ML Aggarwal Solution Class 10 Chapter 7 Ratio and Proportion Test
Test
Question 1
Find the compound ratio of:
Sol :
(a+b)2:(a−b)2
(a2−b2):(a2+b2)
(a4−b4):(a+b)4
=(a+b)2(a−b)2×a2−b2a2+b2×a4−b4(a+b)4
=(a+b)2(a−b)2×(a+b)(a−b)a2+b2×(a2+b2)(a+b)(a−b)(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
⇒129p−86q=14p+6q
⇒129p−14p=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
⇒ 3a + 5n : 7a – 2b
Dividing each term by b
⇒3×35+5:7×35−2
⇒(95+5):(215−2)
⇒9+255:21−105
⇒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)
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,
∴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
⇒ 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+x)(38+x)=(34+x)(17+x)
⇒570+53x+x2=578+51x+x2
⇒x2+53x−x2−51x=578−570
⇒2x=8⇒x=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+ca−c=a−ca−2b+c
⇒(a+2b+c)(a−2b+c)=(a−c)2
⇒a2−2ab+ac+2ab−4b2+2bc+ac−2bc+c2=a2−2ac+c2
⇒a2−2ab+ac+2ab−4b2+2bc+ac−2bc+c2−a2+2ac−c2=0
⇒4ac−4b2=0⇒ac−b2=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
(i) ∵26=6p then
2p=36
⇒p=18
(ii) p54=54q⇒pq=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
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
=k16−12=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
⇒x=256y...(i)
and y2x=128⇒x=y2128...(ii)
From (i) and (ii)
256y=y2128
⇒y3=256×128=32768
⇒y3=(32)3⇒y=32
∴x=256y=25632=8
∴ Numbers are 8,32
Question 12
If q is the mean proportional between p and r, prove that:
Sol :
q is mean proportional between p and r
q² = pr
=(q2)2(1p2−3q2+1r2)=(pr)2(1p2−3q2+1r2)
=(pr)2(1p2−3pr+1r2)
=p2r2(r2−3pr+p2p2r2)=r2−3pr+p2
∴L.H.S=R.H.S
Question 13
If ab=cd=ef , prove that each ratio is
(i) √3a2−5c2+7e23b2−5d2+7t2
(ii) [2a3+5c3+7e32b3+5d3+7f3]13
Sol :
ab=cd=ef=k(say)
∴ a = k, c = dk, e = fk
Question 14
Question 15
=(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
Question 16
If xb+c−a=yc+a−b=za+b−c prove that each ratio’s equal to :
Sol :
xb+c−a=yc+a−b=za+b−c=k( say )
x = k(b + c – a),
y = k(c + a – b),
z = k(a + b – c)
=k(b+c−a)+k(c+a−b)+k(a+b−c)a+b+c
=k(b+c−a+c+a−b+a+b−c)a+b+c
=k(a+b+c)a+b+c=k
Hence proved
Question 17
If a : b = 9 : 10, find the value of
Sol :
a : b = 9 : 10
=5×910+35×910−3
=92+392−3
=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}
=2×81100−32×81100+3
=8150−38150+3
=81−1505081+15050
=−6950×50231=−69231=−2377
Question 18
If (3x² + 2y²) : (3x² – 2y²) = 11 : 9, find the value of 3x4+25y43x4−25y4
Sol :
3x4+25y43x4−25y4=119
Applying componendo and dividendo
⇒6x24y2=202
⇒32x2y2=10
⇒x2y2=10×23=203
3x4+25y43x4−25y4
=3x4y4+25y4y43x4y4−25y4y4
=3(x2y2)2+253(x2y2)2−25
=3×(203)2+253(203)2−25
=3×4009+253×4009−25
=4003+2514003−251
=400+753400−753
=4753×3325=1913
Question 19
If x=2maba+b , find the value of
Sol :
x=2maba+b
⇒xma+2ba+b
Applying componendo and dividendo
=2a−2ba−b=2(a−b)a−b=2
Question 20
If x=paba+b, prove that x+pax−pa−x+pbx−pb=2(a2−b2)ab
Sol :
x=paba+b
⇒xpa+ba+b
Applying componendo and dividendo
Again xpb=aa+b
Applying componendo and dividendo,
x+pbx−pb=a+a+ba−a−b=2a+b−b..(ii)
L.H.S.=x+pax−pa−x+pbx−pb
=a+2b−a−2a+b−b
=a+2b−a+2a+bb
=ab+2b2−2a2−ab−ab
=2b2−2a2−ab
=−2a2+2b2−ab
=−2(a2−b2)−ab=2(a2−b2)ab
=R.H.S
Question 21
Find x from the equation a+x+√a2−x2a+x−√a2−x2=bx
Sol :
a+x+√a2−x2a+x−√a2−x2=bx
Applying componendo and dividendo,
=b+xb−x
⇒2(a+x)2√a2−x2=b+xb−x
⇒a+x√a2−x2=b+xb−x
Squaring both sides,
(a+x)2a2−x2=(b+x)2(b−x)2
⇒(a+x)2(a+x)(a−x)=(b+x)2(b−x)2
⇒a+xa−x=(b+x)2(b−x)2
Again applying componendo and dividendo,
a+x+a−xa+x−a+x=(b+x)2+(b−x)2(b+x)2−(b−x)2
⇒2a2x=2(b2+x2)4bx
⇒ax=b2+x22bx
⇒2a2x=2(b2+x2)4bx
⇒ax=b2+x22bx
⇒x2=2ab−b2
x=√2ab−b2
Question 22
If x=3√a+1+3√a−13√a+1−3√a−1, prove that
x3−3ax2+3x−a=0
Sol :
x=3√a+1+3√a−13√a+1−3√a−1
Applying componendo and dividendo,
x+1x−1=23√a+123√a−1
⇒x+1x−1=3√a+13√a−1
Cubing both sides
(x+1)3(x−1)3=a+1a−1
Again applying componendo and dividendo,
(x+1)3+(x−1)3(x+1)3−(x−1)3=a+1+a−1a+1−a+1
⇒2(x3+3x)2(3x2+1)=2a2
⇒x3+3x3x2+1=a1
⇒x3+3x=3ax2+a
⇒x3−3ax2+3x−a=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+cz−ax−by−czax+by+cz
=b2+c2−a2−b2−c2a2+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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