Question

11. Simplify the following.

$\sf (a) \: \bigg( \frac{2}{3} \bigg) + \bigg( \frac{ - 4}{5} \bigg) + 1 + \bigg( \frac{ - 2}{3} \bigg) + \bigg( \frac{ - 11}{5} \bigg) \\ \\ \sf (b) \: \bigg( \frac{5}{8} \bigg) + \bigg( \frac{ - 8}{9} \bigg) + 0 + \bigg( \frac{ - 13}{3} \bigg) + \bigg( \frac{17}{24} \bigg)$


12. Subtract.

$\sf (a) \: \bigg( \frac{ - 3}{4} \bigg) \: from \: \bigg( \frac{1}{2} \bigg) \\ \\ \sf (b) \: \bigg( \frac{5}{8} \bigg) \: from \: \bigg( \frac{ - 3}{14} \bigg)$


13. What should be added to ( -7/20 ) to get ( -2/5 ) ?

14. The sum of two rational numbers is ( -3/7 ). If one of the number is ( -5/8 ) find the other.

15. The sum of two rational numbers is ( -5/8 ). If one of the number is ( -6/11 ), find the other number.​

Answer 1

11. To simplify the following:

$\bf (a)\ \left(\dfrac{2}{3}\right)\ +\ \left(\dfrac{-4}{5}\right)\ +\ 1\ +\ \left(\dfrac{-2}{3}\right)\ +\ \left(\dfrac{-11}{5}\right)$

Let's first find the LCM.

The common denominators are 3 and 5, hence the LCM would be 3 × 5 = 15.

Converting all the fractions to fractions with 15 as the LCM,

$\implies\ \sf \left(\dfrac{2\ \times\ 5}{3\ \times\ 5}\right)\ +\ \left(\dfrac{-4\ \times\ 3}{5\ \times 3}\right)\ +\ \dfrac{1\ \times 15}{1\ \times15}\ +\ \left(\dfrac{-2\ \times\ 5}{3\ \times\ 5}\right)\ +\ \left(\dfrac{-11\ \times\ 3}{5\ \times\ 3}\right)\\\\\\\\=\ \left(\dfrac{10}{15}\right)\ +\ \left(\dfrac{-12}{15}\right)\ +\ \dfrac{15}{15}\ +\ \left(\dfrac{-10}{15}\right)\ +\ \left(\dfrac{-33}{15}\right)\\\\\\\\=\ \dfrac{-30}{15}\\\\\\\\=\ \bf -2$

$\bf (b)\ \left(\dfrac{5}{8}\right)\ +\ \left(\dfrac{-8}{9}\right)\ +\ 0\ +\ \left(\dfrac{-13}{3}\right)\ +\ \left(\dfrac{17}{24}\right)$

Let's first find the LCM.

Since there's isn't a common denominator, 8 × 9 × 3 × 24 = 72 will be the denominator.

Converting the fractions into fractions with 72 as the denominator,

$\implies\ \sf \left(\dfrac{5\ \times\ 9}{8\ \times\ 9}\right)\ +\ \left(\dfrac{-8\ \times\ 8}{9\ \times 8}\right)\ +\ 0\ +\ \left(\dfrac{-13\ \times\ 24}{3\ \times\ 24}\right)\ +\ \left(\dfrac{17\ \times\ 3}{24\ \times\ 3}\right)\\\\\\\\=\ \left(\dfrac{45}{72}\right)\ +\ \left(\dfrac{-64}{72}\right)\ +\ 0\ +\ \left(\dfrac{-312}{72}\right)\ +\ \left(\dfrac{51}{72}\right)\\\\\\\\=\ \dfrac{-280}{72}\\\\\\\\=\ \bf -3.8$

12. To subtract one fraction from the other.

$\bf (a)\ \left(\dfrac{-3}{4}\right)\ from\ \left(\dfrac{1}{2}\right)$

To subtract 'x' from 'y' implies 'y - x'. So in this case,

$\implies\ \sf \left(\dfrac{1}{2}\right)\ -\ \left(\dfrac{-3}{4}\right)$

The LCM is 4. Converting 'x' into a fraction with 4 as the denominator,

$\implies\ \sf \left(\dfrac{1\ \times\ 2}{2\ \times\ 2}\right)\ -\ \left(\dfrac{-3}{4}\right)\\\\\\=\ \dfrac{2}{4}\ -\ \dfrac{-3}{4}\\\\\\=\ \dfrac{5}{4}\\\\\\=\ \bf 1.25$

$\bf (b)\ \left(\dfrac{5}{8}\right)\ from\ \left(\dfrac{-3}{14}\right)$

The LCM is 8 × 14 = 112. Converting 'x' and 'y' into fractions with 112 as the denominator,

$\implies\ \sf \left(\dfrac{-3\ \times\ 8}{14\ \times\ 8}\right)\ -\ \left(\dfrac{5\ \times\ 14}{8\ \times\ 14}\right)\\\\\\=\ \dfrac{-24}{112}\ -\ \dfrac{70}{112}\ \\\\\\=\ \dfrac{-94}{112}\\\\\\=\ \bf 0.839$

13. Let the unknown number be 'x'.

According to the question, -7/20 + x = -2/5.

x = -2/5 - (-7/20)

x = -2 × 4/5 × 4 - (-7/20)  [∵ LCM = 20]

x = -8/20 - (-7/20)

x = -1/20

14. Let the other rational number be 'x'.

According to the question, -5/8 + x = -3/7.

x = -3/7 - (-5/8)

x = -3 × 8/7 × 8 - (-5 × 7/8 × 7)  [∵ LCM = 56]

x = -24/56 - (-35/56)

x = 11/56

15. Let the other rational number be 'x'.

According to the question, -6/11 + x = -5/8.

x = -5/8 - (-6/11)

x = -5 × 11/8 × 11 - (-6 × 8/11 × 8)  [∵ LCM = 88]

x = -55/88 - (-48/88)

x = -7/88

Answer 2

Required Answers :-

11.

a]

$\sf \dfrac{2}{3} + \dfrac{-4}{5} + 1 + \dfrac{-2}{3} + \dfrac{-11}{5}$

$\sf\dfrac{2}{3}+\dfrac{-4}{5} + \dfrac{1}{1} + \dfrac{-2}{3} + \dfrac{-11}{5}$

$\sf\dfrac{10+ (-12) + 15 + (-10) + (-33)}{15}$

$\sf\dfrac{10 - 12+ 15 -10 -33}{15}$

$\sf\dfrac{25 -55}{15}$

$\sf\dfrac{-30}{15}$

$\sf-2$

b]

$\sf\dfrac{5}{8} + \dfrac{-8}{9} + \dfrac{-13}{3} + \dfrac{17}{24}$

$\sf \dfrac{45 -64 -312 + 51}{72}$

$\sf\dfrac{96 - 376}{72}$

$\sf\dfrac{-280}{72}$

12.

a]

$\sf\dfrac{1}{2} - \dfrac{-3}{4}$

$\sf\dfrac{2 - (-3)}{4}$

$\sf\dfrac{2+3}{4}$

$\sf\dfrac{5}{4}$

b]

$\sf\dfrac{-3}{14} - \dfrac{5}{8}$

$\sf\dfrac{-24 -70}{112}$

$\sf\dfrac{-94}{112}$

13.

Let the number to be added be a

$\sf a + \dfrac{-7}{20} = \dfrac{-2}{5}$

$\sf a = \dfrac{-2}{5}-\dfrac{7}{20}$

$\sf a = \dfrac{-8 + 7}{20}$

$\sf a = \dfrac{-1}{20}$

14.

Let the number be a

$\sf a+ \dfrac{-5}{8} = \dfrac{-3}{7}$

$\sf a = \dfrac{-3}{7} - \dfrac{5}{8}$

$\sf a = \dfrac{-24 +35}{56}$

$\sf a =\dfrac{11}{56}$

15.

Let the number be a

$\sf a + \dfrac{-6}{11} = \dfrac{-5}{8}$

$\sf a = \dfrac{-5}{8} - \dfrac{6}{11}$

$\sf a = \dfrac{-55 -48}{88}$

$\sf a = \dfrac{-103}{88}$