Question

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Answer 1

Solution :

[ SECTION A ]

1. Define improper fraction and mixed fraction .

Answer :

Define an improper fraction to be a number of the form p/q where p & q € I where p ≥ q .

p and q may or may not be in their lowest terms i.e , ( p, q ) = or ≠ 1 .

Examples of improper fractions are 3/2 , 9/7 , etc .

A mixed fraction refers to the combination of a whole number and a fraction .

Example : 1 ¾ is a mixed fraction .

Conversion from improper fraction to mixed fraction :

Let the fraction be p/q , p > q and (p,q) ≠ 1 .

Let p = q k + c , where k and c are both constants , k & c ≥ 1 .

Fraction becomes :

→ ( qk + c )/q

→ k + (c/q)

→ k (c/q) .

This , k (c/q) is the mixed fraction . Expanding this , multiply k with the q and add c , divided all over by q .

2. Change the following improper fraction into mixed fraction :

• 15/4 & 98/9

Answer :

In the fraction, 15/4 , the numerator is 15 . This can be expressed as 4 × 3 + 3 .

So , the mixed fraction becomes 3 ¾ .

For 98/9 , the numerator here is 98 .

This can be written as :

98 = 9 × 10 + 8 .

So , the mixed fraction becomes :

10 ⁸/9 .

3. Reduce the given fractions into its lowest terms :

• 21/49 & 30/65

Answer :

For 21/49 , we can clearly notice that there is a multiple of 7 in both the numerator and denominator .

Cancelling it ;

=> 3/7 .

This is the required fraction in its lowest terms .

For 30/65 , we can clearly notice that there is a multiple of 5 in both the numerator and denominator .

Cancelling it ;

=>6/13 .

This is the required fraction in its lowest terms .

4. Which fraction is greater :

• 3/5 or 4/7 • 2/5 or 4/9

Answer :

3/5

= 3 × 7 / 5 × 7

= 21/35

4/7

= 4 × 5/ 7 × 5

= 20/35

21/35 > 20/35 .

So, 3/5 > 4/7 .

2/5

= 2 × 9/ 5 × 9 =

18/45

4/9

=> 4 × 5/ 9 × 5

=> 20/45

Now , 20/45 > 18/45

So, 4/9 > 2/5 .

5. Add the following :

• ¾ & 2/3. • 3/7 & 5/4

Answer :

The LCM of 3 and 4 is 12 .

3/4 + 2/3

=> [ 3 × 3 ] + [ 2 × 4 ] / 12

=> [ 9 + 8 ]/12

=> 17/12

=> 1 ⁵/12 ( answer )

The LCM of 4 and 7 is 28 .

3/7 + 5/4

=> [ 3 × 4 ] + [ 5 × 7 ]/28

=> ( 12 + 35 )/28

=> 47/28

=> 1 19/28 ( answer ) .

6. Subtract the following :

• 1/7 - 3/4 • 2/11 - 1/5

Answer :

1/7 - 3/4

=> [ 1 × 4 ] - [ 3 × 7 ]/12

=> { 4 - 21 }/12

=> - 17/ 12

=> - 1 5/12

2/11 - 1/5

=> [ 2 × 5 ] - [ 1 × 11 ]/55

=> { 10 - 11 }/55

=> -1 / 55 .

___________________________

Answer 2

Solution :

[ SECtion A CONTINUED ]

7. Two supplementary angles are in the ratio 4 : 5 . Find the angles .

Answer :

The supplementary angles are in the ratio 4 : 5 .

Let us assume that these angles are 4k and 5k respectively where k is any arbitrary constant > 1 .

Now, we know that , supplementary angles always add up to 180° .

So,

4k + 5k = 180°

=> 9k = 180°

=> k = 20°

Angle 1 = 4k = 4 × 20° = 80°

Angle 2 = 5k = 5 × 20° = 100°

Verification : Angle 1 + Angle 2 = 80° + 100° = 180°

Hence Verified

8. Two complementary angles are in the ratio 2 : 3 . Find the angles .

Answer :

The supplementary angles are in the ratio 2 : 3 .

Let us assume that these angles are 2k and 3k respectively where k is any arbitrary constant > 1 .

Now, we know that , complementary angles always add up to 90° .

So,

2k + 3k = 90°

=> 5k = 90°

=> k = 18°

Angle 1 = 2k = 2 × 18° = 36°

Angle 2 = 4k = 3 × 18° = 54°

Verification : Angle 1 + Angle 2 = 36° + 54° = 90°

Hence Verified

9. The measure of two complementary angles are (3x + 8) and (4x -2) . Find x .

Answer :

Here , the complementary angles are ( 3x + 8 )° and ( 4x - 2 ) ° respectively.

Complementary angles add up to 90°

→ ( 3x + 8 ) + (4x - 2 ) = 90°

→ 7x + 6 = 90°

→ 7x = 84°

→ x = 12°

Angle 1 = 3x + 8 = 36 + 8 = 44°

Angle 2 = 46°

Verification : Angle 1 + Angle 2 = 44° + 46° = 90°

10. Find the value of the angle if it's supplementary angle is ⅔ times is it .

Answer :

Let us assume that the required angle is x .

So , the supplementary angle becomes ⅔ x .

Supplementary angles add up to 180°

→ x + ⅔ x = 180°

→ 5/3 x = 180°

→ x = 108°

→ ⅔ x = 72°

Thus , the required angle is 108°

______________________

Section B :

1. In the adjoining figure , find the value of a and b if a : b = 4 : 5 .

Answer :

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0){\thicklines}\put(2,2){\line(2,0){5}}\put(4,2){\line(1,2){1}}\put(3.5,3){\bf{a}}\put(5.3,3){\bf{b}}\put(2,2){\circle*{0.1}}\put(7,2){\circle*{0.1}}\put(5,4){\circle*{0.1}}\end{picture}$

Here , a : b = 4 : 5 .

Let a and b be 4x and 5x respectively .

From the figure , it is clear that a and b are supplementary angles .

So ,

a + b = 180°

=> 4x + 5x = 180°

=> 9x = 180°

=> x = 20°

a = 4x = 80°

b = 5x = 100°

Verification : a + b = 80° + 100° = 180°

2. Two complementary angles differ by 40°. Find the angles .

Answer :

Let the two complementary angles be x and y respectively .

As per the data given ;

x - y = 40° ......... (1)

Also ,

x + y = 90° ........... (2)

Adding both the equations ;

2x = 130°

=> x = 65°

y = 25° .

Thus , x and y are 65° and 25° respectively .

_. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _. _.

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