Question paper, Set 9

Unshaded parts $= 14 - 11 = 3$

Each part $\displaystyle = \frac{1}{14}$

Unshaded region $\displaystyle = \frac{3}{14}$

Answer: $\displaystyle \frac{3}{14}$

$\displaystyle \frac{22}{15} = \frac{15 + 7}{15}$

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

$\displaystyle = 1 + \frac{7}{15}$

Answer: $\displaystyle \frac{22}{15} = 1 + \frac{7}{15}$

Proper fraction: numerator $<$ denominator.

$6 < 13$, $4 < 9$, $1 < 15$

Proper fractions: $\displaystyle \frac{6}{13}, \frac{4}{9}, \frac{1}{15}$

Answer: $\displaystyle \frac{6}{13}, \frac{4}{9}, \frac{1}{15}$

Divide $271$ by $16$: $16 \times 16 = 256$, $271 - 256 = 15$

Quotient $= 16$, remainder $= 15$

$\displaystyle \therefore \frac{271}{16} = 16 + \frac{15}{16} = 16\frac{15}{16}$

Answer: $\displaystyle 16\frac{15}{16}$

$\displaystyle 23\frac{4}{7} = \frac{(23 \times 7) + 4}{7}$

$\displaystyle = \frac{161 + 4}{7} = \frac{165}{7}$

Answer: $\displaystyle \frac{165}{7}$

Like fractions have the same denominator.

$\displaystyle \frac{5}{13}, \frac{20}{13}, \frac{23}{13}$ all have denominator $13$.

Answer: $\displaystyle \frac{5}{13}, \frac{20}{13}, \frac{23}{13}$

$\displaystyle \frac{6}{18} = \frac{6 \div 6}{18 \div 6} = \frac{1}{3}$

$\displaystyle \frac{6}{18} = \frac{6 \times 3}{18 \times 3} = \frac{18}{54}$

Answer: $\displaystyle \frac{1}{3}, \frac{18}{54}$

(i) $\displaystyle \frac{162}{135}$

H.C.F of $162, 135 = 27$; $\displaystyle \frac{162 \div 27}{135 \div 27} = \frac{6}{5}$

(ii) $\displaystyle \frac{-30}{-24} = \frac{30}{24}$, since $(-)/(-) = (+)$

H.C.F of $30, 24 = 6$; $\displaystyle \frac{30 \div 6}{24 \div 6} = \frac{5}{4}$

(iii) $\displaystyle \frac{72}{-63} = -\frac{72}{63}$, since $(+)/(-) = (-)$

H.C.F of $72, 63 = 9$; $\displaystyle -\frac{72 \div 9}{63 \div 9} = -\frac{8}{7}$

Answer: (i) $\displaystyle \frac{6}{5}$ (ii) $\displaystyle \frac{5}{4}$ (iii) $\displaystyle -\frac{8}{7}$

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: $-51 < -49 < -31$

Descending order: $59, 50, 10, -31, -49, -51$

Answer: $59, 50, 10, -31, -49, -51$

$\displaystyle \frac{16}{-9} = -\frac{16}{9}$, since $(+)/(-) = (-)$

L.C.M of $3, 2, 9, 8, 4 = 2 \times 2 \times 3 \times 3 \times 2 = 72$

$\displaystyle \frac{5}{3} = \frac{5 \times 24}{3 \times 24} = \frac{120}{72}$

$\displaystyle \frac{3}{2} = \frac{3 \times 36}{2 \times 36} = \frac{108}{72}$

$\displaystyle -\frac{16}{9} = \frac{-16 \times 8}{9 \times 8} = -\frac{128}{72}$

$\displaystyle \frac{9}{8} = \frac{9 \times 9}{8 \times 9} = \frac{81}{72}$

$\displaystyle -\frac{1}{4} = \frac{-1 \times 18}{4 \times 18} = -\frac{18}{72}$

Comparing numerators: $\displaystyle \frac{120}{72} > \frac{108}{72} > \frac{81}{72} > -\frac{18}{72} > -\frac{128}{72}$

$\displaystyle \Rightarrow \frac{5}{3} > \frac{3}{2} > \frac{9}{8} > -\frac{1}{4} > -\frac{16}{9}$

Answer: $\displaystyle \frac{5}{3}, \frac{3}{2}, \frac{9}{8}, -\frac{1}{4}, -\frac{16}{9}$

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

$1 \times 9 = 9$ and $14 \times 1 = 14$

$\displaystyle 9 < 14 \Rightarrow \frac{1}{14} < \frac{1}{9}$

Greater: $\displaystyle \frac{1}{9}$

Answer: $\displaystyle \frac{1}{9}$

$\displaystyle = \frac{-39 - 23 + 36 + 35}{26}$

$\displaystyle = \frac{71 - 62}{26}$

$\displaystyle = \frac{9}{26}$

Answer: $\displaystyle \frac{9}{26}$

L.C.M of $9, 15, 3 = 3 \times 3 \times 5 = 45$

$\displaystyle = \frac{35 + 48 - 30}{45}$

$\displaystyle = \frac{83 - 30}{45}$

$\displaystyle = \frac{53}{45}$

Answer: $\displaystyle \frac{53}{45}$

L.C.M of $16, 30, 24 = 2 \times 2 \times 2 \times 3 \times 2 \times 5 = 240$

$\displaystyle \frac{9}{16} = \frac{9 \times 15}{16 \times 15} = \frac{135}{240}$

$\displaystyle -\frac{41}{30} = \frac{-41 \times 8}{30 \times 8} = -\frac{328}{240}$

$\displaystyle -\frac{5}{24} = \frac{-5 \times 10}{24 \times 10} = -\frac{50}{240}$

$\displaystyle = \frac{135 - 328 - 50}{240}$

$\displaystyle = \frac{135 - 378}{240}$

$\displaystyle = -\frac{243}{240} = -\frac{81}{80}$

Answer: $\displaystyle -\frac{81}{80}$

$\displaystyle = \frac{4}{1} - \frac{11}{6}$

$\displaystyle = \frac{24 - 11}{6}$

$\displaystyle = \frac{13}{6}$

Answer: $\displaystyle \frac{13}{6}$

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

$\displaystyle = -\frac{7 \times 9 \times 20}{3 \times 4 \times 21}$

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

Cancelling the common factors $2, 2, 3, 3, 7$: $\displaystyle = -\frac{5}{1} = -5$

Answer: $-5$

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

$\displaystyle = \frac{10 \times 76 \times 21}{19 \times 35 \times 32}$

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

Cancelling the common factors $2, 2, 2, 5, 7, 19$: $\displaystyle = \frac{3}{2 \times 2} = \frac{3}{4}$

Answer: $\displaystyle \frac{3}{4}$

$\displaystyle 4\frac{3}{4} = \frac{(4 \times 4) + 3}{4} = \frac{19}{4}$

$\displaystyle 2\frac{2}{9} = \frac{(2 \times 9) + 2}{9} = \frac{20}{9}$

$\displaystyle 4\frac{3}{4} \times 2\frac{2}{9} = \frac{19}{4} \times \frac{20}{9} = \frac{19 \times 20}{4 \times 9}$

Cancelling common factors: $\displaystyle = \frac{95}{9}$

$95 \div 9$: quotient $10$, remainder $5$; $\displaystyle = 10\frac{5}{9}$

Answer: $\displaystyle \frac{95}{9} = 10\frac{5}{9}$

$\displaystyle 11\frac{6}{7} = \frac{(11 \times 7) + 6}{7} = \frac{83}{7}$

$\displaystyle 11\frac{6}{7} \div 3 = \frac{83}{7} \div \frac{3}{1} = \frac{83}{7} \times \frac{1}{3}$

$\displaystyle = \frac{83}{7 \times 3} = \frac{83}{21}$

Answer: $\displaystyle \frac{83}{21}$

$\displaystyle \frac{18}{5} = 3 + \frac{3}{5}$, so it lies between $3$ and $4$.

Divide each unit into $5$ equal parts of $\displaystyle \frac{1}{5}$; $\displaystyle 3 = \frac{15}{5}$.

Count $18$ parts to the right of $0$ and mark $\displaystyle \frac{18}{5}$.

Answer: Divide each unit into $5$ equal parts; $\displaystyle \frac{18}{5} = 3\frac{3}{5}$ is the 3rd point after $3$ (the 18th point from $0$).