[5 marks sum]

Question 15 Marks

A sum of $₹ 500$ is in the form of notes of denominations of $₹ 5$ and $₹ 10.$ If the total number of notes is $90,$ find the number of notes of each type.

Answer

Let the number of $₹ 5$ notes $=x$
$\therefore$ The number of $\text₹ 10$ notes $=90-x$
Value of $₹ 10$ notes = $x \times ₹5 = ₹ 3 x$
and value of $₹ 10$ notes $== (90-x) x ₹10 = (900 - 10x)$
$\therefore$ Total value of all the notes $=₹ 500$
$\therefore 5 x+(900-10 x)=500$
$\Rightarrow 5 x+900-10 x=500 $
$\Rightarrow-5 x=500-900$
$ \Rightarrow x=\frac{400}{5}$
$\Rightarrow x=80$
$\therefore$ The number of $₹ 5$ notes $= x = 80$
and the number of $₹10$ notes $= 90 - X$
$= 90 - 80 =10$

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Question 25 Marks

A man is thrice as old as his son. After 12 years, he will be twice as old as his son at that time. Find their present ages.

Answer

Let the present age of the son = x years
and the present age of the father = 3x years
After 12 years,
Son’s age will be (x + 12) years
and father’s age will be (3x + 12) years
According to the condition,
3x + 12 = 2 (x + 12)
3x + 12 = 2x+ 24
3x – 2x = 24 – 12
x = 12
∴Present age of the son = 12 years
and Present age of the father = 3×12 years
= 36 years

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Question 35 Marks

The numerator of a fraction is four less than its denominator. If 1 is added to both, is numerator and denominator, the fraction becomes $\frac{1}{2}$ Find the fraction.

Answer

Let the numerator of a fraction $=x$
and the denominator of a fraction $=y$
According to the condition,
$ x=y-4 $...(i)
and $\frac{x+1}{y+1}=\frac{1}{2}$
$\Rightarrow 2(x+1)=y+1$
$\Rightarrow 2 x+2=y+1$
$\Rightarrow 2 x-y=-1$...(ii)
Substitute the eq.(i) in eq.(ii)
$2(y-4)-y=-1$
$2 y-8-y=-1$
$y=-1+8$
$y=7$
Now, put the value of $y$ in eq.(i), we get
$ x=7-4=3 $
$\therefore$ The numerator of a fraction is 3
and denominator is 7 and the fraction is $\frac{3}{7}$

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Question 45 Marks

The length of a rectangular plot exceeds its breadth by 5 m. If the perimeter of the plot is 142 m, find the length and the breadth of the plot.

Answer

Let the length of a rectangular plot $=x$
and the breadth of a rectangular plot $=y$
According to the condition,
$ x=y+5 $...(i)
and $2(x+y)=142$
$ \Rightarrow x+y=\frac{142}{2}=71 $
$ \Rightarrow x+y=71 $...(ii)
Now, substitute the value of eq. (i) in eq (ii)
$ \begin{aligned} & y+5+y=71 \\ & \Rightarrow 2 y=71-5 \\ & \Rightarrow y=\frac{66}{2}=33 \end{aligned} $
Now, put the value of $y$ in eq. (i)
$ x=33+5=38 $
$\therefore$ The length of rectangular plot is $38 \mathrm{~m}$ and breadth is $33 \mathrm{~m}$.

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Question 55 Marks

The difference between two numbers is 7. Six times the smaller plus the larger is 77. Find the numbers.

Answer

Let the smallest number $=x$
and the largest number $=\mathrm{y}$
According to the condition,
$ y-x=7 $...(i)
$ \text { and } 6 x+y=77 $....(ii)
From eq. (i)
$ y=7+x $...(ii)
Substitute the eq. (iii) in eq. (ii)
$ \begin{aligned} & 6 x+7+x=77 \\ & \Rightarrow 7 x=77-7 \\ & \Rightarrow x=\frac{70}{7}=10 \end{aligned} $
Now, substitute the value of $x$ in eq. (iii)
$ y=7+10=17 $
$\therefore$ The smallest number 10 and the largest number is 17 .

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MATHS [5 marks sum]

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