{1}{2}-\frac{2}{5} \right)}

\latex{\frac{2}{3} -\left(\frac{1}{2}-\frac{2}{5} \right)}

\latex{\frac{2}{3} -\left(\frac{1}{2}+\frac{2}{5} \right)}

\latex{\frac{2}{5} -0,1\times \left(10+\frac{200}{2} \right)}

\latex{(-5)\times \left(\frac{1}{2}-\frac{2}{8} \right)-\frac{1}{8}}

\latex{\left(\frac{1}{2}-0,4 \right) \times \left(0,4-\frac{1}{10} \right)}

{{exercise_number}}. Perform the following operations.

\latex{\frac{5}{9}\times (-18)+\frac{1}{3} }

\latex{\left(-\frac{4}{9}+\frac{2}{3} \right)\times \left(-\frac{1}{2} \right) }

\latex{\left[-\frac{5}{3}-\left(-\frac{3}{4} \right) \right]\div \left(-\frac{2}{9} \right)}

\latex{\frac{4}{9}\times \left(\frac{3}{2}-\frac{4}{5} \right) \div \left(-\frac{7}{5} \right)}

{{exercise_number}}. Moro Bear really likes honey. He eats honey for three \latex{ days }, a quarter of a honey pot each \latex{ day }. On every fourth \latex{ day }, he eats raspberries. On 'raspberry \latex{ days }', he refills two-thirds of a pot with honey. This morning, he has a pot completely full of honey left.

How much honey will be left in the pot four \latex{ days } later?

How much honey will be left in the pot eight \latex{ days } later?

How many \latex{ days } later will the honey not be enough for Moro Bear?

{{exercise_number}}. The king commissioned the construction of a new castle. During the day, \latex{\frac{1}{30}} of the castle is built; however, during the night, \latex{\frac{1}{3}} of the part built on the given day collapses. When will the construction of the castle be completed?

{{exercise_number}}. You can use the following numbers and signs:

\latex{2}

\latex{1}

\latex{\frac{1}{3} }

\latex{- }

\latex{+ }

\latex{\times }

\latex{( }

\latex{) }

Use them to make a series of operations and calculate the result. Find several solutions.

{{exercise_number}}. Which one is greater?

\latex{A=\frac{\frac{2}{3} }{5}} or \latex{B=\frac{2}{\frac{3}{5} } }

\latex{A=\frac{\frac{5}{3} }{5}} or \latex{B=\frac{5}{\frac{3}{5} }}

\latex{A=\frac{\frac{5}{3} }{2}} or \latex{B=\frac{5}{\frac{3}{2} }}

\latex{A=\frac{-\frac{2}{3} }{5}} or \latex{B=\frac{2}{-\frac{3}{5} } }

{{exercise_number}}. Observe the following fractions. Calculate their values.

\latex{\frac{\left(\frac{1}{3}-0.25+1 \right)\div \left(\frac{1}{3}-0.25+1 \right) }{\left(0.5-1\right)\div \left(0.751-25\right) }}

\latex{\frac{\left(\frac{1}{3}-0.25+1.8 \right)\div \left(\frac{1}{3}-0.25+1.8 \right) }{\left(0.25-1\right)\div \left(0.51- 0.25\right) }}

\latex{\frac{\left(\frac{2}{3}-0.5+2 \right)\div \left(\frac{1}{3}-0.25+1\right) }{\left(0.5-1\right)\div \left(0.751-0.25\right) }}

\latex{\frac{\left(\frac{1}{3}-0.25+1 \right)\div \left(\frac{1}{3}-0.25+1\right) }{\left(-1.2\right)\div \left(0.75-1- 0.25\right) }}

{{exercise_number}}. Write the correct signs between the numbers to make the equalities true.

\latex{\frac{1}{2}\bigcirc \frac{1}{3}=\frac{5}{6}}

\latex{\frac{3}{5}\triangle\frac{12}{6}=0.3}

\latex{0.7\triangledown\frac{1}{2}=\frac{1}{5}}

\latex{\frac{13}{70} \Box\frac{39}{10}=\frac{1}{21}}

{{exercise_number}}. Simplify the following compound fractions.

\latex{\frac{\frac{1}{2}-\frac{1}{3}}{5}}

\latex{\frac{2-\frac{2}{3}}{\frac{2}{3}-2}}

\latex{1+\frac{1}{3+\frac{4}{5} }}

\latex{\frac{1+\large{\frac{1}{1+\frac{1}{2}} } }{2}}

{{exercise_number}}. What distance did you cover if, after completing \latex{\frac{2}{3}} of the road, you still had \latex{\frac{2}{3}} of a \latex{ kilometre } left?

{{exercise_number}}. Carl participated in a cycling race. After the race, he told his parents:

"One-third of the competitors finished before me, while half of them finished after me. There were no ties."

What place did Carl finish in?

{{exercise_number}}. Substitute the number cards 1 2 3 into the compound fraction \latex{\frac{\Box}{\Box}} to get the

greatest;

smallest possible number.

Quiz

Write signs between the following fractions and use brackets to make the equalities correct.

\latex{\frac{1}{2}\div \frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} =\frac{1}{2} ;}

\latex{\frac{1}{2}\div \frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} =1;}

\latex{\frac{1}{2}\div \frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} =0.}

Mathematics 7.

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