[3 marks sum]

Question 13 Marks

The sum of four times the first number and three times the second number is $15.$ The difference of three times the first number and twice the second number is $7.$ Find the numbers.

Answer

Let the two numbers be $x$ and $y$ respectively.
Then, we have
$4x + 3y = 15 ....(i)$
$3x - 2y = 7 ....(ii)$
Multiplying eqn. $(i)$ by $2$ and eqn. $(ii)$ by $3,$
we get
$8x + 6y = 30 ....(iii)$
$9x - 6y = 21 ....(iv)$
Adding eqns. $(iii)$ and $(iv),$ we get
$17x = 51$
$\Rightarrow x = 3$
$\Rightarrow 4(3) + 3y = 15$
$\Rightarrow 12 + 3y = 15$
$\Rightarrow 3y = 3$
$\Rightarrow y = 1$
Thus, the two numbers are $3$ and $1$ respectively.

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Question 23 Marks

If a number is thrice the other and their sum is $68,$ find the numbers.

Answer

Let the two numbers be $x$ and $y$ respectively.
Then, we have
$x = 3y ....(i)$
And,
$x + y = 68$
$\Rightarrow 3y + y = 68$
$\Rightarrow 4y = 68$
$\Rightarrow y = 17$
$\Rightarrow x = 3 \times 17$
$= 51$
Thus, the two numbersare $51$ and $17$ respectively.

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Question 33 Marks

The difference of two numbers is $3,$ and the sum of three times the larger one and twice the smaller one is $19.$ Find the two numbers.

Answer

Let the larger number be $x$ and the smaller number be $y.$
According to given information, we have
$x - y = 3$
$\Rightarrow x - 3 + y ....(i)$
Also, $3x + 2y = 19$
$\Rightarrow 3(3 + y) + 2y = 19 ....[$From $(i)]$
$\Rightarrow 9 + 3y + 2y = 19$
$\Rightarrow 5y = 10$
$\Rightarrow y = 2$
$\Rightarrow x = 3 + 2$
$= 5$
Thus, the required numbers are $5$ and $2$ respectively.

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Question 43 Marks

If $1$ is added to the denominator of a fraction, the fraction becomes $\frac{1}{2}$. If $1$ is added to the numerator of the fraction, the fraction becomes $1$ . Find the fraction.

Answer

Let the numerator and denominator of a fraction be $x$ and $y$ respectively.
Fraction $ =\frac{x}{y} $
According to given information, we have
$ \frac{x}{y+1}=\frac{1}{2}$
$\Rightarrow 2 x=y+1$
$\Rightarrow 2 x-y=1 ....(i) $
Also,
$ \frac{x+1}{y}=1 $
$ \Rightarrow x+1=y$
$\Rightarrow x-y=-1 ....(ii) $
Subtracting eqn. $(ii)$ from $(i), $ we get
$ x=2$
$\Rightarrow 2-y=-1$
$\Rightarrow y=3$
Required fraction $=\frac{2}{3} . $

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Question 53 Marks

The sum of the numerator and denominator of a fraction is $12$ . If the denominator is increased by $3 ,$ the fraction becomes $\frac{1}{2}$. Find the fraction.

Answer

Let the numerator and denominator of a fraction be $x$ and $y$ respectively.
Fraction$=\frac{x}{y} $
According to given information, we have
$ x+y=12 ....(i) $
And,
$ \frac{x}{y+3}=\frac{1}{2}$
$\Rightarrow 2 x=y+3$
$\Rightarrow 2 x-y=3 ....(ii) $
Adding eqns. $(i)$ and $(ii),$ we get
$ 3 x=15$
$\Rightarrow x=5$
$\Rightarrow 5+y=12$
$\Rightarrow y=7 $
Required fraction $ =\frac{5}{7} \text {. } $

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Question 63 Marks

If $2x + y = 23$ and $4x - y = 19 :$ find the value of $x - 3y$ and $5y - 2x.$

Answer

$2x + y = 23 ........(1)$
$4x - y = 19 ........(2)$
Adding $(1)$ and $(2),$
$6x = 42$
$\Rightarrow x = 7$
$\therefore y = 23 - 2x $
$= 23 - 14$
$= 9$
$\therefore x - 3y $
$= 7 - 3(9)$
$= 7 - 27$
$= -20$
$5y - 2x$
$= 5(9) - 2(7)$
$= 45 - 14$
$= 31.$

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MATHEMATICS [3 marks sum]

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