We have, \(\frac { 3\times { 7 }^{ 2 }\times { 11 }^{ 8 } }{ 21\times { 11 }^{ 3 } } \) = \(\frac { { 3 }^{ 1 }\times { 7 }^{ 2 }\times { 11 }^{ 8 } }{ { 3 }^{ 1 }\times 7\times { 11 }^{ 3 } } \)   [\(\because\)21 = 3 x 7]
=3^1-1 x7^2-1 x11^8-3                      [\(\because\)a^m x aⁿ = a^m-n]
=3⁰ x 7¹ x11⁵ = 1x7x11⁵
= 7x11⁵                     [\(\because\)a⁰ =1]

Let the number be x.
\(\therefore\) 75% of \( x =15 \Rightarrow {75\over100}\times x =15\)
\(\Rightarrow x=15\times {100\over75} \Rightarrow x=20\)
Hence, the number is 20.

(i) In ΔPQR and ΔDEF, ㄥQ = ㄥE = 90°
Hypotenuse PR = Hypotenuse DF = 6 cm
Side PQ ≠ Side DE
[∵ PQ = 3 cm and DE = 25 cm]
Therefore, RHS congruence rule is not satisfies.
Hence, ΔPQR and ΔDEF are not congruent.

subtracted from y is 6.
\(\therefore\) y - 2 = 6.

45342

Let x%of 1 h is 30 min.
So,  \(\frac { x }{100} \) x1 h = 30min
∴ \(\frac { x }{100} \) x 60min = 30 min  [∵ 1 h =60 min]
x = \(\frac { 30 ｘ100 }{60} \)
⇒   x =\(\frac { 100 }{ 2 } \) = 50%
Hence, 50% of 1 his 30 min

In the given figure,
ㄥB= 90⁰, ㄥO = 90⁰
AC = DE            [hypotenuse of right angled triangle]
Also, AB = EO    [height of right angled triangle]
Since, two sides and one angle are equal in ΔABC and ΔEOD.
So, ΔABC ≅ ΔDOD 
[by RHS congruence criterion]

Let x be the number.
As per the given information, we have
         \(\frac{9}{2}x-7=26\)
Adding 7 to both the sides, we get
        \(\frac{9}{2}x-7+7=26+7\quad \Rightarrow \frac{9}{2}x=33\)
Now, dividing both sides by \(\frac{9}{2}\), we get
       \(\frac{9/2x}{9/2}=\frac{33}{9/2}\)
\(\Rightarrow\quad x=\frac{33}{1}\times \frac{2}{9}=\frac{11\times 2}{3}\)
So, \(x=\frac{22}{3}\)
Hence, the required number is \(\frac{22}{3}\)

98

The ratio of present ages of A and B is 3 : 2.
Let 3x and 2x be the ages of A and B.
5 yr after, their ages ratio wi II be 14 :11.
\(\therefore {3x+5\over2x+5}={14\over11}\)
11(3x + 5) = 14 (2x + 5) \(\Rightarrow\) 33x + 55 = 28x + 70
33x - 28x = 70 - 55 - 5x = 15 \(\Rightarrow\) x = 3
Hence, present age of A = 3 x 3 = 9 yr
and present age of B = 3 x 2 = 6 yr

In the given figure, we have
BD = CE, EC 丄 AB and AB 丄 AC,
So, ㄥBEC = 90⁰  ⇒ ㄥBDC = 90⁰
So, ㄥBEC = ㄥBDC
and BC = BC
[common side is hypotenuse of right angled triangle]
Since, two sides and one angle of ΔBDC and ΔBEC are equal. So, ΔBCD ≌ ΔCBE
[by RHS congruence criterion]

Let x be the age of Amit.
So, age of Sohan Lal, the father of Amit = 4x yr
If the difference of their ages is 27 yr.
Then, 4x- x = 27 \(\Rightarrow\) 3x = 27 \(\Rightarrow x=\frac{27}{3}=9\)
Hence, age of Amit is 9 yr.