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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Integers (revision)
The addition and subtraction of integers (revision)
Combining
The multiplication of integers
The division of integers
Combining decimals
Multiplication of decimals
Division with decimals
Mixed exercises
Axial symmetry
Rational numbers II.
Proportion
Percentages
Probability, statistics
Supplementary material
Multiplication of decimals
\latex{ 14:26.54 }
\latex{ 0.53 }
\latex{ kg }
\latex{ 173.9 }
\latex{ 0033.25 }
\latex{ 0057.82 }
€
Litre
UNIT
\latex{ 2 }
\latex{ 1.5 }
km
Cent/Litre
TARE
Multiplying a decimal by an integer
Example 1
The first three steps of a pyramid were covered in desert sand. One step is \latex{ 7.18 \;m } high.
Have a discussion about the measuring instruments shown in the images. Write down some multiplications related to them.
  1. How high is the top of the pyramid from the sandy surface if it consists of \latex{ 7 } steps?
  2. How deep is the bottom of the pyramid from ground level?
Solution
  1. The pyramid consists of \latex{ 7 } steps; therefore, \latex{ 4 } steps are above the sandy surface.
\latex{7.18 + 7.18 + 7.18 + 7.18 =\\ = 4\times7.18 = 28.72.}
The top of the pyramid is \latex{ 28.72\; m } above the sandy surface.
\latex{7.}
\latex{2}
\latex{2}
\latex{8.}
\latex{\times}
\latex{4}
\latex{\textcolor{0a92ca}{1}}
\latex{\textcolor{0a92ca}{8}}
\latex{\textcolor{0a92ca}{7}}
\latex{\textcolor{0a92ca}{2}}
  1. The bottom of the first step below the sandy surface is at a depth of \latex{ –7.18 }  \latex{ metres }.
\latex{(-7.18) + (-7.18) + (-7.18) = 3\times(-7.18) = -21.54.}
The base of the pyramid is \latex{ 21.54 \;metres } below the sandy surface.
\latex{7.}
\latex{2}
\latex{1.}
\latex{3}
\latex{\times}
\latex{\textcolor{0092ca}{1}}
\latex{\textcolor{0092ca}{8}}
\latex{\textcolor{0092ca}{5}}
\latex{\textcolor{0092ca}{4}}
When multiplying a decimal by an integer, the product should contain the same number of decimal places as the original decimal. The sign is determined in the same way as in the case of integers.
Multiplying decimals by decimals
Example 2
The figure shows a rectangular terrace. How many \latex{ square } \latex{ metres } is the area of the terrace?
Solution
\latex{5.6{\,m}}
\latex{b}
\latex{a}
\latex{3.2{\,m}}
\latex{a = 5.6 {\,m}}
\latex{\underline{b=3.2{\,m}}}
\latex{A = ?}
\latex{A = a\times b}
\latex{A = 5.6\times 3.2 {\,m}^{2}}
Estimation:
\latex{A\approx 6\times3 {\,m}^{2} = 18 {\,m}^{2}}
Use what you have learned about multiplying integers. Observe how the product changes.
According to the relationship between units, the area is:
\latex{A = 56\times 32 { \,dm}^2}
\latex{A = 1,792 {\,dm}^2}
\latex{A = 17.92 {\,m}^2}
\latex{56\times32 =1,792}
\latex{5.6\times32 =179.2}
\latex{5.6\times3.2 =17.92}
The area of the terrace is \latex{17.92 {\,m}^{2}}.
\latex{\textcolor{00a2e4}{÷10}}
\latex{\textcolor{00a2e4}{÷10}}
\latex{\textcolor{00a2e4}{÷10}}
\latex{\textcolor{00a2e4}{÷10}}
\latex{\textcolor{00a2e4}{÷100}}
\latex{\underline{ 56}\times32}
\latex{168}
\latex{\underline{112}}
\latex{1,792}
Calculating with a decimal:
\latex{A = 5.6\times3.2 {\,m}^{2}}
\latex{A = 17.92 {\,m}^{2}}
\latex{5.}
\latex{\times}
\latex{3.}
\latex{1}
\latex{6}
\latex{8}
\latex{1}
\latex{1}
\latex{2}
\latex{1}
\latex{7.}
\latex{\textcolor{0f92c9}{9}}
\latex{\textcolor{0f92c9}{2}}
\latex{\textcolor{0f92c9}{6}}
\latex{\textcolor{0f92c9}{2}}
Example 3
The owners of the terrace want to cover its side near the garden with green tiles, while the rest of the sides with brown tiles. How many \latex{ square \;metres } will be covered by green tiles if the width of one tile is \latex{ 32 } \latex{ cm }?
Solution
\latex{5.6{\,m}}
\latex{b}
\latex{a}
\latex{3.2{\,m}}
\latex{a = 5.6 {\,m}}
\latex{\underline{c=32{\,cm}=0.32{\,m}}}
\latex{A = ?}
\latex{A = a\times c}
\latex{A = 5.6\times 0.32 {\,m}^{2}}
Estimation:
\latex{A\approx 6\times 0.3 {\,m}^{2} = 1.8 {\,m}^{2}}
\latex{32{\,cm}}
\latex{c}
One of the sides of the green rectangle is \latex{ 5.6 \;m } long, while the other is \latex{ 32 \;cm } long.
Based on the relationships between units, the area is:
\latex{A = 560\times 32 {\,cm}^2}
\latex{A = 17,920 {\,cm}^2}
\latex{A = 1.792 {\,m}^2}
\latex{56\space\times\space\space32 =\space\space\space1,792}
\latex{5.6\times0.32 =1.792}
The area of the part covered with green tiles is \latex{1.792 {\,m}^{2}}.
\latex{\textcolor{00a2e4}{÷10}}
\latex{\textcolor{00a2e4}{÷100}}
\latex{\textcolor{00a2e4}{÷1,000}}
Calculating with a decimal:
\latex{A = 5.6\times0.32 {\,m}^{2}}
\latex{A = 1.792 { \,m}^{2}}
\latex{5.}
\latex{\times}
\latex{0.}
\latex{1}
\latex{6}
\latex{8}
\latex{1}
\latex{1}
\latex{2}
\latex{1.}
\latex{\textcolor{0f92c9}{9}}
\latex{\textcolor{0f92c9}{2}}
\latex{\textcolor{0f92c9}{6}}
\latex{\textcolor{0f92c9}{2}}
Based on the changes
of the product:
\latex{\textcolor{0f92c9}{3}}
\latex{\textcolor{0f92c9}{7}}
When multiplying decimal numbers by decimal numbers, the decimal places in the product must be the same as the sum of the decimal digits of the terms.
Example:
\latex{3.}
\latex{\times}
\latex{1}
\latex{2}
\latex{7}
\latex{4}
\latex{8}
\latex{4}
\latex{4}
\latex{4.}
\latex{\textcolor{009acb}{7}}
\latex{\textcolor{009acb}{4}}
\latex{\textcolor{009acb}{2}}
\latex{\textcolor{009acb}{9}}
\latex{\textcolor{009acb}{0}}
\latex{\textcolor{009acb}{4}}
\latex{1}
\latex{1}
\latex{8}
\latex{8}
\latex{3}
\latex{9.}
\latex{6}
\latex{1.}
\latex{2}
\latex{4}
\latex{\times}
\latex{3.}
\latex{\textcolor{009acb}{4}}
\latex{\textcolor{009acb}{0}}
\latex{\textcolor{009acb}{8}}
\latex{\textcolor{009acb}{3}}
\latex{\textcolor{009acb}{8}}
\latex{\textcolor{009acb}{1}}
\latex{1}
\latex{6.}
\latex{\times}
\latex{\times}
\latex{0.}
\latex{6}
\latex{6}
\latex{0}
\latex{0}
\latex{1}
\latex{3}
\latex{3}
\latex{1.}
\latex{7}
\latex{1}
\latex{\textcolor{009acb}{5}}
\latex{\textcolor{009acb}{4}}
\latex{\textcolor{009acb}{2}}
\latex{\textcolor{009acb}{9}}
\latex{\textcolor{009acb}{3}}
\latex{\textcolor{009acb}{0}}
When there are various terms, multiply them one by one.
\latex{\underbrace{(-0.7)\times5.12}_{\text{-3.584}} \times (-4.1)=\underbrace{(-3.584)\times(-4.1)}_{\text{+14.6944}}=14.6944}
Exercises
{{exercise_number}}. How many decimal places are in the following products without simplifying or expanding them? Estimate the products. Perform the multiplications.
  1. \latex{3.72\times4}
  1. \latex{0.107\times106}
  1. \latex{5.4\times(-2)}
  1. \latex{31.31\times0}
  1. \latex{3.6\times3.14}
  1. \latex{2.8\times2.5}
  1. \latex{0.62\times1.5}
  1. \latex{10.25\times10}
  1. \latex{4.04\times4.04}
  1. \latex{4.04\times100}
  1. \latex{1.5\times1.2\times1.8}
  1. \latex{1.3\times0.1416}
{{exercise_number}}. Calculate the products. What do you notice?
  1. \latex{168\times24\\16.8\times24\\ 1.68\times24}
  1. \latex{16.8\times24\\1.68\times 24\\ 0.168\times24}
  1. \latex{16.8\times2.4\\1.68\times 2.4\\ 0.168\times2.4}
  1. \latex{16.8\times7\\1.68\times 0.7\\ 0.168\times0.07}
  1. \latex{1.68\times7\\1.68\times 0.07\\ 1.68\times0.007}
{{exercise_number}}. Estimate the products before performing the multiplications.
  1. \latex{(+7.25)\times(+0.8)\times(-2)}
  1. \latex{(-1.25)\times(+12)\times(-0.08)\times(-1,000)}
{{exercise_number}}. Multiply \latex{ 3.45 } by \latex{ 2.4 }. Change one of the factors so that the product becomes
  1. two times;
  1. four times;
  1. ten times greater.
{{exercise_number}}. Arrange the products in descending order. How many times greater is the largest product than the smallest one? (Try to answer without performing the multiplications.)
  1. A) \latex{\fcolorbox{f6b900}{fef6e3}{$3.72\times1.8$}\qquad} B) \latex{\fcolorbox{f6b900}{fef6e3}{$37.2\times1.8$}\qquad} C) \latex{\fcolorbox{f6b900}{fef6e3}{$-3.72\times1.8$}\qquad} D) \latex{\fcolorbox{f6b900}{fef6e3}{$0.372\times1.8$}}
  1. A) \latex{\fcolorbox{f6b900}{fef6e3}{$0.25\times4\times3.2$}\qquad} B) \latex{\fcolorbox{f6b900}{fef6e3}{$-0.25\times3.2$}\qquad} C) \latex{\fcolorbox{f6b900}{fef6e3}{$0.25\times4\times0.32$}\qquad} D) \latex{\fcolorbox{f6b900}{fef6e3}{$0.25\times0.32\times0$}}
  1. A) \latex{\fcolorbox{f6b900}{fef6e3}{$-5.6\times8\times12.5$}\qquad} B) \latex{\fcolorbox{f6b900}{fef6e3}{$-5.6\times(-0.8)\times12.5$}\qquad} C) \latex{\fcolorbox{f6b900}{fef6e3}{$-5.6\times(-0.8)\times(-12.5)$}}
{{exercise_number}}. Decide which sum or product is larger. Check your answer by calculating.
  1. \latex{4.5\times12 \text{\quad or\quad }4.05\times120}
  1. \latex{6.2\times0.54 \text{\quad or\quad }0.62\times5.4}
  1. \latex{26.8\times1.1 \text{\quad or\quad }2.68\times11}
  1. \latex{-3.4 + 1.5\times2.4 \text{\quad or\quad }(-3.4 + 1.5)\times2.4}
{{exercise_number}}. When preparing medications, pharmacists work with very small amounts. One of the pills contains \latex{ 25 } \latex{ mg } of active ingredient and \latex{ 47.715 } \latex{ mg } of lactose. How many \latex{ grams } of active ingredient and lactose does a patient take in one year if the doctor prescribed \latex{ 2 } pills per day?
{{exercise_number}}. Peter and his dad ride bicycles from the village to the town. They leave home at the same time. Peter covers a distance of \latex{ 18.4 } \latex{ km\;per\;hour }, while his father covers \latex{ 16.8 } \latex{ km\;per\;hour }. Peter arrives in the city in \latex{ 1.5 } \latex{ hours }. How far is Peter's father from the city when his son arrives there?
{{exercise_number}}. Extension cables are manufactured at a factory. Each cable is \latex{ 3.2\; m } long, while \latex{ 8 \;cm } of additional wire is also needed to fit the plug. \latex{ 78 } extension cables are manufactured during a shift.
How many wires are needed during a shift? Calculate in several ways.
{{exercise_number}}. You are buying drapery and ready-made, sheer curtains. You buy \latex{ 5.7 } \latex{ m } of drapery, \latex{ 2.7 } \latex{ m } less than sheer curtains. \latex{ 1 } \latex{ metre } of drapery costs €\latex{ 16 }, while \latex{ 1 } \latex{ metre } of sheer curtain costs €\latex{ 23}. How much do you have to pay?
{{exercise_number}}. The sides of a square tablecloth are \latex{ 1.6 \;m } long. A lace is sewn on its sides.
How many \latex{ metres } of lace are needed if an extra \latex{ 2 \;cm } must be left at every corner?
{{exercise_number}}. 
A house is built on a rectangular plot. One side of the plot is \latex{ 12.4 \,m } long, while the other is \latex{ 2.5 } times longer.
How long will the fence be if the walls of the house take up a \latex{ 25.6 } \latex{ m } long section and the gate is \latex{ 4.5 } \latex{ m } wide?
How many \latex{ square \,metres } is the area of the plot?
Gate
House
\latex{a = 12.4 {\,m}}
{{exercise_number}}. Calculate the areas of the rectangles in \latex{ square \;metres } if the lengths of their sides are
  1. \latex{a = 6 {\,m }\,5 {\, cm};\\\;\;b=10{\,m }\,3{\,cm};}
  1. \latex{a =1.9 {\,m};\\\;\;\;b=372{ \,cm};}
  1. \latex{a =520{\,cm};\\\;\;\;b=3.8{ \,m}.}
{{exercise_number}}. How many \latex{ square \;centimetres } is the surface area, and how many \latex{ cubic \;centimetres } is the volume of a wooden cube with \latex{ 1.5 \;cm } long edges? A large cube is built using \latex{ 27 } of these small cubes.
What is the surface area and the volume of the large cube?
{{exercise_number}}. 
The length of a fish tank is \latex{ 0.82 \;m }, its width is \latex{ 35 \;cm }, and its height is \latex{ 474 \;mm }.
How many \latex{ square \;centimetres } of glass sheet were used to make the fish tank if it does not have a top?
How many \latex{ litres } is the volume of the fish tank? (Ignore the thickness of the glass sheets.)
Quiz
For which two-digit number is it valid that the number \latex{ 1.2} times  larger consists of the same digits as the original?

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