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Example 1. The scale for a map is 5 cm to 1 km. Express this scale as a ratio.
Solution. It is best to express both quantities in the smaller of the two units, in this case centimeters. The two quantities are 5 cm and 100,000 cm, since 1 km = 100,000 cm. Write
Ratio = 1005000 A ratio is the division of one quantity by another
Ratio = 2oT)oo Canceling the fraction
The scale of the map is 1:20,000.
Ratios are frequently used in everyday life, as illustrated in the following examples.
Example 2. Jacob and Nathan buy a $10 Lotto ticket. Jacob contributes $6.50 and Nathan $3.50.
(a) Write their contributions as a ratio.
(b) If they win $600, how much of the winnings should Jacob receive?
Solution. For (a), write Ratio
3.5 X 2
For (b), write
13 + 7 = 20. $600 h- 20 = $30
This ratio is not written with counting numbers Rearranging the ratio to obtain counting numbers
Adding the numbers in the ratio
The winnings are divided into 20 equal parts of $30
Jacob gets 13 lots of $30 = $390 Nathan gets 7 lots of $30 = $210 Total = $600
Jacob should receive $390.
Example 3. William and his classmates are going on a camping holiday. The school regulations state that the teacher-to-student ratio must be 1:5. If there are 30 students in William’s class, how many teachers must go with them?
Number of teachers Number of students
Arranging the fraction with number of students = 30
Six teachers are needed.
References: Equivalent Fractions, Gears, Proportion, Rate.
References: Algebraic Fractions, Canceling.
One number is the reciprocal of another number if their product is equal to 1. The word product here means that we multiply. For example, the reciprocal of the number | is |, because | x | = 1. The easiest way to obtain the reciprocal of a fraction is to turn it upside down. The reciprocal of 2| is -jy, because 2| = The reciprocal of 5 is 5-
Reciprocals can be used to solve equations, as set out in the following example. Example. Solve the equation
1 _ 2
1 _ 2
— = - Taking the reciprocal of both sides of the equation
x = 1.5
References: Fractions, Recurring Decimal, Solving an Equation.
A rectangle is a four-sided polygon, called a quadrilateral, which has its opposite sides parallel and all its four angles are right angles. An everyday word for rectangle is oblong. A rectangle has opposite sides equal in length. It also has two axes of symmetry and rotational symmetry of order two.
The two diagonals of the rectangle are equal in length and bisect each other (see figure a). Both pairs of “opposite” triangles formed by the two diagonals are congruent isosceles triangles, as shown in figure b. A rectangle that has all four sides equal in length is called a square.
If the lengths of two adjacent sides of a rectangle are known, we can find the perimeter of the rectangle, and we can also find the area of the rectangle.
Example. Find the perimeter and the area of the rectangle drawn in figure c.
Solution. The opposite sides of a rectangle are equal in length, so the other two sides are of lengths 3 cm and 4 cm, respectively. Write
Perimeter =3+4+3+4 = 14 cm
The perimeter is 14 cm.
Alternatively, we could find the sum of the two adjacent sides, which is 7 cm, and double the answer.
For the area, write
Area = 4x3 Area of rectangle = length x width
= 12 cm2
The area is 12 cm2.
The length of the diagonal of the rectangle can be calculated using the theorem of Pythagoras. If the length of the diagonal is denoted by d, then
d2 — 32 + 42 Pythagoras’ theorem = 9+16 = 25
= V25 Using square roots = 5
The length of the diagonal of the rectangle is 5 cm.
References: Algebra, Congruent Figures, Constructions, Diagonal, Isosceles Triangle, Linear Equation, Polygon, Pythagoras’ Theorem, Quadrilateral, Square, Square Root.
This is another name for a cuboid.
For re-entrant quadrilaterals see the entry Acute Angle.
References: Acute Angle, Concave.
Reflection is a geometrical transformation that occurs when a shape, called the object, is reflected in a mirror to obtain the image. Suppose we reflect the triangle labeled
ABC in figure a in the mirror m to obtain the image A'B'C'. The object on the left of the mirror is labeled with the letters A, B, and C in a clockwise direction, but the image is labeled A', B', and C in a counterclockwise direction. The orientation, or sense, has changed. Reflection is the only transformation that changes sense. We sometimes say that reflection is an indirect transformation, because it changes sense.