Exampler Problems on Rational Numbers - Part 1

Commutative Property of Rational Numbers

Commutative Property of Rational Numbers over Addition, Subtraction, Multiplication and Multiolication

Closure Property of Rational Numbers

Closure Properties on Rational Numbers

Rational Numbers are Closed Under Subtraction!

Closure Property of Rational Numbers during Multiplication & Division

More Problems on Properties of Triangles

Solve the Problems using the Properties of Triangles

Solution for Question 5 (a) & (b):

Questions 7 to 12:

Solving More
Problems on Simple Equation

1.  Pranay’s father is seven times his age now. Two years later his father will be five times as old as his son will be. Find their present ages.  

2.  A number consists of two digits in which the tens digit exceeds the units digit by 6. The number itself is equal to ten times the sum of digits. Find the number. 

3.  The present ages of A and B are in the ratio 7:5. Ten years later, their ages will be in the ratio 9:7. Find their present ages.

Valid Points and Exercise on Triangles and Its Properties

Points to Remember on Triangles

1. The six elements of a triangle are its three angles and the three sides.

2. The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle.   A triangle has 3 medians. 

3. The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle.    A triangle has 3 altitudes. 

4. An exterior angle of a triangle is formed, when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle. 

5. A property of exterior angles: The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles. 

6. The angle sum property of a triangle: The total measure of the three angles of a triangle is 180°. 

7. A triangle is said to be equilateral, if each one of its sides has the same length. In an equilateral triangle, each angle has measure 60°. 

8. A triangle is said to be isosceles, if at least any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure. 

9. Property of the lengths of sides of a triangle: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference between the lengths of any two sides is smaller than the length of the third side.  This property is useful to know if it is possible to draw a triangle when the lengths of the three sides are known. 

10. In a right angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs. 

11. Pythagoras property: In a right-angled triangle, the square on the hypotenuse = the sum of the squares on its legs. If a triangle is not right-angled, this property does not hold good.  This property is useful to decide whether a given triangle is right-angled or not.

Pythagoras property may be stated as follows:

In a right-angled triangle, the square on the hypotenuse = sum of the squares on the legs.

7. Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm. 

8. The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.

Decimal Number System

Rounding of Decimals 

Rounding of decimals would be really useful for estimating amounts of money, duration of time, measure of distances and many other physical quantities.

To round a decimal : 

1. First underline the digit that is to be rounded.  Then look at the digit to the right of the underlined digit.  

2. If that digit is less than 5, then the underlined digit remains the same. 

3. If that digit is greater than or equal to 5, add 1 to the underlined digit. 

4. After rounding of, ignore all the digits after the underlined digit.

Example 1: 

Round 2.367 to the nearest whole number. 

Solution:  Underline the digit to be rounded - 2.367.  Since the digit next to the underlined digit is 3 which is less than 5, the underlined digit 2 remains the same. 

Example 2: 

Round 99.95 to the nearest tenth place. 

Solution: Underline the digit to be rounded 99.95. Since the digit right to the tenth place is 5, we add 1 to the tenth place (underlined digit) of 99.95 and we get 100.0. 

Example 3: 

Round 52.6583 upto 2 places of decimal. 

Solution:  Round 52.6583 upto 2 places of decimal means round to the nearest hundredths place. Underline the digit in the hundredth place of 52.6583 gives 52.6583.  

We observe that the digit after the hundredth place value is 8 which is more than 5. Therefore, we should add 1 to the underlined digit. Hence, we get 52.66. So the rounded value of 52.6583 upto 2 places of decimal is 52.66. 

Example 4: 

Round 189.0007 upto 3 places of decimal. 

Solution:  Round 189.0007 upto 3 places of decimal means rounding to the nearest thousandth place.  Underline the digit in the thousandth place of 189.0007 gives 189.0007  

In 189.0007 we observe that the digit next to the thousandth place value is 7, which is greater than 5.  Therefore, we should add 1 to the underlined digit. Hence, we get 189.001.  

So the rounded value of 189.0007 upto 3 places of decimal is 189.001 

Exercise 1.1

1. Round each of the following decimals to the nearest whole number. (i) 8.71 (ii) 26.01 (iii) 69.48 (iv) 103.72 (v) 49.84 (vi) 101.35 (vii) 39.814 (viii) 1.23 

2. Round each decimal number to the given place value. (i) 5.992 to tenth place (ii)  21.805 to hundredth place (iii) 35.0014 to thousandth place 

3. Round the following decimal numbers upto 1 places of decimal (i) 123.37  (ii) 19.99  (iii) 910.546 

4. Round the following decimal numbers upto 2 places of decimal (i) 87.755  (ii) 301.513  (iii) 79.997 

5. Round the following decimal numbers upto 3 place of decimal (i) 24.4003  (ii) 1251.2345  (iii)61.00203