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Basic Maths: Test 36

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Question 1
$\begin{align} & ifI=\frac{2}{3}\div \frac{7}{6}, \\ & II=2\div \left[ \left( 7\div 8 \right)\div (1\div 4) \right], \\ & III=\left[ 6\div \left( 7\div 5 \right) \right]\div 6, \\ & IV=3\div 2\left( 5\div 4 \right) \\ & then \\ \end{align}$
A
I and II are equal
B
I and IV are equal
C
I and III are equal
D
All are equal
Question 1 Explanation:
$\begin{align} & I.\,=\frac{2}{3}\times \frac{6}{7}=\frac{4}{7} \\ & II.\,\,=2\div \left[ \frac{7}{8}\times 4 \right] \\ & =2\div \frac{7}{2}=\frac{4}{7} \\ & III.\,\,\,=\left[ 6\div \frac{7}{5} \right]\div 6=\frac{30}{7}\div 6=\frac{5}{7} \\ & IV.\,\,=3\div 2\times \frac{5}{4}=3\div \frac{5}{2}=\frac{6}{5} \\ \end{align}$
Obviously, (I) and (II) are equal
Question 2
The value of $\frac{0.062\times 0.062\times 0.062+0.033\times 0.033\times 0.033}{0.062\times 0.062-0.062\times 0.033+0.033\times 0.033}$ is:
A
0.95
B
0.095
C
0.0095
D
0.00095
Question 2 Explanation:
Let 0.062=x and 0.033= y
Therefore given expression
$\begin{align} & =\frac{{{x}^{3}}+{{y}^{3}}}{{{x}^{2}}-xy+{{y}^{2}}} \\ & =\frac{\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right)}{{{x}^{2}}-xy+{{y}^{2}}} \\ & =x+y=0.062+0.033 \\ & =0.095 \\ \end{align}$
Question 3
The simplification of $1.\overline{38}-3.\overline{15}+2.\overline{53}$ equals:
A
0.76
B
$0.\overline{76}$
C
2.64
D
$2.\overline{64}$
Question 3 Explanation:
$\begin{align} & 1.\overline{38}-3.\overline{15}+2.\overline{53} \\ & =1\frac{38}{99}-3\frac{15}{99}+2\frac{53}{99} \\ & =1+\frac{38}{99}-3-\frac{15}{99}+2+\frac{53}{99} \\ & =\left( 1-3+2 \right)+\left( \frac{38}{99}-\frac{15}{99}+\frac{53}{99} \right) \\ & =0+\left( \frac{38-15+53}{99} \right) \\ & =\frac{76}{99} \\ & =0.\overline{76} \\ \end{align}$
Question 4
The value of \[\frac{0.8\times 0.8\times 0.8+0.1\times 0.1\times 0.1+0.7\times 0.7\times 0.7-3\times 0.8\times 0.1\times 0.7}{0.8\times 0.8+0.1\times 0.1+0.7\times 0.7-0.8\times 0.1-0.1\times 0.7-0.7\times 0.8}\] is
A
1.6
B
0.6
C
0.16
D
1.0
Question 4 Explanation:
Let 0.8= x, 0.1= y and 0.7= z
Then, the given expression
$\begin{align} & \frac{x\times x\times x+y\times y\times y+z\times z\times z-3\times x\times y\times z}{x\times x+y\times y+z\times z-x\times y-y\times z-z\times x} \\ & =\frac{{{x}^{3}}+{{y}^{3}}+{{x}^{3}}-3xyz}{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx} \\ & =\frac{\left( x+y+z \right)\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx \right)}{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx} \\ & =x+y+z \\ & =0.8+0.1+0.7 \\ & =1.6 \\ \end{align}$
Question 5
The simplified value of $\left( 1+\frac{1}{3} \right)\,\left( 1+\frac{1}{4} \right)\,\left( 1+\frac{1}{5} \right)\,.....\left( 1+\frac{1}{99} \right)\,\left( 1+\frac{1}{100} \right)$ is
A
$\frac{3}{99}$
B
$\frac{1}{101}$
C
$\frac{101}{3}$
D
$\frac{3}{100}$
Question 5 Explanation:
$\begin{align} & \frac{4}{3}\times \frac{5}{4}\times \frac{6}{5}\times .......\times \frac{100}{99}\times \frac{101}{100} \\ & =\frac{101}{3} \\ \end{align}$
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