5 Marks Questions

Question 15 Marks

Divide the sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their difference.

Answer

$\Big(\frac{65}{12}+\frac{8}{3}\Big)\div\Big(\frac{65}{12}-\frac{8}{3}\Big)$
$=\Big(\frac{65}{12}+\frac{8\times4}{3\times4}\Big)\div\Big(\frac{65}{12}-\frac{8\times4}{3\times4}\Big)$
$=\Big(\frac{65}{12}+\frac{32}{12}\Big)\div\Big(\frac{65}{12}-\frac{32}{12}\Big)$
$=\Big(\frac{65+32}{132}\Big)\div\Big(\frac{65-32}{12}\Big)$
$=\Big(\frac{65+32}{132}\Big)\div\Big(\frac{12}{65-32}\Big)$
$=\frac{65+32}{65-32}$
$=\frac{97}{33}$

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Question 25 Marks

Simplify: $\Big(-5\times\frac{2}{15}\Big)-\Big(-6\times\frac{2}{9}\Big)$

Answer

$\Big(-5\times\frac{2}{15}\Big)-\Big(-6\times\frac{2}{9}\Big)$
$=\frac{-10}{15}-\Big(-\frac{12}{9}\Big)$
$=\frac{-10\times3}{15\times3}-\Big(\frac{-12\times5}{9\times5}\Big)$
$=\frac{-30}{45}-\Big(\frac{-60}{45}\Big)$
$=\frac{-30+60}{45}$
$=\frac{30}{45}$
$=\frac{2}{3}$

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Question 35 Marks

Divide the sum of $\frac{-13}{5}$ and $\frac{12}{7}$ by the product of $\frac{-31}{7}$ and $\frac{1}{2}.$

Answer

$\Big(\frac{-13}{5}+\frac{12}{7}\Big)\div\Big(\frac{-31}{7}\times\frac{-1}{2}\Big)$
$=\Big(\frac{-13\times7}{5\times7}+\frac{12\times5}{7\times5}\Big)\div\Big(\frac{-31}{7}\times\frac{-1}{2}\Big)$
$=\Big(\frac{-91}{35}+\frac{60}{35}\Big)\div\Big(\frac{31}{14}\Big)$
$=\Big(-\frac{91+60}{35}\Big)\div\Big(\frac{31}{14}\Big)$
$=\Big(\frac{-31}{35}\Big)\div\Big(\frac{31}{14}\Big)$
$=\Big(\frac{-31}{35}\Big)\times\Big(\frac{14}{31}\Big)$
$=\frac{-14}{35}$
$=\frac{-2}{5}$

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Question 45 Marks

Simplify: $\Big(\frac{3}{11}\times\frac{5}{6}\Big)-\Big(\frac{9}{12}\times\frac{4}{3}\Big)+\Big(\frac{5}{13}\times\frac{6}{15}\Big)$

Answer

$\Big(\frac{3}{11}\times\frac{5}{6}\Big)-\Big(\frac{9}{12}\times\frac{4}{3}\Big)+\Big(\frac{5}{13}\times\frac{6}{15}\Big)$
$=\Big(\frac{1}{11}\times\frac{5}{2}\Big)-\Big(\frac{3}{3}\times\frac{1}{1}\Big)+\Big(\frac{1}{13}\times\frac{6}{3}\Big)$
$=\Big(\frac{1}{11}\times\frac{5}{2}\Big)-\Big(\frac{3}{3}\times\frac{1}{1}\Big)+\Big(\frac{1}{13}\times\frac{2}{1}\Big)$
$=\Big(\frac{5}{22}\Big)-(1)+\Big(\frac{2}{13}\Big)$
$=\Big(\frac{5\times13}{22\times13}\Big)-\Big(\frac{1\times286}{1\times286}\Big)+\Big(\frac{2\times22}{13\times22}\Big)$
$=\Big(\frac{65-286+44}{286}\Big)$
$=\Big(-\frac{177}{286}\Big)$

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Question 55 Marks

Simplify: $\Big(\frac{-9}{4}\times\frac{5}{3}\Big)+\Big(\frac{13}{2}\times\frac{5}{6}\Big)$

Answer

$\Big(\frac{-9}{4}\times\frac{5}{3}\Big)+\Big(\frac{13}{2}\times\frac{5}{6}\Big)$
$=\Big(\frac{-3}{4}\times5\Big)+\Big(\frac{65}{12}\Big)$
$=\Big(\frac{-15}{4}\Big)+\Big(\frac{65}{12}\Big)$
$=\frac{-15\times3}{4\times3}+\frac{65}{12}$
$=\frac{-45}{12}+\frac{65}{12}$
$=\frac{-45+65}{12}$
$=\frac{20}{12}$
$=\frac{10}{6}$
$=\frac{5}{3}$

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Question 65 Marks

What should be subtracted from $\Big(\frac{3}{4}-\frac{2}{3}\Big)$ to get $\frac{-1}{6}?$

Answer

Let the required number be $x$ $\Big(\frac{3}{4}-\frac{2}{3}\Big)-\text{x}=\frac{-1}{6}$
$\Rightarrow\Big(\frac{3\times3}{4\times3}-\frac{2\times4}{3\times4}\Big)-\text{x}=-\frac{1}{6}$
$\Rightarrow\Big(\frac{9}{12}-\frac{8}{12}\Big)-\text{x}=\frac{-1}{6}$
$\Rightarrow\frac{1}{12}-\text{x}=\frac{-1}{6}$
$\Rightarrow-\text{x}=-\frac{1}{6}-\frac{1}{12}$
$\Rightarrow-\text{x}=\frac{-1\times2}{6\times2}-\frac{1}{12}$
$\Rightarrow-\text{x}=\frac{-2}{12}-\frac{1}{12}$
$\Rightarrow-\text{x}=\frac{-2-1}{12}$
$\Rightarrow-\text{x}=-\frac{3}{12}$
$\Rightarrow\text{x}=\frac{3}{12}$
$\text{x}=\frac{1}{4}$
The required number is $\frac{1}{4}$

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Question 75 Marks

Simplify: $\Big(\frac{13}{9}\times\frac{-15}{2}\Big)+\Big(\frac{7}{3}\times\frac{8}{5}\Big)+\Big(\frac{3}{5}\times\frac{1}{2}\Big)$

Answer

$\Big(\frac{13}{9}\times\frac{-15}{2}\Big)+\Big(\frac{7}{3}\times\frac{8}{5}\Big)+\Big(\frac{3}{5}\times\frac{1}{2}\Big)$
$=\Big(\frac{13}{3}\times\frac{-5}{2}\Big)+\Big(\frac{56}{15}\Big)+\Big(\frac{3}{10}\Big)$
$=\Big(\frac{-65}{6}\Big)+\Big(\frac{56}{15}\Big)+\Big(\frac{3}{10}\Big)$
$=\Big(\frac{-65\times5}{6\times5}\Big)-\Big(\frac{56\times2}{15\times2}\Big)+\Big(\frac{3\times3}{10\times3}\Big)$
$=\Big(\frac{-325}{30}\Big)+\Big(\frac{112}{30}\Big)+\Big(\frac{9}{30}\Big)$
$=\Big(\frac{-204}{30}\Big)$
$=\Big(\frac{-34}{5}\Big)$

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Question 85 Marks

Add the following rational numbers: $\frac{6}{13}\text{ and }\frac{-9}{13}$

Answer

$\frac{6}{13}+\frac{-9}{13}$ $=\frac{6}{13}-\frac{9}{13}$ $=\frac{6-9}{13}$ $\frac{-3}{13}$

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Question 95 Marks

What should be added to $\Big(\frac{1}{2}+\frac{1}{3}+\frac{1}{5}\Big)$ to get $3? $

Answer

Let the required number be x $\Big(\frac{1}{2}+\frac{1}{3}+\frac{1}{5}\Big)+\text{x}=3$
$\Rightarrow\Big(\frac{1\times15}{2\times15}+\frac{1\times10}{3\times10}+\frac{1\times6}{5\times6}\Big)+\text{x}=3$
$\Rightarrow\Big(\frac{15+10+6}{30}\Big)+\text{x}=3$
$\Rightarrow\frac{31}{30}+\text{x}=3$
$\Rightarrow\text{x}=3-\frac{31}{30}$
$\Rightarrow\text{x}=\frac{3\times30}{1\times30}-\frac{31}{30}$
$\Rightarrow\text{x}=\frac{90}{30}-\frac{31}{30}$
$\text{x}=\frac{59}{30}$
The required number is $\frac{59}{30}$

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Question 105 Marks

What should be added to $\Big(\frac{2}{3}+\frac{3}{5}\Big)$ to get $\frac{-2}{15}?$

Answer

Let the required number be x
$\Big(\frac{2}{3}+\frac{3}{5}\Big)+\text{x}=\frac{-2}{15}$
$\Rightarrow\Big(\frac{2\times5}{3\times5}+\frac{3\times3}{5\times3}\Big)+\text{x}=\frac{-2}{15}$
$\Rightarrow\Big(\frac{10}{15}+\frac{9}{15}\Big)+\text{x}=\frac{-2}{15}$
$\Rightarrow\frac{19}{15}+\text{x}=\frac{-2}{15}$
$\Rightarrow\text{x}=\frac{-2}{15}-\frac{19}{15}$
$\Rightarrow\frac{-2-19}{15}$
$\Rightarrow\text{x}=\frac{-21}{15}$
$\text{x}=\frac{-7}{5}$
The required number is $\frac{-7}{5}$

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Question 115 Marks

If 24 trousers of equal size can be prepared in 54 metres of cloth, what length of cloth is required for each trouser?

Answer

$\frac{9}{4}$ metres.

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Question 125 Marks

Divide the sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their difference.

Answer

$\frac{97}{33}$

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Question 135 Marks