Question paper, Set 1

Each part $\displaystyle = \frac{1}{13}$

Shaded region $\displaystyle = \frac{1}{13} + \frac{1}{13} + \frac{1}{13}$ $\displaystyle = \frac{3}{13}$

Answer: $\displaystyle \frac{3}{13}$

First figure: $\displaystyle \frac{11}{11} = 1$ (whole)

Second figure: $\displaystyle \frac{1}{11}$

Total shaded region $\displaystyle = 1 + \frac{1}{11}$

$\displaystyle = \frac{11}{11} + \frac{1}{11} = \frac{12}{11}$

Answer: $\displaystyle 1 + \frac{1}{11} = \frac{12}{11}$

Proper fraction: numerator $<$ denominator.

$7 < 9$, $4 < 5$, $4 < 11$

Proper fractions: $\displaystyle \frac{7}{9}, \frac{4}{5}, \frac{4}{11}$

Answer: $\displaystyle \frac{7}{9}, \frac{4}{5}, \frac{4}{11}$

Divide $66$ by $5$: $5 \times 13 = 65$, $66 - 65 = 1$

Quotient $= 13$, remainder $= 1$

$\displaystyle \therefore \frac{66}{5} = 13 + \frac{1}{5} = 13\frac{1}{5}$

Answer: $\displaystyle 13\frac{1}{5}$

$\displaystyle 9\frac{3}{8} = \frac{(9 \times 8) + 3}{8}$

$\displaystyle = \frac{72 + 3}{8} = \frac{75}{8}$

Answer: $\displaystyle \frac{75}{8}$

Like fractions have the same denominator.

$\displaystyle \frac{1}{12}, \frac{11}{12}, \frac{7}{12}$ all have denominator $12$.

Answer: $\displaystyle \frac{1}{12}, \frac{11}{12}, \frac{7}{12}$

$\displaystyle \frac{14}{11} = \frac{14 \times 2}{11 \times 2} = \frac{28}{22}$

$\displaystyle \frac{14}{11} = \frac{14 \times 3}{11 \times 3} = \frac{42}{33}$

$\displaystyle \frac{14}{11} = \frac{14 \times 4}{11 \times 4} = \frac{56}{44}$

$\displaystyle \frac{14}{11} = \frac{28}{22} = \frac{42}{33} = \frac{56}{44}$

Answer: $\displaystyle \frac{28}{22}, \frac{42}{33}, \frac{56}{44}$

(i) $\displaystyle \frac{63}{54}$

H.C.F of $63, 54 = 9$; $\displaystyle \frac{63 \div 9}{54 \div 9} = \frac{7}{6}$

(ii) $\displaystyle \frac{-153}{-81} = \frac{153}{81}$, since $(-)/(-) = (+)$

H.C.F of $153, 81 = 9$; $\displaystyle \frac{153 \div 9}{81 \div 9} = \frac{17}{9}$

(iii) $\displaystyle \frac{-5}{20} = -\frac{5}{20}$, since $(-)/(+) = (-)$

H.C.F of $5, 20 = 5$; $\displaystyle -\frac{5 \div 5}{20 \div 5} = -\frac{1}{4}$

Answer: (i) $\displaystyle \frac{7}{6}$ (ii) $\displaystyle \frac{17}{9}$ (iii) $\displaystyle -\frac{1}{4}$

Every negative number is less than $0$ and less than every positive number.

Among negative numbers, the bigger the number after the minus sign, the smaller it is: $-20 < -10 < -2 < -1$

Ascending order: $-20, -10, -2, -1, 11, 30$

Answer: $-20, -10, -2, -1, 11, 30$

$\displaystyle \frac{6}{-5} = -\frac{6}{5}$, since $(+)/(-) = (-)$

L.C.M of $10, 2, 5, 12, 3 = 2 \times 3 \times 5 \times 2 = 60$

$\displaystyle -\frac{13}{10} = \frac{-13 \times 6}{10 \times 6} = -\frac{78}{60}$

$\displaystyle \frac{3}{2} = \frac{3 \times 30}{2 \times 30} = \frac{90}{60}$

$\displaystyle -\frac{19}{10} = \frac{-19 \times 6}{10 \times 6} = -\frac{114}{60}$

$\displaystyle -\frac{6}{5} = \frac{-6 \times 12}{5 \times 12} = -\frac{72}{60}$

$\displaystyle \frac{23}{12} = \frac{23 \times 5}{12 \times 5} = \frac{115}{60}$

$\displaystyle -\frac{5}{3} = \frac{-5 \times 20}{3 \times 20} = -\frac{100}{60}$

Comparing numerators: $\displaystyle \frac{115}{60} > \frac{90}{60} > -\frac{72}{60} > -\frac{78}{60} > -\frac{100}{60} > -\frac{114}{60}$

$\displaystyle \Rightarrow \frac{23}{12} > \frac{3}{2} > -\frac{6}{5} > -\frac{13}{10} > -\frac{5}{3} > -\frac{19}{10}$

Answer: $\displaystyle \frac{23}{12}, \frac{3}{2}, -\frac{6}{5}, -\frac{13}{10}, -\frac{5}{3}, -\frac{19}{10}$

Rule: $\displaystyle \frac{p}{q} > \frac{r}{s}$ if $ps > qr$.

$14 \times 8 = 112$ and $9 \times 13 = 117$

$\displaystyle 112 < 117 \Rightarrow \frac{14}{9} < \frac{13}{8}$

Greater: $\displaystyle \frac{13}{8}$

Answer: $\displaystyle \frac{13}{8}$

$\displaystyle = \frac{21 - 49 - 23}{40}$

$\displaystyle = -\frac{51}{40}$

Answer: $\displaystyle -\frac{51}{40}$

L.C.M of $5, 6, 12 = 2 \times 3 \times 5 \times 2 = 60$

$\displaystyle = \frac{72 + 50 - 65}{60}$

$\displaystyle = \frac{122 - 65}{60}$

$\displaystyle = \frac{57}{60} = \frac{19}{20}$

Answer: $\displaystyle \frac{19}{20}$

$\displaystyle = -\frac{9}{16} + \frac{11}{12} - \frac{5}{4}$, since $-(-x) = +x$ and $+(-x) = -x$

L.C.M of $16, 12, 4 = 2 \times 2 \times 4 \times 3 = 48$

$\displaystyle -\frac{9}{16} = \frac{-9 \times 3}{16 \times 3} = -\frac{27}{48}$

$\displaystyle \frac{11}{12} = \frac{11 \times 4}{12 \times 4} = \frac{44}{48}$

$\displaystyle -\frac{5}{4} = \frac{-5 \times 12}{4 \times 12} = -\frac{60}{48}$

$\displaystyle = \frac{-27 + 44 - 60}{48}$

$\displaystyle = \frac{44 - 87}{48}$

$\displaystyle = -\frac{43}{48}$

Answer: $\displaystyle -\frac{43}{48}$

$\displaystyle = \frac{7}{1} - \frac{11}{5}$

$\displaystyle = \frac{35 - 11}{5}$

$\displaystyle = \frac{24}{5}$

Answer: $\displaystyle \frac{24}{5}$

Sign: $(-) \times (+) \times (+) = (-)$

$\displaystyle = -\frac{15 \times 8 \times 5}{7 \times 25 \times 12}$

$\displaystyle = -\frac{(3 \times 5) \times (2 \times 2 \times 2) \times 5}{7 \times (5 \times 5) \times (2 \times 2 \times 3)}$

Cancelling the common factors $2, 2, 3, 5, 5$: $\displaystyle = -\frac{2}{7}$

Answer: $\displaystyle -\frac{2}{7}$

Sign: $(-) \times (-) \times (+) = (+)$

$\displaystyle = \frac{14 \times 13 \times 36}{117 \times 12 \times 35}$

$\displaystyle = \frac{(2 \times 7) \times 13 \times (2 \times 2 \times 3 \times 3)}{(3 \times 3 \times 13) \times (2 \times 2 \times 3) \times (5 \times 7)}$

Cancelling the common factors $2, 2, 3, 3, 7, 13$: $\displaystyle = \frac{2}{3 \times 5} = \frac{2}{15}$

Answer: $\displaystyle \frac{2}{15}$

$\displaystyle 6\frac{1}{5} = \frac{(6 \times 5) + 1}{5} = \frac{31}{5}$

$\displaystyle 4\frac{1}{6} = \frac{(4 \times 6) + 1}{6} = \frac{25}{6}$

$\displaystyle 6\frac{1}{5} \times 4\frac{1}{6} = \frac{31}{5} \times \frac{25}{6} = \frac{31 \times 25}{5 \times 6}$

Cancelling common factors: $\displaystyle = \frac{155}{6}$

$155 \div 6$: quotient $25$, remainder $5$; $\displaystyle = 25\frac{5}{6}$

Answer: $\displaystyle \frac{155}{6} = 25\frac{5}{6}$

$\displaystyle 2\frac{5}{6} = \frac{(2 \times 6) + 5}{6} = \frac{17}{6}$

$\displaystyle 5\frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{16}{3}$

$\displaystyle 2\frac{5}{6} \div 5\frac{1}{3} = \frac{17}{6} \div \frac{16}{3} = \frac{17}{6} \times \frac{3}{16}$ (reciprocal of $\displaystyle \frac{16}{3}$ is $\displaystyle \frac{3}{16}$)

$\displaystyle = \frac{17 \times 3}{6 \times 16} = \frac{17}{32}$

Answer: $\displaystyle \frac{17}{32}$

$\displaystyle \frac{21}{8} = 2 + \frac{5}{8}$, so it lies between $2$ and $3$.

Divide each unit into $8$ equal parts of $\displaystyle \frac{1}{8}$; $\displaystyle 2 = \frac{16}{8}$.

Count $21$ parts to the right of $0$ and mark $\displaystyle \frac{21}{8}$.

Answer: Divide each unit into $8$ equal parts; $\displaystyle \frac{21}{8} = 2\frac{5}{8}$ is the 5th point after $2$ (the 21st point from $0$).