Exercise 4C
Question 1
(i) $\frac{4}{9} \text{ by } \frac{5}{7}$
Sol :
$=\frac{4}{9} \times \frac{5}{7}=\frac{20}{63}$
(ii) $\frac{-7}{10} \text{ by } \frac{5}{8}$
Sol :
$=\frac{-7}{10} \times \frac{5}{8}=\frac{-7}{16}$
(iii) $\frac{11}{12} \text{ by } \frac{-8}{15}$
Sol :
$=\frac{11}{12} \times \frac{-8}{15}=\frac{-22}{45}$
(iv) $\frac{-9}{13} \text{ by } \frac{65}{-3}$
Sol :
$=\frac{-9}{13} \times \frac{65}{-3}=15$
Question 2
Multiplying :
(i) $\frac{-9}{16} \text{ by } 60$
Sol :
$=\frac{-9}{16} \times 60=\frac{-135}{4}$
(ii) $\frac{-9}{7} \text{ by } -14$
Sol :
$=\frac{-9}{7} \times -14 =18$
(iii) $\frac{28}{-9} \text{ by } -27$
Sol :
$=\frac{28}{-9} \times -27=84$
(iv) $-48 \text{ by } \frac{17}{-16}$
Sol :
$=-48 \times \frac{17}{-16}=51$
Question 3
Simplify and express in standard form :
(i) $\frac{-9}{7} \times \frac{-35}{6}$
Sol :
$=\frac{-3}{1} \times \frac{-5}{2}=\frac{15}{2}$
(ii) $\frac{-115}{28} \times \frac{-49}{69}$
Sol :
$=\frac{-5}{4} \times \frac{-7}{3}=\frac{35}{12}$
(iii) $\frac{-15}{14} \times \frac{112}{-105}$
Sol :
$=\frac{8}{7}$
(iv) $\frac{-39}{3} \times \frac{19}{5} \times \frac{-45}{38}$
Sol :
$=\frac{117}{2}$
Question 4
(i) $\frac{11}{18} \times (-9) +\frac{-3}{4} \times \frac{5}{6}$
Sol :
$=\frac{-11}{2}-\frac{5}{8}=\frac{-44-5}{8}=\frac{-49}{8}$
(ii) $\left(\frac{26}{9} \times \frac{3}{7}\right)-\left(\frac{13}{14} \times \frac{-2}{3}\right)$
Sol :
$=\frac{26}{21}+\frac{13}{21}=\frac{26+13}{21}=\frac{39}{21}$
(iii) $\left(\frac{-3}{2} \times \frac{4}{5}\right)+\left(\frac{9}{5}\times \frac{10}{3}\right)-\left(\frac{1}{2} \times \frac{3}{4}\right)$
Sol :
$=\frac{-6}{5}+\frac{6}{1}-\frac{3}{8}$
$=\frac{-48+240-15}{40}=\frac{177}{40}=4\frac{17}{40}$
Question 5
Write the reciprocal of :
(i) $\frac{6}{23}$
Sol :
$=\frac{23}{6}$
(ii) -8
Sol :
$=\frac{1}{-8}$
(iii) $\frac{-10}{21}$
Sol :
$=\frac{-21}{10}$
(iv) $\mid \frac{-4}{5} \mid $
Sol :
$=\frac{4}{5}$
$\frac{5}{4}$
(v) $\frac{2}{3}-\frac{4}{5}$
Sol :
$=\frac{10-12}{15}=\frac{-2}{15}=\frac{15}{-2}$
(v) $\frac{-7}{-8}$
Sol :
$=\frac{7}{8}=\frac{8}{7}$
Question 6
(i) $1 \text{ by } \frac{1}{8}$
Sol :
$=1 \div \frac{1}{8}=1 \times \frac{8}{1}$
=8
(ii) $8 \text{ by } \frac{-4}{9}$
Sol :
$=8 \div \frac{-4}{9}=8 \times \frac{-9}{4}$
=-18
(iii) $\frac{1}{-4} \text{ by } \frac{1}{12}$
Sol :
$=\frac{1}{-4} \div \frac{1}{12}$
$=\frac{1}{-4} \times \frac{12}{1}$
=-3
(iv) $\frac{1}{2} \text{ by } \frac{1}{-4}$
Sol :
$=\frac{1}{2} \div \frac{1}{-4}$
$=\frac{1}{2} \times \frac{-4}{1}$
=-2
(v) $\frac{-5}{6} \text{ by } \frac{1}{-2}$
Sol :
$=\frac{-5}{6} \div \frac{1}{-2}$
$=\frac{-5}{6} \times \frac{-2}{1}$
$=\frac{5}{3}$
(vi) $\frac{65}{12} \text{ by } \frac{13}{15}$
Sol :
$=\frac{65}{12} \div \frac{13}{15}$
$=\frac{65}{12} \times \frac{15}{13}$
$=\frac{5}{4} \times \frac{5}{1}=\frac{25}{4}$
(vii) $\frac{50}{77} \text{ by } (-25)$
Sol :
$=\frac{50}{77} \times \frac{1}{-25}=\frac{-2}{77}$
Question 7
Verify that $(x \div y)^{-1}=x^{-1} \div y^{-1}$ by taking $x=\frac{7}{11}$ , $y=\frac{-22}{21}$
Sol :
$\left(\frac{7}{11} \div \frac{-22}{21}\right)^{-1}=\left(\frac{7}{11}\right)^{-1} \div \left(\frac{-22}{21}\right)^{-1}$
$\left(\frac{7}{11} \times \frac{-21}{22}\right)^{-1}=\frac{11}{7} \div \left(\frac{-21}{22}\right)$
$\left(\frac{-147}{242} \right)^{-1}=\frac{11}{7} \times \left(\frac{-22}{21}\right)$
$\frac{-242}{147}=\frac{-242}{147}$
Question 8
Divide the reciprocal of $\frac{2}{5} \times \frac{-10}{3} \text{ by } \left(\frac{1}{2}-\frac{1}{3}\right)$
Sol :
$=\frac{2}{5} \times \frac{-10}{3}=\frac{-4}{3}$
Reciprocal$=\frac{-3}{4}$
A.T.Q
$=\frac{2}{5} \times \frac{-10}{3} \text{ by } \left(\frac{1}{2}-\frac{1}{3}\right)$
$=\frac{-3}{4} \div \left(\frac{1}{2}-\frac{1}{3}\right)$
$=\frac{-3}{4} \div \left(\frac{1 \times 3-1 \times 2}{6}\right)$
$=\frac{-3}{4} \div \left(\frac{3-2}{6}\right)$
$=\frac{-3}{4} \times \left(\frac{6}{1}\right)$
$=\frac{-3}{4} \times \left(\frac{6}{1}\right)$
$=\frac{-3 \times 3}{2}=\frac{-9}{2}$
Question 9
The product of two rational numbers is $\frac{-35}{18}$. If one of the numbers is $\frac{5}{12}$ , find the other.
Sol :
Let the other number is x
A.T.Q
$x \times \frac{5}{12}=\frac{-35}{18}$
$x=\frac{-35}{18} \times \frac{12}{5}$
$x=\frac{-7}{9} \times \frac{6}{1}$
$x=\frac{-7}{3} \times \frac{2}{1}=\frac{-7 \times 2}{3}=\frac{-14}{3}$
Question 10
By what number should $\frac{-55}{18}$ be divided to get $\frac{-22}{9}$ ?
Sol :
Let that number be x
$\frac{-55}{18} \div x=\frac{-22}{9}$
$\frac{-55}{18} \times \frac{1}{x}=\frac{-22}{9}$
$ \frac{1}{x}=\frac{-22}{9}\times \frac{-18}{55} $
$ \frac{1}{x}=\frac{-2}{1}\times \frac{-2}{5} $
$ \frac{1}{x}=\frac{-2 \times -2}{1 \times 5}=\frac{4}{5} $
$x=\frac{5}{4}$
Question 11
How many pieces of tape $3\frac{4}{7}$ cm long can be cut from a long tape , which is 1 meter 75 cm?
Sol :
[1 meter = 100 cm]
1 meter 75 cm = 100 cm + 75 cm =175 cm
Number of pieces=Total length ÷ small portion
$=175 \div 3\frac{4}{7}$
$=175 \div \frac{3 \times 7+4}{7}$
$=175 \div \frac{25}{7}=175 \times \frac{7}{25}$
=7×7=49 pieces
Question 12
The cast of $5\frac{1}{3}$ m of cloth is $85\frac{1}{3}$. Find the cost of cloth per meter.
Sol :
$5\frac{1}{3}$ m of cloth is ₹$85\frac{1}{3}$
$\frac{5 \times 3+1}{3}$ m=₹$\frac{85 \times 3+1}{3}$
$\frac{16}{3}m=₹ \frac{256}{3}$
16 m = ₹ 256
$1 m =\frac{256}{16}$
1m=₹ 16
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