Question paper, Set 6

Unshaded parts $= 11 - 2 = 9$

Each part $\displaystyle = \frac{1}{11}$

Unshaded region $\displaystyle = \frac{9}{11}$

Answer: $\displaystyle \frac{9}{11}$

$\displaystyle \frac{19}{11} = \frac{11 + 8}{11}$

$\displaystyle = \frac{11}{11} + \frac{8}{11}$

$\displaystyle = 1 + \frac{8}{11}$

Answer: $\displaystyle \frac{19}{11} = 1 + \frac{8}{11}$

Improper fraction: numerator $\ge$ denominator.

$8 = 8$, $13 > 6$, $7 > 6$

Improper fractions: $\displaystyle \frac{8}{8}, \frac{13}{6}, \frac{7}{6}$

Answer: $\displaystyle \frac{8}{8}, \frac{13}{6}, \frac{7}{6}$

Divide $32$ by $9$: $9 \times 3 = 27$, $32 - 27 = 5$

Quotient $= 3$, remainder $= 5$

$\displaystyle \therefore \frac{32}{9} = 3 + \frac{5}{9} = 3\frac{5}{9}$

Answer: $\displaystyle 3\frac{5}{9}$

$\displaystyle 22\frac{5}{7} = \frac{(22 \times 7) + 5}{7}$

$\displaystyle = \frac{154 + 5}{7} = \frac{159}{7}$

Answer: $\displaystyle \frac{159}{7}$

Denominators: $11, 11, 11$

They are the same, so these are like fractions.

Answer: Like fractions

$\displaystyle \frac{49}{70} = \frac{49 \div 7}{70 \div 7} = \frac{7}{10}$

$\displaystyle \frac{49}{70} = \frac{49 \times 3}{70 \times 3} = \frac{147}{210}$

Answer: $\displaystyle \frac{7}{10}, \frac{147}{210}$

(i) $\displaystyle \frac{112}{88}$

H.C.F of $112, 88 = 8$; $\displaystyle \frac{112 \div 8}{88 \div 8} = \frac{14}{11}$

(ii) $\displaystyle \frac{-8}{-24} = \frac{8}{24}$, since $(-)/(-) = (+)$

H.C.F of $8, 24 = 8$; $\displaystyle \frac{8 \div 8}{24 \div 8} = \frac{1}{3}$

(iii) $\displaystyle \frac{56}{-32} = -\frac{56}{32}$, since $(+)/(-) = (-)$

H.C.F of $56, 32 = 8$; $\displaystyle -\frac{56 \div 8}{32 \div 8} = -\frac{7}{4}$

Answer: (i) $\displaystyle \frac{14}{11}$ (ii) $\displaystyle \frac{1}{3}$ (iii) $\displaystyle -\frac{7}{4}$

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: $-50 < -38 < -23$

Ascending order: $-50, -38, -23, 13, 15, 20$

Answer: $-50, -38, -23, 13, 15, 20$

$\displaystyle \frac{1}{-8} = -\frac{1}{8}$, since $(+)/(-) = (-)$

L.C.M of $8, 12, 2, 3 = 2 \times 2 \times 3 \times 2 = 24$

$\displaystyle \frac{15}{8} = \frac{15 \times 3}{8 \times 3} = \frac{45}{24}$

$\displaystyle -\frac{1}{8} = \frac{-1 \times 3}{8 \times 3} = -\frac{3}{24}$

$\displaystyle \frac{13}{12} = \frac{13 \times 2}{12 \times 2} = \frac{26}{24}$

$\displaystyle -\frac{3}{2} = \frac{-3 \times 12}{2 \times 12} = -\frac{36}{24}$

$\displaystyle -\frac{1}{3} = \frac{-1 \times 8}{3 \times 8} = -\frac{8}{24}$

Comparing numerators: $\displaystyle -\frac{36}{24} < -\frac{8}{24} < -\frac{3}{24} < \frac{26}{24} < \frac{45}{24}$

$\displaystyle \Rightarrow -\frac{3}{2} < -\frac{1}{3} < -\frac{1}{8} < \frac{13}{12} < \frac{15}{8}$

Answer: $\displaystyle -\frac{3}{2}, -\frac{1}{3}, -\frac{1}{8}, \frac{13}{12}, \frac{15}{8}$

$21 \times 40 = 840$ and $24 \times 35 = 840$

$\displaystyle 840 = 840 \Rightarrow \frac{21}{24} = \frac{35}{40}$

Answer: Equal: $\displaystyle \frac{21}{24} = \frac{35}{40}$

$\displaystyle = \frac{-73 + 32 + 52 - 58}{58}$

$\displaystyle = \frac{84 - 131}{58}$

$\displaystyle = -\frac{47}{58}$

Answer: $\displaystyle -\frac{47}{58}$

L.C.M of $9, 3, 6 = 3 \times 3 \times 2 = 18$

$\displaystyle = \frac{14 - 12 + 3}{18}$

$\displaystyle = \frac{17 - 12}{18}$

$\displaystyle = \frac{5}{18}$

Answer: $\displaystyle \frac{5}{18}$

L.C.M of $10, 14, 20 = 2 \times 5 \times 7 \times 2 = 140$

$\displaystyle \frac{3}{10} = \frac{3 \times 14}{10 \times 14} = \frac{42}{140}$

$\displaystyle \frac{17}{14} = \frac{17 \times 10}{14 \times 10} = \frac{170}{140}$

$\displaystyle -\frac{19}{20} = \frac{-19 \times 7}{20 \times 7} = -\frac{133}{140}$

$\displaystyle = \frac{42 + 170 - 133}{140}$

$\displaystyle = \frac{212 - 133}{140}$

$\displaystyle = \frac{79}{140}$

Answer: $\displaystyle \frac{79}{140}$

$\displaystyle = \frac{13}{12} - \frac{6}{1}$

$\displaystyle = \frac{13 - 72}{12}$

$\displaystyle = -\frac{59}{12}$

Answer: $\displaystyle -\frac{59}{12}$

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

$\displaystyle = -\frac{16 \times 9 \times 5}{15 \times 32 \times 12}$

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

Cancelling the common factors $2, 2, 2, 2, 3, 3, 5$: $\displaystyle = -\frac{1}{2 \times 2 \times 2} = -\frac{1}{8}$

Answer: $\displaystyle -\frac{1}{8}$

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

$\displaystyle = -\frac{63 \times 19 \times 55}{40 \times 132 \times 133}$

$\displaystyle = -\frac{(3 \times 3 \times 7) \times 19 \times (5 \times 11)}{(2 \times 2 \times 2 \times 5) \times (2 \times 2 \times 3 \times 11) \times (7 \times 19)}$

Cancelling the common factors $3, 5, 7, 11, 19$: $\displaystyle = -\frac{3}{2 \times 2 \times 2 \times 2 \times 2} = -\frac{3}{32}$

Answer: $\displaystyle -\frac{3}{32}$

$\displaystyle 2\frac{1}{9} = \frac{(2 \times 9) + 1}{9} = \frac{19}{9}$

$\displaystyle 2\frac{5}{8} = \frac{(2 \times 8) + 5}{8} = \frac{21}{8}$

$\displaystyle 2\frac{1}{9} \times 2\frac{5}{8} = \frac{19}{9} \times \frac{21}{8} = \frac{19 \times 21}{9 \times 8}$

Cancelling common factors: $\displaystyle = \frac{133}{24}$

$133 \div 24$: quotient $5$, remainder $13$; $\displaystyle = 5\frac{13}{24}$

Answer: $\displaystyle \frac{133}{24} = 5\frac{13}{24}$

$\displaystyle 2\frac{3}{10} = \frac{(2 \times 10) + 3}{10} = \frac{23}{10}$

$\displaystyle 2\frac{3}{10} \div 9 = \frac{23}{10} \div \frac{9}{1} = \frac{23}{10} \times \frac{1}{9}$

$\displaystyle = \frac{23}{10 \times 9} = \frac{23}{90}$

Answer: $\displaystyle \frac{23}{90}$

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

Points to the left of $0$: $\displaystyle -\frac{1}{9}, -\frac{2}{9}, -\frac{3}{9}, -\frac{4}{9}, \ldots$

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

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