ML Aggarwal Solution Class 10 Chapter 7 Ratio and Proportion Test

 Test

Question 1

Find the compound ratio of:

(a+b)2:(ab)2
(a2b2):(a2+b2)
(a4b4):(a+b)4

Sol :

(a+b)2:(ab)2

(a2b2):(a2+b2)

(a4b4):(a+b)4

=(a+b)2(ab)2×a2b2a2+b2×a4b4(a+b)4

=(a+b)2(ab)2×(a+b)(ab)a2+b2×(a2+b2)(a+b)(ab)(a+b)4

=11 = 1 : 1


Question 2

If (7 p + 3 q) : (3 p – 2 q) = 43 : 2 find p : q

Sol :

(7p + 3q) : (3p – 2q) = 43 : 2

7p+3q3p2q=432

129p86q=14p+6q

129p14p=6q+86q

115p=92q

pq=92115=45

∴p : q= 4 : 5 


Question 3

If a : b = 3 : 5, find (3a + 5b): (7a – 2b).

Sol :

a : b = 3 : 5

ab=35

⇒ 3a + 5n : 7a – 2b

Dividing each term by b

3ab+5:3ab2

3×35+5:7×352

(95+5):(2152)

9+255:21105

345:115

=34:11


Question 4

The ratio of the shorter sides of a right angled triangle is 5 : 12. If the perimeter of the triangle is 360 cm, find the length of the longest side.

Sol :

Let the two shorter sides of a right-angled triangle be 5x and 12x.

Third (longest side)

=(5x)2+(12x)2
=25x2+144x2
=169x2=13x

But 5x+12x+13x=360cm

⇒30x=360

x=36030=12

∴Length of the longest side=13x

=13×12cm=156cm


Question 5

The ratio of the pocket money saved by Lokesh and his sister is 5 : 6. If the sister saves Rs 30 more, how much more the brother should save in order to keep the ratio of their savings unchanged?

Sol :

Let the savings of Lokesh and his sister are 5x and 6x.

and the Lokesh should save Rs y more Now, according to the problem,

5x+y6x+30=56
⇒ 30x + 6y = 30x + 150
⇒ 6y=150

y=1506=25

Hence, Lokesh should save 25 more


Question 6

In an examination, the number of those who passed and the number of those who failed were in the ratio of 3 : 1. Had 8 more appeared, and 6 less passed, the ratio of passed to failures would have been 2 : 1. Find the number of candidates who appeared.

Sol :

Let number of passed = 3 x

and failed = x

Total candidates appeared = 3x + x = 4x.

In second case

No. of candidates appeared = 4 x + 8

and No. of passed = 3 x – 6

and failed = 4x + 8 – 3x + 6 = x + 14

then ratio will be = 2 : 1

Now according to the condition

3x6x+14=21

⇒ 3x-6=2x+28

⇒ 3x-2x=28+6

⇒ x=34

∴No. of candidates appeared

=4x=4×34=136


Question 7

What number must be added to each of the numbers 15, 17, 34 and 38 to make them proportional ?

Sol :

Let x be added to each number, then numbers will be

15 + x, 17 + x, 34 + x, and 38 + x.

Now according to the condition

15+x17+x=34+x38+x

(15+x)(38+x)=(34+x)(17+x)

570+53x+x2=578+51x+x2

x2+53xx251x=578570

2x=8x=4

4 is to be added.


Question 8

If (a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion, prove that b is the mean proportional between a and c.

Sol :

(a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion

a+2b+cac=aca2b+c

a+2b+cac=aca2b+c

(a+2b+c)(a2b+c)=(ac)2

a22ab+ac+2ab4b2+2bc+ac2bc+c2=a22ac+c2

a22ab+ac+2ab4b2+2bc+ac2bc+c2a2+2acc2=0

4ac4b2=0acb2=0

b2=ac

Here b is the mean proportional between a and c


Question 9

If 2, 6, p, 54 and q are in continued proportion, find the values of p and q.

Sol :

2, 6, p, 54 and q are in continued proportional then

26=6p=p54=549

(i) 26=6p then 

2p=36

p=18


(ii) p54=54qpq=54×54

18q=54×54

q=54×5418=162

Hence , p=18, q=162


Question 10

If a, b, c, d, e are in continued proportion, prove that: a:e=a4:b4

Sol :

a, b, c, d, e are in continued proportion

ab=bc=cd=de=k( say )

d=ek,c=ek2,b=ek3 and a=ek4

Now, L.H.S.=ae=ek4e=k4

R.H.S a4b4=(ek4)4(ek3)4=e4k16e4k12

=k1612=k4

∴L.H.S=R.H.S


Question 11

Find two numbers whose mean proportional is 16 and the third proportional is 128.

So :

Let x and y be two numbers

Their mean proportion = 16

and third proportion = 128

xy=16xy=256

x=256y...(i)

and y2x=128x=y2128...(ii)

From (i) and (ii)

256y=y2128

y3=256×128=32768

y3=(32)3y=32

x=256y=25632=8

Numbers are 8,32


Question 12

If q is the mean proportional between p and r, prove that:

p23q2+r2=q4(1p23q2+1r2)

Sol :

q is mean proportional between p and r

q² = pr

L.H.S. =p23q2+r2=p23pr+r2

R.H.S.=q4(1p23q2+1r2)

=(q2)2(1p23q2+1r2)=(pr)2(1p23q2+1r2)

=(pr)2(1p23pr+1r2)

=p2r2(r23pr+p2p2r2)=r23pr+p2

∴L.H.S=R.H.S


Question 13

If ab=cd=ef , prove that each ratio is 

(i) 3a25c2+7e23b25d2+7t2

(ii) [2a3+5c3+7e32b3+5d3+7f3]13

Sol :

ab=cd=ef=k(say)

∴ a = k, c = dk, e = fk

(i) 3a25c2+7e23b25d2+7f2

=3b2k25d2k2+7f2k23b25d27f2

=k3b25d2+7f23b25d2+7f2=k
hence proved

(ii) [2a3+5c3+7e32b3+5d3+7f3]13

=[2b3k3+5d3k3+7f3k32b3+5d3+7f3]13

=k[2b3+5d3+7f32b3+5d3+7f3]13=k
hence proved

Question 14

If xa=yb=zc, prove that 3x35y3+4z33a35b3+4c3=(3x5y+4z3a5b+4c)3
Sol :
xa=yb=zc=k( say )
x=a k, y=b k, z=c k

L.H.S. =3x35y3+4z33a35b3+4c3
=3a3k35b3k3+4c3k33a35b3+4c3
=k3(3a35b3+4c3)3a35b3+4c3=k3

R.H.S. =(3x5y+4z3a5b+4c)3

=(3ak5bk+4ck3a5b+4c)3=(k(3a5b+4c)(3a5b+4c))3
=(k)3=k3
∴L.H.S=R.H.S

Question 15

If x : a = y : b, prove that
x4+a4x3+a3+y4+b4y3+b3=(x+y)4+(a+b)4(x+y)3+(a+b)3
Sol :
xa=yb=k(say)

x = ak, y = bk

L.H.S. =x4+a4x3+a3+y4+b4y3+b3

=a4k4+a4a3k3+a3+b4k4+b4b3k3+b3
=a4(k4+1)a3(k3+1)+b4(k4+1)b3(k3+1)
=a(k4+1)k3+1+b(k4+1)k3+1
=a(k4+1)+b(k4+1)k3+1=(k4+1)(a+b)k3+1

R.H.S. =(x+y)4+(a+b)4(x+y)3+(a+b)3

=(ak+bk)4+(a+b)4(ak+bk)3+(a+b)3

=k4(a+b)4+(a+b)4k3(a+b)3(a+b)3=(a+b)4(k4+1)(a+b)3(k3+1)

=(a+b)(k4+1)k3+1=(k4+1)(a+b)k3+1

∴L.H.S=R.H.S
Hence proved

Question 16

If xb+ca=yc+ab=za+bc prove that each ratio’s equal to :

x+y+za+b+c

Sol :

xb+ca=yc+ab=za+bc=k( say )

x = k(b + c – a),

y = k(c + a – b),

z = k(a + b – c)

x+y+za+b+c

=k(b+ca)+k(c+ab)+k(a+bc)a+b+c

=k(b+ca+c+ab+a+bc)a+b+c

=k(a+b+c)a+b+c=k

Hence proved


Question 17

If a : b = 9 : 10, find the value of

(i) 5a+3b5a3b
(ii) 2α23b22a2+3b2

Sol :

a : b = 9 : 10

ab=910

(i) 5a+3b5a3b=5ab+3bb5ab3bb=5ab+35ab3

(dividing by b)

=5×910+35×9103

=92+3923

=15232

(Substituting the value of ab)

=152×23=5


(ii) \frac{2 a^{2}-3 b^{2}}{2 a^{2}+3 b^{2}}$

$=\frac{\frac{2 a^{2}}{b^{2}}-\frac{3 b^{2}}{b^{2}}}{\frac{2 a^{2}}{b^{2}}-\frac{3 b^{2}}{h^{2}}}

(Dividing by b2)

=\frac{2\left(\frac{a}{b}\right)^{2}-3}{2\left(\frac{a}{b}\right)^{2}+3}$

$=\frac{2\left(\frac{9}{10}\right)^{2}-3}{2\left(\frac{9}{10}\right)^{2}+3}

=2×8110032×81100+3

=815038150+3

=811505081+15050

=6950×50231=69231=2377


Question 18

If (3x² + 2y²) : (3x² – 2y²) = 11 : 9, find the value of  3x4+25y43x425y4

Sol :

3x4+25y43x425y4=119

Applying componendo and dividendo

3x2+2y2+3x22y23x2+2y23x2+2y2=11+9119

6x24y2=202

32x2y2=10

x2y2=10×23=203

3x4+25y43x425y4

=3x4y4+25y4y43x4y425y4y4

=3(x2y2)2+253(x2y2)225

=3×(203)2+253(203)225

=3×4009+253×400925

=4003+2514003251

=400+753400753

=4753×3325=1913


Question 19

If x=2maba+b , find the value of

x+maxma+x+mbxmb

Sol :

x=2maba+b

xma+2ba+b

Applying componendo and dividendo

x+maxma=2b+a+b2bab=3b+aba...(i)

Again, xmb=2aa+b

Applying componendo and dividendo,

x+mbxmb=2a+a+b2aab=3a+bab..(ii)

Adding (i) and (ii)

x+maxma+x+mbxmb=3b+aba+3a+bab

=3b+aab+3a+bab=3ba+3a+bab

=2a2bab=2(ab)ab=2


Question 20

If x=paba+b, prove that x+paxpax+pbxpb=2(a2b2)ab

Sol :

x=paba+b

xpa+ba+b

Applying componendo and dividendo

x+paxpa=b+a+bbab=a+2ba..(i)

Again xpb=aa+b

Applying componendo and dividendo,

x+pbxpb=a+a+baab=2a+bb..(ii)

L.H.S.=x+paxpax+pbxpb

=a+2ba2a+bb

=a+2ba+2a+bb

=ab+2b22a2abab

=2b22a2ab

=2a2+2b2ab

=2(a2b2)ab=2(a2b2)ab

=R.H.S


Question 21

Find x from the equation a+x+a2x2a+xa2x2=bx

Sol :

a+x+a2x2a+xa2x2=bx

Applying componendo and dividendo,

a+x+a2x2+a+xa2x2a+x+a2x2ax+a2x2

=b+xbx

2(a+x)2a2x2=b+xbx

a+xa2x2=b+xbx

Squaring both sides,

(a+x)2a2x2=(b+x)2(bx)2

(a+x)2(a+x)(ax)=(b+x)2(bx)2

a+xax=(b+x)2(bx)2

Again applying componendo and dividendo,

a+x+axa+xa+x=(b+x)2+(bx)2(b+x)2(bx)2

2a2x=2(b2+x2)4bx

ax=b2+x22bx

2a2x=2(b2+x2)4bx

ax=b2+x22bx

x2=2abb2

x=2abb2 


Question 22

If x=3a+1+3a13a+13a1, prove that

x33ax2+3xa=0

Sol :

x=3a+1+3a13a+13a1

Applying componendo and dividendo,

x+1x1=3a+1+3a1+3a+13a13a+1+3a13a+1+3a1

x+1x1=23a+123a1

x+1x1=3a+13a1

Cubing both sides

(x+1)3(x1)3=a+1a1

Again applying componendo and dividendo,

(x+1)3+(x1)3(x+1)3(x1)3=a+1+a1a+1a+1

2(x3+3x)2(3x2+1)=2a2

x3+3x3x2+1=a1

x3+3x=3ax2+a

x33ax2+3xa=0


Question 23

If by+czb2+c2=cz+axc2+a2=ax+bya2+b2, prove that each of these ratio is equal to 

Sol :

by+czb2+c2=cz+αxc2+a2=ax+bya2+b2

=2(ax+by+cz)2(a2+b2+c2)=ax+by+cza2+b2+c2 (Adding)

Now by+czb2+c2=ax+by+cza2+b2+c2

by+czax+by+cz=b2+c2a2+b2+c2 (By alternendo)

by+czaxbyczax+by+cz

=b2+c2a2b2c2a2+b2+c2

axax+by+cz=a2a2+b2+c2

xax+by+cz=aa2+b2+c2

xa=ax+by+cza2+b2+c2 ..(i)

Similarly we can prove that

yb=ax+by+cza2+b2+c2..(ii)

and zc=ax+by+cza2+b2+c2...(iii)

From (i), (ii) and (iii)

Hence xa=yb=zc

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