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Question

What is the sum of two numbers?
I. The bigger of these two numbers is 6 more than the smaller number.
II. 40% of the smaller number is equal to 30% of the bigger number.
III. The ratio between half of the bigger number and one-third of the smaller number is 2 : 1


A
Only II and III are sufficient
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B
Only I and II are sufficient
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C
I and either II or III is sufficient
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D
All, II and III together are sufficient
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Solution

The correct option is C I and either II or III is sufficient
Let the two numbers be $$x$$ and $$y$$ such that $$x<y$$

Given: 

I.  $$ y=x+6$$                $$...(1)$$
  
II.  $$40\%$$ of $$x=30\%$$ of $$y$$

$$\Rightarrow \dfrac{40}{100}\times x=\dfrac{30}{100}\times y$$

$$\Rightarrow 0.4x=0.3y$$

$$\Rightarrow \dfrac {x}{y}=\dfrac {3}{4}$$           $$...(2)$$
IIII.  $$ \bigg(\dfrac{1}{2}\times y\bigg): \bigg(\dfrac{1}{3}\times x\bigg)= 2:1$$

$$\Rightarrow \dfrac{\ \dfrac y2\ }{\ \dfrac x3\ }=\dfrac{2}{1}$$

$$\Rightarrow \dfrac{3y}{2x}=\dfrac{2}{1}$$

$$\Rightarrow \dfrac {y}{x}=\dfrac {4}{3}$$

$$\Rightarrow \dfrac {x}{y}=\dfrac {3}{4}$$

$$\Rightarrow 4x=3y$$         $$...(3)$$

Let us multiply the equation $$(1)$$ with $$3$$

$$\Rightarrow 3y=3x+18$$

$$\Rightarrow 4x=3x+18$$
$$\therefore x=18$$

Substitute the value of $$x$$ in equation $$1$$
$$\Rightarrow y=18+6$$
$$\Rightarrow y=24$$
$$\therefore$$ Sum of $$x$$ and $$y$$ is $$18+24=42$$

Mathematics

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