Exercise 1 Archives - Noon Academy

Exercise 1

If
$x=\frac{\sqrt[3]{a+1}+\sqrt[3]{a-1}}{\sqrt[3]{a+1}-\sqrt[3]{a-1}}$
prove that :
${{x}^{3}}-3a{{x}^{2}}+3x-a=0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By cross multiplication \[\begin{array}{*{35}{l}} {{x}^{3}}~+\text{ }3x\text{ }=\text{ }3a{{x}^{2}}~+\text{ }a \\ {{x}^{3}}~\text{ }3a{{x}^{2}}~+\text{ }3x\text{ }\text{ }a\text{...

Find a from the equation
$\frac{a+x+\sqrt{{{a}^{2}}-{{x}^{2}}}}{a+x-\sqrt{{{a}^{2}}-{{x}^{2}}}}=\frac{b}{x}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that

If
$x=\frac{pab}{a+b}$
, prove that
$\frac{x+pa}{x-pa}-\frac{x+pb}{x-pb}=\frac{2({{a}^{2}}-{{b}^{2}})}{ab}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that = RHS

If
$x=\frac{2mab}{a+b}$
, find the value of
$\frac{x+ma}{x-ma}+\frac{x+mb}{x-mb}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that

If
$(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}):\text{ }(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{11}:\text{ }\mathbf{9}$
, find the value of
$\frac{3{{x}^{4}}+25{{y}^{4}}}{3{{x}^{4}}-25{{y}^{4}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}):\text{ }(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{...

If a: b =
$9:10$
, find the value of (i)
$\frac{5a+3b}{5a-3b}$
(ii)
$\frac{2{{a}^{2}}-3{{b}^{2}}}{2{{a}^{2}}+3{{b}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a: b = \[9:10\] So we get a/b = \[9/10\] = \[5\]

If
$\frac{x}{b+c-a}=\frac{y}{c+a-b}=\frac{z}{a+b-c}$
prove that each ratio’s equal to:
$\frac{x+y+z}{a+b+c}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider So we get x = k (b + c – a) y = k (c + a – b) z = k (a + b – a) Here = k Therefore, it is proved.

If x: a = y: b, prove that
$\frac{{{x}^{4}}+{{a}^{4}}}{{{x}^{3}}+{{a}^{3}}}+\frac{{{y}^{4}}+{{b}^{4}}}{{{y}^{3}}+{{b}^{3}}}=\frac{{{(x+y)}^{4}}+{{(a+b)}^{4}}}{{{(x+y)}^{3}}+{{(a+b)}^{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

We know that x/a = y/b = k So we get x = ak, y = bk Here Here LHS = RHS Therefore, it is proved.

If x/a = y/b = z/c, prove that
$\frac{3{{x}^{3}}-5{{y}^{3}}+4{{z}^{3}}}{3{{a}^{3}}-5{{b}^{3}}+4{{c}^{3}}}={{\left( \frac{3x-5y+4z}{3a-5b+4c} \right)}^{3}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that x/a = y/b = z/c = k So we get x = ak, y = bk, z = ck Here = \[{{k}^{3}}\] Hence, LHS = RHS.

If a/b = c/d = e/f, prove that each ratio is (i)
$\sqrt{\frac{3{{a}^{2}}-5{{c}^{2}}+7{{e}^{2}}}{3{{b}^{2}}-5{{d}^{2}}+7{{f}^{2}}}}$
(ii)
${{\left[ \frac{2{{a}^{3}}+5{{c}^{2}}+7{{e}^{2}}}{2{{b}^{3}}+5{{d}^{3}}+7{{f}^{3}}} \right]}^{\frac{1}{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a/b = c/d = e/f = k So we get a = k, c = dk, e = fk Therefore, it is proved. = k Therefore, it is proved.

If q is the mean proportional between p and r, prove that:
${{p}^{2}}-3{{q}^{2}}+{{r}^{2}}={{q}^{4}}(\frac{1}{{{p}^{2}}}-\frac{3}{{{q}^{2}}}+\frac{1}{{{r}^{2}}})$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that q is the mean proportional between p and r q2 = pr Here LHS = \[{{p}^{2}}~\text{ }3{{q}^{2}}~+\text{ }{{r}^{2}}\] We can write it as \[=\text{ }{{p}^{2}}~\text{ }3pr\text{ }+\text{...

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

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x and y as the two numbers Mean proportion = \[16\] Third proportion = \[128\] \[\begin{array}{*{35}{l}} \surd xy\text{ }=\text{ }16 \\ xy\text{ }=\text{ }256 \\ \end{array}\] Here...

If a, b, c, d, e are in continued proportion, prove that:
$\mathbf{a}:\text{ }\mathbf{e}\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{4}}}:\text{ }{{\mathbf{b}}^{\mathbf{4}}}$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d, e are in continued proportion We can write it as a/b = b/c = c/d = d/e = k \[d\text{ }=\text{ }ek,\text{ }c\text{ }=\text{ }e{{k}^{2}},\text{ }b\text{ }=\text{...

If
$\mathbf{2},\text{ }\mathbf{6},\text{ }\mathbf{p},\text{ }\mathbf{54}$
and q are in continued proportion, find the values of p and q.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{2},\text{ }\mathbf{6},\text{ }\mathbf{p},\text{ }\mathbf{54}\] and q are in continued proportion We can write it as \[2/6\text{ }=\text{ }6/p\text{ }=\text{ }p/54\text{...

If
$\left( \mathbf{a}\text{ }+\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right),\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{c} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right)$
are in continued proportion, prove that b is the mean proportional between a and c.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\left( \mathbf{a}\text{ }+\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right),\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{c} \right)\text{ }\mathbf{and}\text{ }\left(...

What number must be added to each of the numbers
$\mathbf{15},\text{ }\mathbf{17},\text{ }\mathbf{34}\text{ }\mathbf{and}\text{ }\mathbf{38}$
to make them proportional?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x be added to each number So the numbers will be \[15\text{ }+\text{ }x,\text{ }17\text{ }+\text{ }x,\text{ }34\text{ }+\text{ }x\text{ }and\text{ }38\text{ }+\text{ }x\] Based on the...

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.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider the number of passed = \[3x\] Number of failed = x So the total candidates appeared = \[3x\text{ }+\text{ }x\text{ }=\text{ }4x\] In the second case Number of candidates appeared =...

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?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider \[5x\] and \[6x\] as the savings of Lokesh and his sister. Lokesh should save Rs y more Based on the problem \[\left( 5x\text{ }+\text{ }y \right)/\text{ }\left( 6x\text{ }+\text{ }30...

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.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider the two shorter sides of a right-angled triangle as \[5x\] and \[12x\] So the third longest side = \[13x\] It is given that \[5x\text{ }+\text{ }12x\text{ }+\text{ }13x\text{ }=\text{...

If a: b =
$3:5$
, find
$\left( \mathbf{3a}\text{ }+\text{ }\mathbf{5b} \right):\text{ }\left( \mathbf{7a}\text{ }\text{ }\mathbf{2b} \right)$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a: b = \[3:5\] We can write it as a/b = \[3/5\] Here \[\left( 3a\text{ }+\text{ }5b \right):\text{ }\left( 7a\text{ }\text{ }2b \right)\] Now dividing the terms by b Here \[\left(...

If
$\left( \mathbf{7p}\text{ }+\text{ }\mathbf{3q} \right):\text{ }\left( \mathbf{3p}\text{ }\text{ }\mathbf{2q} \right)\text{ }=\text{ }\mathbf{43}:\text{ }\mathbf{2}$
, find p: q.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\left( 7p\text{ }+\text{ }3q \right):\text{ }\left( 3p\text{ }\text{ }2q \right)\text{ }=\text{ }43:\text{ }2\] We can write it as \[\left( 7p\text{ }+\text{ }3q \right)/\text{...

Find the compound ratio of
${{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{2}}}:\text{ }{{\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right)}^{\mathbf{2}}},\text{ }({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}}):\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{a}}^{\mathbf{4}}}~\text{ }{{\mathbf{b}}^{\mathbf{4}}}):\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{4}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

\[\begin{array}{*{35}{l}} {{\left( a\text{ }+\text{ }b \right)}^{2}}:\text{ }{{\left( a\text{ }\text{ }b \right)}^{2}} \\ ({{a}^{2}}~\text{ }{{b}^{2}}):\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}) \\...

If
$\frac{x+y}{ax+by}=\frac{y+z}{ay+bz}=\frac{z+x}{az+bx}$
, prove that each of these ratio is equal to
$\frac{2}{a+b}$
unless x+y+z=0

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that If \[x\text{ }+\text{ }y\text{ }+\text{ }z\text{ }\ne \text{ }0\] Therefore, it is proved.

Using the properties of proportion, solve the following equation for x; given
$\frac{{{x}^{3}}+3x}{3{{x}^{2}}+1}=\frac{341}{91}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By cross multiplication \[\begin{array}{*{35}{l}} 6x\text{ }\text{ }6\text{ }=\text{ }5x\text{ }+\text{ }5 \\ 6x\text{ }\text{ }5x\text{ }=\text{ }5\text{ }+\text{ }6 \\ ...

Given
$\frac{{{x}^{3}}+12x}{6{{x}^{2}}+8}=\frac{{{y}^{3}}+27y}{9{{y}^{2}}+27}$
Using componendo and dividendo find x y.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By further calculation \[\begin{array}{*{35}{l}} 2x/4\text{ }=\text{ }2y/3 \\ x/2\text{ }=\text{ }y/3 \\ \end{array}\] By cross multiplication \[x/y\text{ }=\text{ }2/3\] Hence,...

Given that
$\frac{{{a}^{3}}+3a{{b}^{2}}}{{{b}^{3}}+3{{a}^{2}}b}=\frac{63}{62}$
. Using componendo and dividendo find a : b.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that By cross multiplication \[a\text{ }+\text{ }b\text{ }=\text{ }5a\text{ }\text{ }5b\] We can write it as \[\begin{array}{*{35}{l}} 5a\text{ }\text{ }a\text{ }\text{ }5b\text{ }\text{...

Given
$x=\frac{\sqrt{{{a}^{2}}+{{b}^{2}}}+\sqrt{{{a}^{2}}-{{b}^{2}}}}{\sqrt{{{a}^{2}}+{{b}^{2}}}-\sqrt{{{a}^{2}}-{{b}^{2}}}}$
use componendo and dividend to prove that
${{b}^{2}}=\frac{2{{a}^{2}}x}{{{x}^{2}}+1}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

If
$x=\frac{\sqrt{a+x}+\sqrt{a-1}}{\sqrt{a+1}-\sqrt{a-1}}$
, using properties of proportion, show that
${{x}^{2}}-2ax+1=0$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that We get \[\begin{array}{*{35}{l}} 2ax\text{ }=\text{ }{{x}^{2}}~+\text{ }1 \\ {{x}^{2}}~\text{ }2ax\text{ }+\text{ }1\text{ }=\text{ }0 \\ \end{array}\] Therefore, it is...

Solve for x:
$16{{(\frac{a-x}{a+x})}^{3}}=\frac{a+x}{a-x}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

So we get \[\begin{array}{*{35}{l}} 3x\text{ }=\text{ }a \\ x\text{ }=\text{ }a/3 \\ \end{array}\] So we get x = \[3a\] Therefore, x= \[a/3,3a\].

Solve
$\frac{1+x+{{x}^{2}}}{1-x+{{x}^{2}}}=\frac{62(1+x)}{63(1-x)}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

x = \[1/5\]

Using properties of proportion solve for x. Given that x is positive. (i)
$\frac{3x+\sqrt{9{{x}^{2}}-5}}{3x-\sqrt{9{{x}^{2}}-5}}=5$
(ii)
$\frac{2x+\sqrt{4{{x}^{2}}-1}}{2x-\sqrt{4{{x}^{2}}-1}}=4$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[\begin{array}{*{35}{l}} 81{{x}^{2}}~\text{ }45\text{ }=\text{ }36{{x}^{2}} \\ 81{{x}^{2}}~\text{ }36{{x}^{2}}~=\text{ }45 \\ \end{array}\] So we get...

Using properties of properties, find x from the following equations: (v)
$\frac{3x+\sqrt{9{{x}^{2}}+5}}{3x-\sqrt{9{{x}^{2}}+5}}=5$
(vi)
$\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}=\frac{c}{d}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[\begin{array}{*{35}{l}} 81{{x}^{2}}~\text{ }45\text{ }=\text{ }36{{x}^{2}} \\ 81{{x}^{2}}~\text{ }36{{x}^{2}}~=\text{ }45 \\ \end{array}\] So we get 45x2 = 45...

Using properties of properties, find x from the following equations: (iii)
$\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}=\frac{a}{b}$
(iv)
$\frac{\sqrt{12x+1}+\sqrt{2x-3}}{\sqrt{12x+1}-\sqrt{2x-3}}=\frac{3}{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[\begin{array}{*{35}{l}} 50x\text{ }\text{ }75\text{ }=\text{ }12x\text{ }+\text{ }1 \\ 50x\text{ }\text{ }12x\text{ }=\text{ }1\text{ }+\text{ }75 \\ \end{array}\] So we...

Using properties of properties, find x from the following equations: (i)
$\frac{\sqrt{2-x}+\sqrt{2+x}}{\sqrt{2-x}-\sqrt{2+x}}=3$
(ii)
$\frac{\sqrt{x+4}+\sqrt{x-10}}{\sqrt{x+4}-\sqrt{x-10}}=\frac{5}{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

By cross multiplication \[8\text{ }+\text{ }4x\text{ }=\text{ }2\text{ }\text{ }x\] So we get \[\begin{array}{*{35}{l}} 4x\text{ }+\text{ }x\text{ }=\text{ }2\text{ }\text{ }8 \\ 5x\text{ }=\text{...

If
$x=\frac{4\sqrt{6}}{\sqrt{2}+\sqrt{3}}$
find the value of
$\frac{x+2\sqrt{2}}{x-2\sqrt{2}}+\frac{x+2\sqrt{3}}{x-2\sqrt{3}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

If
$x=\frac{8ab}{a+b}$
find the value of
$\frac{x+4a}{x-4a}+\frac{x+4b}{x-4b}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

\[\begin{array}{*{35}{l}} =\text{ }2\left( a\text{ }\text{ }b \right)/\text{ }\left( a\text{ }\text{ }b \right) \\ =\text{ }2 \\ \end{array}\]

If
$x=\frac{2ab}{a+b}$
find the value of
$\frac{x+a}{x-a}+\frac{x+b}{x-b}$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

\[\begin{array}{*{35}{l}} =\text{ }2\left( a\text{ }\text{ }b \right)/\text{ }\left( a\text{ }\text{ }b \right) \\ =\text{ }2 \\ \end{array}\]

If
$(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~\text{ }\mathbf{13}{{\mathbf{d}}^{\mathbf{2}}})\text{ }=\text{ }(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{d}}^{\mathbf{2}}})$
, prove that a: b :: c: d.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~\text{...

If (ma + nb): b :: (mc + nd): d, prove that a, b, c, d are in proportion.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that (ma + nb): b :: (mc + nd): d We can write it as (ma + nb)/ b = (mc + nd)/ d By cross multiplication mad + nbd = mbc + nbd Here mad = mbc ad = bc By further calculation a/b = c/d...

If (pa + qb): (pc + qd) :: (pa – qb): (pc – qd) prove that a: b :: c: d.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that (pa + qb): (pc + qd) :: (pa – qb): (pc – qd) We can write it as Therefore, it is proved that a: b :: c: d.

If
$~\left( \mathbf{4a}\text{ }+\text{ }\mathbf{5b} \right)\text{ }\left( \mathbf{4c}\text{ }\text{ }\mathbf{5d} \right)\text{ }=\text{ }\left( \mathbf{4a}\text{ }\text{ }\mathbf{5d} \right)\text{ }\left( \mathbf{4c}\text{ }+\text{ }\mathbf{5d} \right)$
, prove that a, b, c, d are in proportion.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[~\left( \mathbf{4a}\text{ }+\text{ }\mathbf{5b} \right)\text{ }\left( \mathbf{4c}\text{ }\text{ }\mathbf{5d} \right)\text{ }=\text{ }\left( \mathbf{4a}\text{ }\text{ }\mathbf{5d}...

(i) If
$\frac{5x+7y}{5u+7v}=\frac{5x-7y}{5u-7v}$
, show that
$\frac{x}{y}=\frac{u}{v}$
(ii)
$\frac{8a-5b}{8c-5d}=\frac{8a+5b}{8c+5d}$
, prove that
$\frac{a}{b}=\frac{c}{d}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Therefore, it is proved. Therefore, it is proved.

If a: b :: c: d, prove that (iii)
$\left( \mathbf{2a}\text{ }+\text{ }\mathbf{3b} \right)\text{ }\left( \mathbf{2c}\text{ }\text{ }\mathbf{3d} \right)\text{ }=\text{ }\left( \mathbf{2a}\text{ }\text{ }\mathbf{3b} \right)\text{ }\left( \mathbf{2c}\text{ }+\text{ }\mathbf{3d} \right)$
(iv) (la + mb): (lc + mb) :: (la – mb): (lc – mb)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) We know that If a: b :: c: d we get a/b = c/d By multiplying \[2/3\] \[2a/3b\text{ }=\text{ }2c/3d\] By applying componendo and dividendo \[\left( 2a\text{ }+\text{ }3b \right)/\text{ }\left(...

If a: b :: c: d, prove that (i)
$\frac{2a+5b}{2a-5b}=\frac{2c+5d}{2c-5d}$
(ii)
$\frac{5a+11b}{5c+11d}=\frac{5a-11b}{5c-11d}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) We know that If a: b :: c: d we get a/b = c/d By multiplying \[2/5\] \[2a/5b\text{ }=\text{ }2c/5d\] By applying componendo and dividendo \[\left( 2a\text{ }+\text{ }5b \right)/\text{ }\left(...

If a, b, c, d are in continued proportion, prove that: (V)
${{\left( \frac{a-b}{c}+\frac{a-c}{b} \right)}^{2}}-{{\left( \frac{d-b}{c}+\frac{d-c}{b} \right)}^{2}}={{(a-d)}^{2}}\left( \frac{1}{{{c}^{2}}}-\frac{1}{{{b}^{2}}} \right)$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

If a, b, c, d are in continued proportion, prove that: (iii)
$\left( \mathbf{a}\text{ }+\text{ }\mathbf{d} \right)\text{ }\left( \mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{c} \right)\text{ }\left( \mathbf{b}\text{ }+\text{ }\mathbf{d} \right)\text{ }=\text{ }{{\left( \mathbf{b}\text{ }\text{ }\mathbf{c} \right)}^{\mathbf{2}}}$
(iv) a: d = triplicate ratio of (a – b): (b – c)

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

If a, b, c, d are in continued proportion, prove that: (i)
$\frac{{{a}^{3}}+{{b}^{3}}+{{c}^{3}}}{{{b}^{3}}+{{c}^{3}}+{{d}^{3}}}=\frac{a}{d}$
(ii)
$({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }({{\mathbf{c}}^{\mathbf{2}}}~\text{ }{{\mathbf{d}}^{\mathbf{2}}})\text{ }=\text{ }{{({{\mathbf{b}}^{\mathbf{2}}}~\text{ }{{\mathbf{c}}^{\mathbf{2}}})}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

If a, b, c are in continued proportion, prove that: (v)
$\mathbf{abc}\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)}^{\mathbf{3}}}~=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{bc}\text{ }+\text{ }\mathbf{ca} \right)}^{\mathbf{3}}}$
(vi)
$\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion So we get a/b = b/c = k (v) LHS = \[abc\text{ }{{\left( a\text{ }+\text{ }b\text{ }+\text{ }c \right)}^{3}}\] We can write it as \[=\text{...

If a, b, c are in continued proportion, prove that: (iii)
$\mathbf{a}:\text{ }\mathbf{c}\text{ }=\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}):\text{ }({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}})$
(iv)
$~{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}~({{\mathbf{a}}^{-\mathbf{4}}}~+\text{ }{{\mathbf{b}}^{-\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{-\mathbf{4}}})\text{ }=\text{ }{{\mathbf{b}}^{-\mathbf{2}}}~({{\mathbf{a}}^{\mathbf{4}}}~+\text{ }{{\mathbf{b}}^{\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{\mathbf{4}}})$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion So we get a/b = b/c = k (iii) \[~a:\text{ }c\text{ }=\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}):\text{ }({{b}^{2}}~+\text{ }{{c}^{2}})\] We...

If a, b, c are in continued proportion, prove that: (i)
$\frac{a+b}{b+c}=\frac{{{a}^{2}}(b-c)}{{{b}^{2}}(a-b)}$
(ii)
$\frac{1}{{{a}^{3}}}+\frac{1}{{{b}^{3}}}+\frac{1}{{{c}^{3}}}=\frac{a}{{{b}^{2}}{{c}^{2}}}+\frac{b}{{{c}^{2}}{{a}^{2}}}+\frac{c}{{{a}^{2}}{{b}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion So we get a/b = b/c = k Therefore, LHS = RHS. Therefore, LHS = RHS.

If a, b, c are in continued proportion, prove that:
$\frac{p{{a}^{2}}+qab+r{{b}^{2}}}{p{{b}^{2}}+qbc+r{{c}^{2}}}=\frac{a}{c}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c are in continued proportion \[\frac{p{{a}^{2}}+qab+r{{b}^{2}}}{p{{b}^{2}}+qbc+r{{c}^{2}}}=\frac{a}{c}\] Consider a/b = b/c = k So we get a = bk and b = ck ….. (1) From...

If x, y, z are in continued proportion, prove that:
${{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y} \right)}^{\mathbf{2}}}/\text{ }{{\left( \mathbf{y}\text{ }+\text{ }\mathbf{z} \right)}^{\mathbf{2}}}~=\text{ }\mathbf{x}/\mathbf{z}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that x, y, z are in continued proportion Consider x/y = y/z = k So we get y = kz \[x\text{ }=\text{ }yk\text{ }=\text{ }kz\text{ }\times \text{ }k\text{ }=\text{ }{{k}^{2}}z\] Therefore,...

If a, b, c and d are in proportion, prove that: (vii)
$\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}}=\frac{ab+ad-bc}{bc+cd-ad}$
(viii)
$abcd\left[ \frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{{{d}^{2}}} \right]={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk Therefore, LHS = RHS. So we get = d2 (1 + k2) + b2 (1 + k2) = (1 + k2) (b2 + d2) RHS = a2 + b2 + c2 + d2 We can...

If a, b, c and d are in proportion, prove that: (v)
$\frac{{{(a+c)}^{3}}}{{{(b+d)}^{3}}}=\frac{a{{(a-c)}^{2}}}{b{{(b-d)}^{2}}}$
(vi)
$\frac{{{a}^{2}}+ab+{{b}^{2}}}{{{a}^{2}}-ab+{{b}^{2}}}=\frac{{{c}^{2}}+cd+{{d}^{2}}}{{{c}^{2}}-cd+{{d}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk Therefore, LHS = RHS. Therefore, LHS = RHS.

If a, b, c and d are in proportion, prove that: (i)
$\left( \mathbf{5a}\text{ }+\text{ }\mathbf{7v} \right)\text{ }\left( \mathbf{2c}\text{ }\text{ }\mathbf{3d} \right)\text{ }=\text{ }\left( \mathbf{5c}\text{ }+\text{ }\mathbf{7d} \right)\text{ }\left( \mathbf{2a}\text{ }\text{ }\mathbf{3b} \right)$
(ii) (ma + nb): b = (mc + nd): d

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk (i) LHS = \[\left( 5a\text{ }+\text{ }7b \right)\text{ }\left( 2c\text{ }\text{ }3d \right)\] Substituting the...

18. If ax = by = cz; prove that
$\frac{{{x}^{2}}}{yz}+\frac{{{y}^{2}}}{zx}+\frac{{{z}^{2}}}{xy}=\frac{bc}{{{a}^{2}}}+\frac{ca}{{{b}^{2}}}+\frac{ab}{{{c}^{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider ax = by = cz = k It can be written as x = k/a, y = k/b, z = k/c

If a/b = c/d = e/f prove that:
$\frac{{{a}^{2}}}{{{b}^{2}}}+\frac{{{c}^{2}}}{{{d}^{2}}}+\frac{{{e}^{2}}}{{{f}^{2}}}=\frac{ac}{bd}+\frac{ce}{df}+\frac{ae}{df}$
(iv)
$bdf{{\left[ \frac{a+b}{b}+\frac{c+d}{d}+\frac{c+f}{f} \right]}^{3}}=27(a+b)(c+d)(e+f)$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider a/b = c/d = e/f = k So we get a = bk, c = dk, e = fk Therefore, LHS = RHS. So we get \[=\text{ }bdf\text{ }{{\left( k\text{ }+\text{ }1\text{ }+\text{ }k\text{ }+\text{ }1\text{ }+\text{...

If a/b = c/d = e/f prove that: (i)
$({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{d}}^{\mathbf{2}}}~+\text{ }{{\mathbf{f}}^{\mathbf{2}}})\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }{{\mathbf{e}}^{\mathbf{2}}})\text{ }=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{cd}\text{ }+\text{ }\mathbf{ef} \right)}^{\mathbf{2}}}$
(ii)
$\frac{{{({{a}^{3}}+{{c}^{3}})}^{2}}}{{{({{b}^{3}}+{{d}^{3}})}^{2}}}=\frac{{{e}^{6}}}{{{f}^{6}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider a/b = c/d = e/f = k So we get a = bk, c = dk, e = fk (i) LHS = \[({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{e}^{2}})\] We can write it...

16. If x/a = y/b = z/c, prove that (iii)
$\frac{ax-by}{(a+b)(x-y)}+\frac{by-cz}{(b+c)(y-z)}+\frac{cz-ax}{(c+a)(z-x)}=3$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Therefore, LHS = RHS.

16. If x/a = y/b = z/c, prove that (i)
$\frac{{{x}^{3}}}{{{a}^{2}}}+\frac{{{y}^{3}}}{{{b}^{2}}}+\frac{{{z}^{3}}}{{{c}^{2}}}=\frac{{{(x+y+z)}^{3}}}{{{(a+b+c)}^{2}}}$
(ii)
${{\left[ \frac{{{a}^{2}}{{x}^{2}}+{{b}^{2}}{{y}^{2}}+{{c}^{2}}{{z}^{2}}}{{{a}^{3}}x+{{b}^{3}}y+{{c}^{3}}z} \right]}^{3}}=\frac{xyz}{abc}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that x/a = y/b = z/c We can write it as x = ak, y = bk and z = ck Therefore, LHS = RHS. Therefore, LHS = RHS.

If a + c = mb and
$\mathbf{1}/\mathbf{b}\text{ }+\text{ }\mathbf{1}/\mathbf{d}\text{ }=\text{ }\mathbf{m}/\mathbf{c}$
, prove that a, b, c and d are in proportion.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a + c = mb and \[\mathbf{1}/\mathbf{b}\text{ }+\text{ }\mathbf{1}/\mathbf{d}\text{ }=\text{ }\mathbf{m}/\mathbf{c}\] a + c = mb Dividing the equation by b a/b + c/d = m ……. (1)...

If y is mean proportional between x and z, prove that
$\mathbf{xyz}\text{ }{{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{z} \right)}^{\mathbf{3}}}~=\text{ }{{\left( \mathbf{xy}\text{ }+\text{ }\mathbf{yz}\text{ }+\text{ }\mathbf{zx} \right)}^{\mathbf{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that y is mean proportional between x and z We can write it as \[{{y}^{2}}~=\text{ }xz\]…… (1) Consider LHS = \[xyz\text{ }{{\left( x\text{ }+\text{ }y\text{ }+\text{ }z \right)}^{3}}\]...

If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between
$({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}})$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that b is the mean proportional between a and c \[{{b}^{2}}~=\text{ }ac\]…. (1) Here (ab + bc) is the mean proportional between \[({{\mathbf{a}}^{\mathbf{2}}}~+\text{...

If b is the mean proportional between a and c, prove that a, c,
${{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~\mathbf{and}\text{ }{{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~$
are proportional.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Solution: It is given that b is the mean proportional between a and c We can write it as b2 = a × c b2 = ac ….. (1) We know that a, c, \[{{\mathbf{a}}^{\mathbf{2}}}~+\text{...

Find two numbers such that the mean proportional between them is
$28$
and the third proportional to them is
$224$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider a and b as the two numbers It is given that \[28\] is the mean proportional \[a:\text{ }28\text{ }::\text{ }28:\text{ }b\] We get \[ab\text{ }=\text{ }{{28}^{2}}~=\text{ }784\] Here...

What number must be added to each of the numbers
$\mathbf{16},\text{ }\mathbf{26}\text{ }\mathbf{and}\text{ }\mathbf{40}$
so that the resulting numbers may be in continued proportion?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x be added to each number \[16\text{ }+\text{ }x\text{ },\text{ }26\text{ }+\text{ }x\text{ }and\text{ }40\text{ }+\text{ }x\] are in continued proportion It can be written as \[\left(...

If
$\mathbf{x}\text{ }+\text{ }\mathbf{5}$
is the mean proportion between
$\mathbf{x}\text{ }+\text{ }\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{9}$
, find the value of x.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{x}\text{ }+\text{ }\mathbf{5}\] is the mean proportion between \[\mathbf{x}\text{ }+\text{ }\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{9}\]...

If
$\mathbf{k}\text{ }+\text{ }\mathbf{3},\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2},\text{ }\mathbf{3k}\text{ }\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{2k}\text{ }\text{ }\mathbf{3}$
are in proportion, find k.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{k}\text{ }+\text{ }\mathbf{3},\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2},\text{ }\mathbf{3k}\text{ }\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{2k}\text{...

What number should be subtracted from each of the numbers
$\mathbf{23},\text{ }\mathbf{30},\text{ }\mathbf{57}\text{ }\mathbf{and}\text{ }\mathbf{78}$
so that the remainders are in proportion?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x be subtracted from each term \[23\text{ }\text{ }x,\text{ }30\text{ }\text{ }x,\text{ }57\text{ }\text{ }x\text{ }and\text{ }78\text{ }\text{ }x\] are proportional It can be written as...

What number must be added to each of the numbers
$\mathbf{5},\text{ }\mathbf{11},\text{ }\mathbf{19}\text{ }\mathbf{and}\text{ }\mathbf{37}$
so that they are in proportion?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider x to be added to \[\mathbf{5},\text{ }\mathbf{11},\text{ }\mathbf{19}\text{ }\mathbf{and}\text{ }\mathbf{37}\] to make them in proportion \[5\text{ }+\text{ }x:\text{ }11\text{ }+\text{...

If a,
$12,16$
and b are in continued proportion find a and b.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that a, \[12,16\] and b are in continued proportion \[a/12\text{ }=\text{ }12/16\text{ }=\text{ }16/b\] We know that \[a/12\text{ }=\text{ }12/16\] By cross multiplication \[a/12\text{...

Find the mean proportion of: (iii)
$\mathbf{8}.\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{2}.\mathbf{5}$
(iv)
$\left( a\text{ }\text{ }b \right)\text{ }and\text{ }({{a}^{3}}~\text{ }{{a}^{2}}b),\text{ }a>b$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) Consider x as the mean proportion of \[8.1\text{ }and\text{ }2.5\] \[8.1:\text{ }x\text{ }::\text{ }x:\text{ }2.5\] It can be written as \[\begin{array}{*{35}{l}} {{x}^{2}}~=\text{ }8.1\text{...

Find the mean proportion of: (i)
$\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{80}$
(ii)
$\mathbf{1}/\mathbf{12}\text{ }\mathbf{and}\text{ }\mathbf{1}/\mathbf{75}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Consider x as the mean proportion of 5 and 80 \[5:\text{ }x\text{ }::\text{ }x:\text{ }80\] It can be written as \[\begin{array}{*{35}{l}} {{x}^{2}}~=\text{ }5\text{ }\times \text{ }80\text{...

Find the third proportional to (iii)
$\mathbf{Rs}.\text{ }\mathbf{3},\text{ }\mathbf{Rs}.\text{ }\mathbf{12}$
(iv)
$\mathbf{5}\text{ }{\scriptscriptstyle 1\!/\!{ }_4}\text{ }\mathbf{and}\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) Consider x as the third proportional to \[~Rs.\text{ }3\text{ }and\text{ }Rs.\text{ }12\] \[3:\text{ }12\text{ }::\text{ }12:\text{ }x\] It can be written as \[\begin{array}{*{35}{l}} 3\text{...

Find the third proportional to (i)
$5,10$
(ii)
$\mathbf{0}.\mathbf{24},\text{ }\mathbf{0}.\mathbf{6}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Consider x as the third proportional to \[5,10\] \[5:\text{ }10\text{ }::\text{ }10:\text{ }x\] It can be written as \[\begin{array}{*{35}{l}} 5\text{ }\times \text{ }x\text{ }=\text{ }10\text{...

Find the fourth proportional to (iii)
$\mathbf{1}.\mathbf{5},\text{ }\mathbf{2}.\mathbf{5},\text{ }\mathbf{4}.\mathbf{5}$
(iv)
$\mathbf{9}.\mathbf{6}\text{ }\mathbf{kg},\text{ }\mathbf{7}.\mathbf{2}\text{ }\mathbf{kg},\text{ }\mathbf{28}.\mathbf{8}\text{ }\mathbf{kg}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[1.5,\text{ }2.5,\text{ }4.5\] Consider x as the fourth proportional to \[1.5,\text{ }2.5,\text{ }4.5\] \[1.5:\text{ }2.5\text{ }::\text{ }4.5:\text{ }x\] We can write it as \[1.5\text{...

Find the fourth proportional to (i)
$\mathbf{3},\text{ }\mathbf{12},\text{ }\mathbf{15}$
(ii)
$\mathbf{1}/\mathbf{3},\text{ }\mathbf{1}/\mathbf{4},\text{ }\mathbf{1}/\mathbf{5}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{3},\text{ }\mathbf{12},\text{ }\mathbf{15}\] Consider x as the fourth proportional to \[\mathbf{3},\text{ }\mathbf{12},\text{ }\mathbf{15}\] \[3:\text{ }12\text{ }::\text{ }15:\text{...

Find the value of x in the following proportions: (iii)
$\mathbf{2}.\mathbf{5}:\text{ }\mathbf{1}.\mathbf{5}\text{ }=\text{ }\mathbf{x}:\text{ }\mathbf{3}$
(iv)
$\mathbf{x}:\text{ }\mathbf{50}\text{ }::\text{ }\mathbf{3}:\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii)\[~2.5:\text{ }1.5\text{ }=\text{ }x:\text{ }3\] We can write it as \[1.5\text{ }\times \text{ }x\text{ }=\text{ }2.5\text{ }\times \text{ }3\] So we get \[\begin{array}{*{35}{l}} x\text{...

Find the value of x in the following proportions: (i)
$\mathbf{10}:\text{ }\mathbf{35}\text{ }=\text{ }\mathbf{x}:\text{ }\mathbf{42}$
(ii)
$\mathbf{3}:\text{ }\mathbf{x}\text{ }=\text{ }\mathbf{24}:\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i)\[\mathbf{10}:\text{ }\mathbf{35}\text{ }=\text{ }\mathbf{x}:\text{ }\mathbf{42}\] We can write it as \[35\text{ }\times \text{ }x\text{ }=\text{ }10\text{ }\times \text{ }42\] So we get...

In an examination, the ratio of passes to failures was
$4:1$
. If
$30$
less had appeared and
$20$
less passed, the ratio of passes to failures would have been
$5:1$
. How many students appeared for the examination.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

Consider number of passes = \[4x\] Number of failures = x Total number of students appeared = \[4x\text{ }+\text{ }x\text{ }=\text{ }5x\] In case \[2\] Number of students appeared = \[5x\text{...

(i) The monthly pocket money of Ravi and Sanjeev are in the ratio
$5:7$
. Their expenditures are in the ratio
$3:5$
. If each saves Rs
$80$
per month, find their monthly pocket money. (ii) In class X of a school, the ratio of the number of boys to that of the girls is
$4:3$
. If there were
$20$
more boys and
$12$
less girls, then the ratio would have been
$2:1$
. How many students were there in the class?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Consider the monthly pocket money of Ravi and Sanjeev as \[5x\] and \[7x\] Their expenditure is \[3y\] and \[5y\] respectively. \[5x\text{ }\text{ }3y\text{ }=\text{ }80\] …… (1) \[7x\text{...

(i) In a mixture of
$45$
litres, the ratio of milk to water is
$13:2$
. How much water must be added to this mixture to make the ratio of milk to water as
$3:1$
? (ii) The ratio of the number of boys to the numbers of girls in a school of
$560$
pupils is
$5:3$
. If
$10$
new boys are admitted, find how many new girls may be admitted so that the ratio of the number of boys to the number of girls may change to
$3:2$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that Mixture of milk to water = \[45\] litres Ratio of milk to water = \[13:2\] Sum of ratio = \[13\text{ }+\text{ }2\text{ }=\text{ }15\] Here the quantity of milk = \[\left(...

(i) A certain sum was divided among A, B and C in the ratio
$7:5:4$
. If B got Rs
$500$
more than C, find the total sum divided. (ii) In a business, A invests Rs
$50000$
for
$6$
months, B Rs
$60000$
for
$4$
months and C Rs
$80000$
for
$5$
months. If they together earn Rs
$18800$
find the share of each.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that Ratio between A, B and C = \[7:\text{ }5:\text{ }4\] Consider A share = \[7x\] B share = \[5x\] C share = \[4x\] So the total sum =\[~7x\text{ }+\text{ }5x\text{ }+\text{...

Three numbers are in the ratio
$\mathbf{1}/\mathbf{2}:\text{ }\mathbf{1}/\mathbf{3}:\text{ }{\scriptscriptstyle 1\!/\!{ }_4}$
. If the sum of their squares is
$244$
, find the numbers.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that Ratio of three numbers \[=\text{ }1/2:\text{ }1/3:\text{ }1/4\] \[\begin{array}{*{35}{l}} =\text{ }\left( 6:\text{ }4:\text{ }3 \right)/\text{ }12 \\ =\text{ }6:\text{ }4:\text{...

(i) The sides of a triangle are in the ratio
$7:5:3$
and its perimeter is
$30$
cm. Find the lengths of sides. (ii) If the angles of a triangle are in the ratio
$2:3:4$
, find the angles.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that Perimeter of triangle = \[30\] cm Ratio among sides = \[7:5:3\] Here the sum of ratios = \[7\text{ }+\text{ }5\text{ }+\text{ }3\text{ }=\text{ }15\] We know that Length of...

(i) A woman reduces her weight in the ratio
$7:5$
. What does her weight become if originally it was
$91$
kg. (ii) A school collected Rs 2100 for charity. It was decided to divide the money between an orphanage and a blind school in the ratio of 3: 4. How much money did each receive?

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Ratio of original and reduced weight of woman = \[7:5\] Consider original weight = \[7x\] Reduced weight = \[5x\] Here original weight = \[91\] kg So the reduced weight = \[~\left( 91\text{...

(i) Find two numbers in the ratio of
$\mathbf{8}:\text{ }\mathbf{7}$
such that when each is decreased by
$\mathbf{12}\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$
, they are in the ratio
$\mathbf{11}:\text{ }\mathbf{9}$
. (ii) The income of a man is increased in the ratio of
$\mathbf{10}:\text{ }\mathbf{11}$
. If the increase in his income is Rs
$\mathbf{600}$
per month, find his new income.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) Ratio = \[\mathbf{8}:\text{ }\mathbf{7}\] Consider the numbers as \[8x\] and \[7x\] Using the condition \[\left[ 8x\text{ }\text{ }25/2 \right]/\text{ }\left[ 7x\text{ }\text{ }25/2...

(iii) If
$\left( \mathbf{x}\text{ }+\text{ }\mathbf{2y} \right):\text{ }\left( \mathbf{2x}\text{ }\text{ }\mathbf{y} \right)$
is equal to the duplicate ratio of
$\mathbf{3}:\text{ }\mathbf{2}$
, find x: y.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[\left( x\text{ }+\text{ }2y \right)/\text{ }\left( 2x\text{ }\text{ }y \right)\text{ }=\text{ }{{3}^{2}}/\text{ }{{2}^{2}}\] So we get \[\left( x\text{ }+\text{ }2y \right)/\text{ }\left(...

(i) If
$\left( \mathbf{x}\text{ }\text{ }\mathbf{9} \right):\text{ }\left( \mathbf{3x}\text{ }+\text{ }\mathbf{6} \right)$
is the duplicate ratio of
$\mathbf{4}:\text{ }\mathbf{9}$
, find the value of x. (ii) If
$\left( \mathbf{3x}\text{ }+\text{ }\mathbf{1} \right):\text{ }\left( \mathbf{5x}\text{ }+\text{ }\mathbf{3} \right)$
is the triplicate ratio of
$\mathbf{3}:\text{ }\mathbf{4}$
, find the value of x.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\left( x\text{ }\text{ }9 \right)/\text{ }\left( 3x\text{ }+\text{ }6 \right)\text{ }=\text{ }{{\left( 4/9 \right)}^{2}}\] So we get \[\left( x\text{ }\text{ }9 \right)/\text{ }\left( 3x\text{...

(i) If
$(\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{xy}):\text{ }(\mathbf{3xy}\text{ }\text{ }{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{12}:\text{ }\mathbf{5}$
, find
$\left( \mathbf{x}\text{ }+\text{ }\mathbf{2y} \right):\text{ }\left( \mathbf{2x}\text{ }+\text{ }\mathbf{y} \right)$
(ii) If
$\mathbf{y}\text{ }\left( \mathbf{3x}\text{ }\text{ }\mathbf{y} \right):\text{ }\mathbf{x}\text{ }\left( \mathbf{4x}\text{ }+\text{ }\mathbf{y} \right)\text{ }=\text{ }\mathbf{5}:\text{ }\mathbf{12}$
. Find
$({{\mathbf{x}}^{\mathbf{2}}}~+\text{ }{{\mathbf{y}}^{\mathbf{2}}}):\text{ }{{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y} \right)}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[(\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{xy}):\text{ }(\mathbf{3xy}\text{ }\text{ }{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{12}:\text{ }\mathbf{5}\] We can write it...

(i) If
$\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}$
,find x: y. (ii) If
$\mathbf{a}:\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{3}:\text{ }\mathbf{11}$
, find
$\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)$
.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}\] By cross multiplication \[9x\text{ }+\text{ }15y\text{ }=\text{...

(i) If
$\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{1}/\mathbf{4}:\text{ }\mathbf{1}/\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{1}/\mathbf{7}:\text{ }\mathbf{1}/\mathbf{6}$
, find A: B: C. (ii) If
$\mathbf{3A}\text{ }=\text{ }\mathbf{4B}\text{ }=\text{ }\mathbf{6C}$
, find A: B: C

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) We know that \[\begin{array}{*{35}{l}} A:\text{ }B\text{ }=\text{ }1/4\text{ }\times \text{ }5/1\text{ }=\text{ }5/4 \\ B:\text{ }C\text{ }=\text{ }1/7\text{ }\times \text{ }6/1\text{ }=\text{...

(i) If
$\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{4}:\text{ }\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{C}:\text{ }\mathbf{D}\text{ }=\text{ }\mathbf{6}:\text{ }\mathbf{7}$
, find A: D. (ii) If
$\mathbf{x}:\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{2}:\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{y}:\text{ }\mathbf{z}\text{ }=\text{ }\mathbf{4}:\text{ }\mathbf{7}$
, find x: y: z.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) It is given that \[\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{4}:\text{ }\mathbf{5}\text{...

Arrange the following ratios in ascending order of magnitude:
$\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{17}:\text{ }\mathbf{21},\text{ }\mathbf{11}:\text{ }\mathbf{14}\text{ }\mathbf{and}\text{ }\mathbf{5}:\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that \[\mathbf{2}:\text{ }\mathbf{3},\text{ }\mathbf{17}:\text{ }\mathbf{21},\text{ }\mathbf{11}:\text{ }\mathbf{14}\text{ }\mathbf{and}\text{ }\mathbf{5}:\text{ }\mathbf{7}\] We can...

Find the reciprocal ratio of (iii)
$\mathbf{1}/\mathbf{9}:\text{ }\mathbf{2}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[\mathbf{1}/\mathbf{9}:\text{ }\mathbf{2}\] We know that Reciprocal ratio of \[1/9:\text{ }2\text{ }=\text{ }2:\text{ }1/9\text{ }=\text{ }18:\text{ }1\]

Find the reciprocal ratio of (i)
$\mathbf{4}:\text{ }\mathbf{7}$
(ii)
${{\mathbf{3}}^{\mathbf{2}}}:\text{ }{{\mathbf{4}}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{4}:\text{ }\mathbf{7}\] We know that Reciprocal ratio of \[4:\text{ }7\text{ }=\text{ }7:\text{ }4\] (ii) \[{{\mathbf{3}}^{\mathbf{2}}}:\text{ }{{\mathbf{4}}^{\mathbf{2}}}\] We know...

Find the sub-triplicate ratio of (iii)
$\mathbf{27}{{\mathbf{a}}^{\mathbf{3}}}:\text{ }\mathbf{64}{{\mathbf{b}}^{\mathbf{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[27{{a}^{3}}:\text{ }64{{b}^{3}}\] We know that Sub-triplicate ratio of \[27{{a}^{3}}:\text{ }64{{b}^{3}}~=\text{ }{{[{{\left( 3a \right)}^{3}}]}^{1/3}}:\text{ }{{[{{\left( 4b...

Find the sub-triplicate ratio of (i)
$\mathbf{1}:\text{ }\mathbf{216}$
(ii)
$\mathbf{1}/\mathbf{8}:\text{ }\mathbf{1}/\mathbf{125}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{1}:\text{ }\mathbf{216}\] We know that Sub-triplicate ratio of \[1:\text{ }216\text{ }=\sqrt[3]{1}:\sqrt[3]{216}\] By further calculation \[\begin{array}{*{35}{l}} =\text{...

Find the sub-duplicate ratio of (iii)
$\mathbf{9}{{\mathbf{a}}^{\mathbf{2}}}:\text{ }\mathbf{49}{{\mathbf{b}}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[9{{a}^{2}}:\text{ }49{{b}^{2}}\] We know that Sub-duplicate ratio of \[9{{a}^{2}}:\text{ }49{{b}^{2}}~=\text{ }\surd 9{{a}^{2}}:\text{ }\surd 49{{b}^{2}}~=\text{ }3a:\text{ }7b\]

Find the sub-duplicate ratio of (i)
$\mathbf{9}:\text{ }\mathbf{16}$
(ii)
${\scriptscriptstyle 1\!/\!{ }_4}:\text{ }\mathbf{1}/\mathbf{9}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{9}:\text{ }\mathbf{16}\] We know that Sub-duplicate ratio of \[9:\text{ }16\text{ }=\text{ }\surd 9:\text{ }\surd 16\text{ }=\text{ }3:\text{ }4\] (ii) \[{\scriptscriptstyle 1\!/\!{...

Find the triplicate ratio of (iii)
${{\mathbf{1}}^{\mathbf{3}}}:\text{ }{{\mathbf{2}}^{\mathbf{3}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii)\[~{{1}^{3}}:\text{ }{{2}^{3}}\] We know that Triplicate ratio of \[{{1}^{3}}:\text{ }{{2}^{3}}~=\text{ }{{({{1}^{3}})}^{3}}:\text{ }{{({{2}^{3}})}^{3}}~=\text{ }{{1}^{3}}:\text{...

Find the triplicate ratio of (i)
$\mathbf{3}:\text{ }\mathbf{4}$
(ii)
${\scriptscriptstyle 1\!/\!{ }_2}:\text{ }\mathbf{1}/\mathbf{3}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{3}:\text{ }\mathbf{4}\] We know that Triplicate ratio of \[3:\text{ }4\text{ }=\text{ }{{3}^{3}}:\text{ }{{4}^{3}}~=\text{ }27:\text{ }64\] (ii) \[{\scriptscriptstyle 1\!/\!{...

Find the duplicate ratio of (iii)
$\mathbf{5a}:\text{ }\mathbf{6b}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

iii) \[5a:\text{ }6b\] We know that Duplicate ratio of \[5a:\text{ }6b\text{ }=\text{ }{{\left( 5a \right)}^{2}}:\text{ }{{\left( 6b \right)}^{2}}~=\text{ }25{{a}^{2}}:\text{ }36{{b}^{2}}\]

Find the duplicate ratio of (i)
$\mathbf{2}:\text{ }\mathbf{3}$
(ii)
$\surd \mathbf{5}:\text{ }\mathbf{7}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{2}:\text{ }\mathbf{3}\] We know that Duplicate ratio of \[2:\text{ }3\text{ }=\text{ }{{2}^{2}}:\text{ }{{3}^{2}}~=\text{ }4:\text{ }9\] (ii) \[\surd \mathbf{5}:\text{ }\mathbf{7}\] We...

Find the compounded ratio of: (iii)
$\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right):\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right),\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{2}}}:\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{a}}^{\mathbf{4}}}~\text{ }{{\mathbf{b}}^{\mathbf{4}}}):\text{ }{{({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}})}^{\mathbf{2}}}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(iii) \[\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right):\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right),\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}...

Find the compounded ratio of: (i)
$\mathbf{2}:\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{4}:\text{ }\mathbf{9}$
(ii)
$\mathbf{4}:\text{ }\mathbf{5},\text{ }\mathbf{5}:\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{9}:\text{ }\mathbf{11}$

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

(i) \[\mathbf{2}:\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{4}:\text{ }\mathbf{9}\] We know that Compound ratio \[\begin{array}{*{35}{l}} ~=\text{ }2/3\text{ }\times \text{ }4/9 \\...

An alloy consists of
$\mathbf{27}\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$
kg of copper and
$\mathbf{2}\text{ }{\scriptscriptstyle 3\!/\!{ }_4}$
kg of tin. Find the ratio by weight of tin to the alloy.

Class 10, Exercise 1, ICSE, Maths, ML Aggarwal, Ratio and Proportion

It is given that Copper = \[\mathbf{27}\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\] kg = \[55/2\]kg Tin = \[\mathbf{2}\text{ }{\scriptscriptstyle 3\!/\!{ }_4}\] kg = \[11/4\] kg We know that Total...