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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Proportion
Direct proportion
Using direct proportion to solve problems
Inverse proportion
Exercises with inverse proportions
Ratios
Proportional division
Mixed exercises
Percentages
Probability, statistics
Supplementary material
Ratios
Every object and figure on a diorama is a smaller copy of the real ones. In Mark's diorama, \latex{ 1\;cm }, \latex{ 2 \;cm }, and \latex{ 3 \;cm } correspond to \latex{ 120 \;cm, } \latex{ 240 \;cm }, and \latex{ 360 \;cm }, respectively, in real life.
You can notice that the quotients of the corresponding values are equal:
 
\latex{\frac{1\;cm}{120\;cm}=\frac{2\;cm}{240\;cm}=\frac{3\;cm}{360\;cm}=\frac{1}{120}}.
The scale of this diorama is \latex{1\div120}.
Example 1
The height of a Lilliputian adult is \latex{ 15\;cm }, while Gulliver is \latex{ 180\; cm } tall.
  1. How many times taller is Gulliver than the Lilliputians? What is the ratio of Gulliver's height to a Lilliputian's height?
  2. How many times smaller is a Lilliputian than Gulliver? What is the ratio of a Lilliputian's height to Gulliver's height?
During his journey, Gulliver meets the Lilliputians (dwarfs) and the giants as well. Among the dwarfs, he feels tall, while among the giants, he feels tiny.
Solution
  1. When comparing to the height of the Lilliputians, the unit is \latex{ 1\;cm }.
\latex{1\;cm} is \latex{\frac{1}{15}} times \latex{15\;cm} ,

\latex{180\;cm} is \latex{\frac{180}{15}} times \latex{15\;cm} or \latex{\frac{180}{15}=12} times \latex{15\;cm}.

Gulliver is \latex{ 12 } times taller than the Lilliputians.
The ratio of Gulliver's height to the Lilliputians' height is \latex{12:1} (\latex{12} to \latex{1}).
  1. Comparing to Gulliver’s height.
\latex{1\;cm} is \latex{\frac{1}{180}} times \latex{180\;cm} ,

\latex{15\;cm} is \latex{\frac{15}{180}} times \latex{180\;cm} , thus \latex{\frac{15}{180}=\frac{1}{12}} times \latex{180\;cm}.

The height of the Lilliputians is \latex{\frac{1}{12}} of Gulliver's height.
The ratio of a Lilliputian's height to Gulliver's height is \latex{1:12} (\latex{1} to \latex{12}).
\latex{12:1=\frac{12}{1}}
\latex{1:12=\frac{1}{12}}
Example 2
The palm of the king of the Giants is \latex{ 150 \;cm } long, while Gulliver's palm is \latex{ 20\;cm } long.
  1. How many times longer is the king's palm than Gulliver's palm? What is the ratio of the king's palm to Gulliver's palm?
  2. How many times is the length of Gulliver's palm that of the king's? What is the ratio of Gulliver's palm to the king's palm?
Solution
  1. When comparing to Gulliver's palm \latex{ (20\;cm) }, the unit is \latex{ 1 \;cm }.
\latex{1} \latex{ cm } is \latex{\frac{1}{20}} times \latex{20\;cm} ,

a \latex{150\;cm} is \latex{\frac{150}{20}} times \latex{20\;cm}, that is, \latex{\frac{150}{20}=\frac{15}{2}} times \latex{ (20\;cm) }.

The palm of the king of the Giants is \latex{\frac{15}{2}} times longer than Gulliver's palm. The ratio of the king's palm to Gulliver's palm is \latex{15\div2} (\latex{15} to \latex{2}).
  1. When comparing to the king's palm, that is, to \latex{ 150\;cm },
\latex{1\;cm} is \latex{\frac{1}{150}} times \latex{150\;cm},

\latex{20\;cm} is \latex{\frac{20}{150}} times \latex{150\;cm} , that is,  \latex{\frac{20}{150}=\frac{2}{15}} times \latex{150\;cm}.

Gulliver's palm is \latex{\frac{2}{15}} times the length of the king's palm. The ratio of Gulliver's palm to the king's palm is \latex{2:15} (\latex{2} to \latex{15}).
\latex{\frac{150}{20}=150:20}

\latex{150\div 20=15: 2}

\latex{\frac{20}{150}=20: 150}

\latex{20: 150=2: 15}
The ratio of two numbers is the quotient of the two numbers.
The ratio of two numbers shows how much the first number is compared to the second number.
For example, the ratio \latex{3: 7} shows that \latex{3} is \latex{\frac{3}{7}} times \latex{7}, because

\latex{\frac{3}{7}\times 7=\frac{3}{\cancel{7}_{1} }\times \cancel{7}^{1}=3}.
Example 3
Write down the ratio of \latex{12} to \latex{15} and \latex{4} to \latex{5}.
Solution
The ratio of \latex{ 12 } to \latex{ 15 } is \latex{12: 15=\frac{12}{15}=\frac{4}{5}}.
The ratio of \latex{ 4 } to \latex{ 5 } is \latex{4: 5=\frac{4}{5}}.
The two ratios are equal, that is,
\latex{12: 15=4: 5} (\latex{12} to \latex{15} is the same as \latex{4} to \latex{5}).
\latex{ 12:15 = 4:5 } are equivalent ratios.
The ratio of two numbers does not change if both numbers are multiplied or divided by the same number, other than zero.
\latex{4}
\latex{5}
\latex{15}
\latex{12}
\latex{:}
\latex{:}
Ratios can be simplified and expanded.
Example 4
Write down the ratio of \latex{ 1 \;litre } to \latex{ 3 \;decilitres }.
Solution
To determine the ratio of the two quantities, they must be converted to the same unit of measurement.
\latex{ 1\;litre = 10 \;decilitres}
The ratio is \latex{10: 3=\frac{10}{3}}.
This means that \latex{ 10\; dl } is \latex{\frac{10}{3}} times \latex{3\;dl}.
\latex{\frac{10}{\cancel{3}_{1} }\times 3^{1}dl =10\;dl}
Note:
If the exercise asked you to write down the ratio of \latex{3\,dl} and \latex{1\,litre}, the ratio would be \latex{3: 10=\frac{3}{10}}.
\latex{\frac{3}{\cancel{10}^{1} } \times \cancel{10}^{1}dl=3\;dl}
Exercises
{{exercise_number}}. How many times \latex{12} is
  1. \latex{1};
  1. \latex{2};
  1. \latex{6};
  1. \latex{30};
  1. \latex{36};
  1. \latex{60?}
{{exercise_number}}. How many times \latex{24} is
  1. \latex{1};
  1. \latex{6};
  1. \latex{8};
  1. \latex{24};
  1. \latex{36};
  1. \latex{60?}
{{exercise_number}}. Write down the ratios of the following numbers.
  1. \latex{5}  and  \latex{7}
  1. \latex{4}  and  \latex{8}
  1. \latex{9}  and  \latex{6}
  1. \latex{10}  and  \latex{12}
  1. \latex{0.25}  and  \latex{1}
  1. \latex{500}  and  \latex{1000}
  1. \latex{0.5}  and  \latex{0.25}
  1. \latex{0.17}  and  \latex{0.51}
{{exercise_number}}. Write down the ratios of the following quantities.
  1. \latex{3\;kg}  and  \latex{5\;kg}
  1. \latex{8\;cm}  and  \latex{12\;cm}
  1. \latex{2\;dm}  and  \latex{1\;m}
  1. \latex{1\;km}  and  \latex{250\;m}
{{exercise_number}}. List some numbers whose ratio is
  1. \latex{2};
  1. \latex{\frac{1}{2} };
  1. \latex{0.25};
  1. \latex{\frac{5}{2} };
  1. \latex{\frac{7}{11} }.
{{exercise_number}}. Write the following ratios as the ratio of two positive integers.
  1. \latex{\frac{1}{3} } : \latex{\frac{2}{3} };
  1. \latex{\frac{1}{4} } : \latex{\frac{1}{2} };
  1. \latex{\frac{1}{2} } : \latex{\frac{3}{4} };
  1. \latex{\frac{5}{7} } : \latex{\frac{4}{5} };
  1. \latex{\frac{5}{17} } : \latex{\frac{4}{7} };
{{exercise_number}}. The ratio of two numbers is \latex{ 2:3 }. The first number is how many times the second number?
{{exercise_number}}. A car travelled \latex{ 15\;km } out of the \latex{ 60\;km } that separate two cities. Write down the following ratios.
  1. The ratio of the distance covered by the car to the distance between the two cities.
  2. The ratio of the remaining distance to the distance between the two cities.
  3. The ratio of the distance covered to the remaining distance.
{{exercise_number}}. Pete is \latex{ 8 } and Sam is \latex{ 12 } \latex{ years } old. What is the ratio of the ages of the boys? What will the ratio of their ages be in \latex{ 6 } \latex{ years }? When was or will be the ratio of their ages \latex{ 1:2 }?
{{exercise_number}}. In the image, the midpoints of a triangle's sides are connected, forming four triangles with equal areas.
  1. What is the ratio of the area of one of the small triangles to the area of the large triangle?
  2. What is the ratio of the perimeter of one of the small triangles to the perimeter of the large triangle?
  3. What is the ratio of the area of the red triangle to the area of the yellow triangles?
  4. What is the ratio of the perimeter of the yellow triangles to the perimeter of the red triangle?
Quiz
In \latex{ 1,000\;minutes }, how many times more circles does the hour hand of a clock complete than the minute hand?
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