Question paper, Set 2

$3$ parts of $\displaystyle \frac{1}{11}$ each are added.

$\displaystyle \frac{1}{11} + \frac{1}{11} + \frac{1}{11} = \frac{3}{11}$

Answer: $\displaystyle \frac{3}{11}$

First two figures: $\displaystyle \frac{5}{5} + \frac{5}{5} = 1 + 1 = 2$

Third figure: $\displaystyle \frac{3}{5}$

Total shaded region $\displaystyle = 2 + \frac{3}{5}$

$\displaystyle = \frac{10}{5} + \frac{3}{5} = \frac{13}{5}$

Answer: $\displaystyle 2 + \frac{3}{5} = \frac{13}{5}$

Improper fraction: numerator $\ge$ denominator.

$27 > 10$, $25 > 9$, $12 = 12$

Improper fractions: $\displaystyle \frac{27}{10}, \frac{25}{9}, \frac{12}{12}$

Answer: $\displaystyle \frac{27}{10}, \frac{25}{9}, \frac{12}{12}$

Divide $68$ by $7$: $7 \times 9 = 63$, $68 - 63 = 5$

Quotient $= 9$, remainder $= 5$

$\displaystyle \therefore \frac{68}{7} = 9 + \frac{5}{7} = 9\frac{5}{7}$

Answer: $\displaystyle 9\frac{5}{7}$

$\displaystyle 2\frac{4}{5} = \frac{(2 \times 5) + 4}{5}$

$\displaystyle = \frac{10 + 4}{5} = \frac{14}{5}$

Answer: $\displaystyle \frac{14}{5}$

Denominators: $12, 12, 12$

They are the same, so these are like fractions.

Answer: Like fractions

$108 = 12 \times 9$, so $\displaystyle \frac{11}{12} = \frac{11 \times 9}{12 \times 9} = \frac{99}{108}$

$22 = 11 \times 2$, so $\displaystyle \frac{11}{12} = \frac{11 \times 2}{12 \times 2} = \frac{22}{24}$

Answer: Boxes: $99$ and $24$

(i) $\displaystyle \frac{54}{81}$

H.C.F of $54, 81 = 27$; $\displaystyle \frac{54 \div 27}{81 \div 27} = \frac{2}{3}$

(ii) $\displaystyle \frac{-4}{-44} = \frac{4}{44}$, since $(-)/(-) = (+)$

H.C.F of $4, 44 = 4$; $\displaystyle \frac{4 \div 4}{44 \div 4} = \frac{1}{11}$

(iii) $\displaystyle \frac{36}{-45} = -\frac{36}{45}$, since $(+)/(-) = (-)$

H.C.F of $36, 45 = 9$; $\displaystyle -\frac{36 \div 9}{45 \div 9} = -\frac{4}{5}$

Answer: (i) $\displaystyle \frac{2}{3}$ (ii) $\displaystyle \frac{1}{11}$ (iii) $\displaystyle -\frac{4}{5}$

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: $-57 < -51 < -24 < -19$

Ascending order: $-57, -51, -24, -19, 13, 17$

Answer: $-57, -51, -24, -19, 13, 17$

$\displaystyle \frac{17}{-10} = -\frac{17}{10}$, since $(+)/(-) = (-)$

L.C.M of $10, 12, 2, 6, 3 = 2 \times 3 \times 5 \times 2 = 60$

$\displaystyle -\frac{17}{10} = \frac{-17 \times 6}{10 \times 6} = -\frac{102}{60}$

$\displaystyle \frac{17}{12} = \frac{17 \times 5}{12 \times 5} = \frac{85}{60}$

$\displaystyle -\frac{1}{2} = \frac{-1 \times 30}{2 \times 30} = -\frac{30}{60}$

$\displaystyle -\frac{1}{6} = \frac{-1 \times 10}{6 \times 10} = -\frac{10}{60}$

$\displaystyle \frac{1}{2} = \frac{1 \times 30}{2 \times 30} = \frac{30}{60}$

$\displaystyle \frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}$

Comparing numerators: $\displaystyle \frac{85}{60} > \frac{40}{60} > \frac{30}{60} > -\frac{10}{60} > -\frac{30}{60} > -\frac{102}{60}$

$\displaystyle \Rightarrow \frac{17}{12} > \frac{2}{3} > \frac{1}{2} > -\frac{1}{6} > -\frac{1}{2} > -\frac{17}{10}$

Answer: $\displaystyle \frac{17}{12}, \frac{2}{3}, \frac{1}{2}, -\frac{1}{6}, -\frac{1}{2}, -\frac{17}{10}$

$28 \times 18 = 504$ and $24 \times 21 = 504$

$\displaystyle 504 = 504 \Rightarrow \frac{28}{24} = \frac{21}{18}$

Answer: Equal: $\displaystyle \frac{28}{24} = \frac{21}{18}$

$\displaystyle = \frac{51 - 60 + 14}{44}$

$\displaystyle = \frac{5}{44}$

Answer: $\displaystyle \frac{5}{44}$

L.C.M of $3, 5, 15 = 3 \times 5 = 15$

$\displaystyle = \frac{5 - 9 + 11}{15}$

$\displaystyle = \frac{16 - 9}{15}$

$\displaystyle = \frac{7}{15}$

Answer: $\displaystyle \frac{7}{15}$

Group the like fractions: $\displaystyle = \left(-\frac{12}{13} + \frac{4}{13}\right) + \left(\frac{1}{7} - \frac{11}{7}\right)$

$\displaystyle = \frac{-12 + 4}{13} + \frac{1 - 11}{7}$

$\displaystyle = -\frac{8}{13} - \frac{10}{7}$

L.C.M of $13, 7 = 13 \times 7 = 91$

$\displaystyle = \frac{-56 - 130}{91} = -\frac{186}{91}$

Answer: $\displaystyle -\frac{186}{91}$

$\displaystyle = \frac{5}{1} - \frac{9}{8}$

$\displaystyle = \frac{40 - 9}{8}$

$\displaystyle = \frac{31}{8}$

Answer: $\displaystyle \frac{31}{8}$

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

$\displaystyle = -\frac{2 \times 15 \times 14}{21 \times 32 \times 3}$

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

Cancelling the common factors $2, 2, 3, 7$: $\displaystyle = -\frac{5}{2 \times 2 \times 2 \times 3} = -\frac{5}{24}$

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

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

$\displaystyle = \frac{91 \times 95 \times 21}{38 \times 98 \times 130}$

$\displaystyle = \frac{(7 \times 13) \times (5 \times 19) \times (3 \times 7)}{(2 \times 19) \times (2 \times 7 \times 7) \times (2 \times 5 \times 13)}$

Cancelling the common factors $5, 7, 7, 13, 19$: $\displaystyle = \frac{3}{2 \times 2 \times 2} = \frac{3}{8}$

Answer: $\displaystyle \frac{3}{8}$

$\displaystyle 4\frac{1}{7} = \frac{(4 \times 7) + 1}{7} = \frac{29}{7}$

$\displaystyle 5\frac{4}{9} = \frac{(5 \times 9) + 4}{9} = \frac{49}{9}$

$\displaystyle 4\frac{1}{7} \times 5\frac{4}{9} = \frac{29}{7} \times \frac{49}{9} = \frac{29 \times 49}{7 \times 9}$

Cancelling common factors: $\displaystyle = \frac{203}{9}$

$203 \div 9$: quotient $22$, remainder $5$; $\displaystyle = 22\frac{5}{9}$

Answer: $\displaystyle \frac{203}{9} = 22\frac{5}{9}$

Rule: $\displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

$\displaystyle \frac{13}{10} \div \left(-\frac{17}{10}\right) = \frac{13}{10} \times \left(-\frac{10}{17}\right)$ (reciprocal of $\displaystyle -\frac{17}{10}$ is $\displaystyle -\frac{10}{17}$)

$\displaystyle = -\frac{13 \times 10}{10 \times 17}$, since $(+) \times (-) = (-)$

Cancelling common factors: $\displaystyle = -\frac{13}{17}$

Answer: $\displaystyle -\frac{13}{17}$

Each part $\displaystyle = \frac{1}{9}$; $\displaystyle 1 = \frac{9}{9}$ and $\displaystyle -1 = -\frac{9}{9}$.

Count $7$ parts to the right of $0$: $\displaystyle \frac{7}{9}$.

Count $7$ parts to the left of $0$: $\displaystyle -\frac{7}{9}$.

Answer: Divide $0$ to $1$ and $-1$ to $0$ into $9$ equal parts each; $\displaystyle \frac{7}{9}$ is the 7th point to the right of $0$ and $\displaystyle -\frac{7}{9}$ the 7th point to the left.