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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
Division with decimals
Dividing decimals by integers
Example 1
At a ski resort, \latex{ 70 } cups of hot chocolate were sold in one day. The revenue was €\latex{ 189 }. How much does one cup of hot chocolate cost?
Solution
\latex{70} cups \latex{189\,euros}
\latex{\underline{1 \text{ cup } 189\div70}}
Estimation: \latex{2 \lt 189\div70 \lt 3.}
Perform the division using the column method.
After the first step, the remainder is \latex{ 49 }.
Since \latex{49 = 49.\textcolor{199fe4}{0}}, you can continue dividing.
\latex{1}
\latex{8}
\latex{9}
\latex{\div}
\latex{7}
\latex{0}
\latex{=}
\latex{2.}
\latex{7}
\latex{4}
\latex{9}
\latex{\textcolor{0092ca}{0}}
\latex{0}
Check:
\latex{2.}
\latex{7}
\latex{\times}
\latex{7}
\latex{0}
\latex{1}
\latex{8}
\latex{9.}
\latex{0}
When you reach the place where the decimal point is in the dividend, put a decimal point in the quotient as well.
\latex{189\div70 = 2.7}, so a cup of hot chocolate costs \latex{2.7\,euros}.
Example 2
On a trip to Austria, three friends spent \latex{ 92.5 \;euros } on fuel.
How much do they have to pay each if they want to divide this amount equally among themselves?
Solution
\latex{3} persons \latex{92.5\,euros}
\latex{\underline{1 }}  person \latex{\underline{92.5\div3\;euros}}
Estimation: \latex{90\div3 \approx30.}
Perform the division using the column method.
The quotient is a recurring decimal. You must round the amount of money to be paid by one person.
\latex{9'}
\latex{2.}
\latex{5}
\latex{\div}
\latex{3}
\latex{=}
\latex{3}
\latex{0.}
\latex{8}
\latex{3}
\latex{3...}
\latex{\textcolor{0092ca}{0}}
\latex{2}
Check:
\latex{5}
\latex{1}
\latex{1}
\latex{1}
\latex{3}
\latex{0.}
\latex{8}
\latex{3}
\latex{3}
\latex{\times}
\latex{3}
\latex{9}
\latex{2.}
\latex{4}
\latex{9}
\latex{9}
\latex{\textcolor{0092ca}{0}}
\latex{30.8333...=30.8\dot{3}}
\latex{30.8333... \approx 30.83}. One person must pay \latex{ 30.83 \;euros } for the fuel.
Integers or decimals can be divided by an integer. The quotient is either a terminating or recurring decimal.
If the divisor is a decimal, the division can be performed as if it were an integer.
Every integer can be expressed as a decimal.
E.g.: \latex{189 = 189.0.}
Division with decimals
Example 3
Perform the following divisions using the column method.
  1. \latex{1.19\div3.5}
  1. \latex{17.5\div1.25}
Solution
In both cases, the divisor is a decimal. Use the principle that the quotient does not change when the dividend and the divisor are multiplied by the same number other than zero.
  1. Estimation: \latex{1.19\div3.5\approx1\div4; 0\lt1\div4 \lt1.}
If the dividend and the divisor are multiplied by \latex{ 10 }, the divisor becomes an integer.
\latex{1.19}
\latex{\div}
\latex{3.5}
\latex{11.9}
\latex{\div}
\latex{35}
\latex{\textcolor{329fe3}{\times10}}
\latex{\textcolor{329fe3}{\times10}}
\latex{1.19\div3.5 = 11.9\div35.}
Perform the division using the column method.
Answer: \latex{1.19\div3.5 = 0.34.}
\latex{1}
\latex{1'.}
\latex{9}
\latex{\div}
\latex{3}
\latex{=}
\latex{5}
\latex{0.}
\latex{3}
\latex{4}
\latex{\textcolor{0092ca}{0}}
\latex{1}
Check:
\latex{1}
\latex{9}
\latex{1}
\latex{4}
\latex{0}
\latex{0.}
\latex{3}
\latex{4}
\latex{\times}
\latex{3.}
\latex{5}
\latex{1}
\latex{0}
\latex{2}
\latex{1}
\latex{7}
\latex{0}
\latex{1.}
\latex{1}
\latex{9}
\latex{0}
\latex{\frac{1.19}{3.5}=\frac{11.9}{35}}
  1. Estimation: \latex{17.5\div2\lt17.5\div1.25\lt17.5\div1,}
              \latex{8.75\lt17.5\div 1.25\lt17.5.}
If the dividend and the divisor are multiplied by \latex{ 100 }, the divisor becomes an integer.
\latex{17.5}
\latex{\div}
\latex{1.25}
\latex{1,750}
\latex{\div}
\latex{125}
\latex{\textcolor{329fe3}{\times100}}
\latex{\textcolor{329fe3}{\times100}}
\latex{17.5\div1.25 = 1,750\div125.}
Perform the division using the column method.
Answer: \latex{17.5\div1.25 = 14.}
\latex{1,}
\latex{7}
\latex{5'}
\latex{0}
\latex{\div}
\latex{1}
\latex{2}
\latex{5}
\latex{=}
\latex{1}
\latex{4}
\latex{5}
Check:
\latex{0}
\latex{0}
\latex{0}
\latex{1}
\latex{4}
\latex{\times}
\latex{1.}
\latex{2}
\latex{5}
\latex{2}
\latex{8}
\latex{7}
\latex{0}
\latex{1}
\latex{7.}
\latex{5}
\latex{0}
\latex{ · 100 }
\latex{ ÷ 1.25 }
If the divisor is a decimal, multiply both the dividend and the divisor so that the divisor becomes an integer, then perform the division.
Exercises
{{exercise_number}}. What is the quotient? What do you notice?
  1. \latex{168\div24\\16.8\div24\\ 1.68\div24}
  1. \latex{16.8\div24\\1.68\div 24\\ 0.168\div24}
  1. \latex{16.8\div2.4\\1.68\div 2.4\\ 0.168\div2.4}
  1. \latex{16.8\div7\\1.68\div 0.7\\ 0.168\div0.07}
  1. \latex{1.68\div7\\1.68\div 0.07\\ 1.68\div0.007}
{{exercise_number}}. How should the dividend be modified so that the quotient remains unchanged if the divisor is multiplied by
  1. \latex{ 10};
  2. \latex{ 100};
  3. \latex{ 1,000?}
Think of examples for each case.
{{exercise_number}}. Perform the divisions, then check your answers.
  1. \latex{(-41.76)\div6.4}
  1. \latex{(-4.176)\div(-0.64)}
  1. \latex{417.6\div(-6.4)}
  1. \latex{(-4.176)\div(-6.4)}
  1. \latex{(-4.176)\div0.64}
  1. \latex{-41.76\div0.64}
{{exercise_number}}. What is the quotient if \latex{ (–752) } is divided by its tenth part?
{{exercise_number}}. The dividend is \latex{ 72.42 }, and the quotient is \latex{ (–6.035) }. What is the divisor?
{{exercise_number}}. A number multiplied by \latex{ 3.14 } is \latex{ 79.285 }. What is this number?
{{exercise_number}}. Perform the following divisions. What do you notice?
  1. \latex{74.5\div0.1\qquad\qquad9.23\div0.1\qquad\qquad 756.3\div 0.01\qquad\qquad 123.456\div 0.001}
  1. \latex{2\div0.5\qquad\qquad\qquad 97.8\div0.5\qquad\qquad 4.05\div0.5\qquad\qquad 432.12\div0.5}
  1. \latex{1.2\div0.25\qquad\qquad 0.45\div0.25\qquad\qquad 10.2\div0.25\qquad\qquad 109.15\div0.25}
{{exercise_number}}. Express the following in \latex{ minutes. }
  1. \latex{0.1\,hours}
  1. \latex{0.5\,hours}
  1. \latex{1.3\,hours}
  1. \latex{2.5\,hours}
  1. \latex{0.15\,hours}
  1. \latex{0.75\,hours}
  1. \latex{0.25\,hours}
  1. \latex{2.25\,hours}
{{exercise_number}}. Children were asked how long it takes them to arrive at school. Express their answers in \latex{ hours. }
  1. Andrew: \latex{6\,min}
  1. Bob: \latex{3\,min}
  1. Cecilia: \latex{12\,min}
  1. Dora: \latex{24\,min}
  1. Edith: \latex{9\,min}
  1. Flora: \latex{36\,min}
  1. Gabe: \latex{18\,min}
  1. Hedi: \latex{15\,min}
{{exercise_number}}. A train completes a \latex{ 186 \;km } long journey in \latex{ 2.5 \;hours }. What distance does it cover in one \latex{ hour ?}
{{exercise_number}}. For a trip abroad, you exchange \latex{ 200 } \latex{ dollars } for \latex{ euros }. How many \latex{ euros } do you get if the daily exchange rate is \latex{ 1:1.04 } (\latex{ euros } \latex{ to } \latex{ dollars })?
Round to the nearest one. Find the actual exchange rate and solve the exercise based on it.
{{exercise_number}}. The circumference of a bicycle's wheel is \latex{ 1.8 } \latex{ m }. How many times does it turn when riding the bicycle over a distance of \latex{ 1.5 } \latex{ km }? Round the result to the nearest \latex{ metre }.
{{exercise_number}}. Calculate
  1. \latex{1.2} times the sum of \latex{-3.5} and \latex{-4.3};
  1. the difference of the quotient of \latex{ 2.64 } and \latex{ 0.8 } and \latex{ 47.8 }.
{{exercise_number}}. Perform the following operations.
  1. \latex{(62.5-3.75)\div2.5\qquad\qquad62.5 \div2.5-3.75\qquad\qquad62.5\div 2.5-3.75\div2.5}
  1. \latex{420.6\div1.2 + 32.04\div1.2\qquad\qquad(420.6 + 32.04)\div1.2\qquad\qquad420.6 + 32.04\div1.2}
{{exercise_number}}. The Jaragua dwarf gecko is one of the smallest reptiles in the world. It lives on an island in the Dominican Republic. Its body length is \latex{ 0.016 \;m }. The reticulated python, one of the longest reptiles in the world, inhabits Southeast Asia. It can measure up to \latex{ 10 \;m } in length.
How many times longer is the reticulated python than the Jaragua dwarf gecko?
{{exercise_number}}. A group is planning a trip to England. The distances on the map are shown in \latex{ kilometres }. During the trip, the group will cover a distance of \latex{ 2,400 } \latex{ km }. How many \latex{ miles } is \latex{ 2,400 } \latex{ km } if \latex{ 1 } \latex{ mile } equals approximately \latex{ 1.6 } \latex{ km }?
{{exercise_number}}. A supermarket ordered \latex{ 1,000 } \latex{ kg } of apples. How many crates of apples did they get if one crate contained \latex{ 22.8 } \latex{ kg } on average?
{{exercise_number}}. Haruka lives in Japan. Her grandmother gave her $\latex{500} for her \latex{20th} birthday. Since she is going to Paris next month, she decides to exchange the \latex{dollars} for \latex{euros}.
How many \latex{ euros } will Haruka get if \latex{ 1 \;dollar } is \latex{ 155 \;yen } and \latex{ 1 \;euro } is \latex{ 161 \;yen ?}
{{exercise_number}}. After her trip to Paris, Haruka had \latex{ 35.6 \;euros } left. She decided to exchange the remaining \latex{ euros } for \latex{ yen }. How many yen did she get if \latex{ 1 \;yen } is worth \latex{ 0.0062 \;euros ?}
{{exercise_number}}. The length and the width of the sides of a sandpit are \latex{ 1.1 \;m}. What height will the sand reach if \latex{ 0.5 \;cubic \;metres} of sand is distributed uniformly in it?
*{{exercise_number}}. Gabe's three dogs ate \latex{ 16.2 \;kg } of dog food in two weeks. Titan ate \latex{ 2 } times more, while Charlie ate \latex{ 1.5 } times more than Maddock.
How many \latex{ kilograms } of dog food did each of them eat?
Quiz
What is the \latex{ 1,001 }st digit after the decimal point in the quotient of \latex{1\div 1,001}\latex{?}

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