M l aggarwal understanding icse mathematics class 10 Solutions for Chapter 6 ratio and proportion

An alloy consists of 27 ½ kg of copper and 2 ¾ kg of tin. Find the ratio by weight of tin to the alloy.

Find the compounded ratio of:

(i) 2: 3 and 4: 9

(ii) 4: 5, 5: 7 and 9: 11

(iii) (a – b): (a + b), (a+b)^{2}:\left(a^{2}+b^{2}\right) \text { and }\left(a^{4}-b^{4}\right):\left(a^{2}-b^{2}\right)^{2}

Find the duplicate ratio of

(i) 2: 3

(ii) √5: 7

(iii) 5a: 6b

Find the triplicate ratio of

(i) 3: 4

(ii) ½: 1/3

(iii) 1^{3}: 2^{3}

Find the sub-duplicate ratio of

(i) 9: 16

(ii) ¼: 1/9

(iii) 9 a^{2}: 49 b^{2}

Find the sub-triplicate ratio of

(i) 1: 216

(ii) 1/8: 1/125

(iii) 27 a^{3}: 64 b^{3}

Arrange the following ratios in ascending order of magnitude:

2: 3, 17: 21, 11: 14 and 5: 7

(i) If A: B = 2: 3, B: C = 4: 5 and C: D = 6: 7, find A: D.

(ii) If x: y = 2: 3 and y: z = 4: 7, find x: y: z.

(i) If A: B = 1/4: 1/5 and B: C = 1/7: 1/6, find A: B: C.

(ii) If 3A = 4B = 6C, find A: B: C

(i) If 3x + 5y/ 3x – 5y = 7/3, find x: y.

(ii) If a: b = 3: 11, find (15a – 3b): (9a + 5b).

(i) If \left(4 x^{2}+x y\right):\left(3 x y-y^{2}\right) = 12: 5, find (x + 2y): (2x + y).

(ii) If y (3x – y): x (4x + y) = 5: 12. Find \left(x^{2}+y^{2}\right):(x+y)^{2}.

(i) Find two numbers in the ratio of 8: 7 such that when each is decreased by 12 ½, they are in the ratio 11: 9.

(ii) The income of a man is increased in the ratio of 10: 11. If the increase in his income is Rs 600 per month, find his new income.

(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?

(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.

Three numbers are in the ratio 1/2: 1/3: ¼. If the sum of their squares is 244, find the numbers.

(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.

(i) In a mixture of 45 liters, 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 number 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.

(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 fewer girls, then the ratio would have been 2: 1. How many students were there in the class?

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.

Find the value of x in the following proportions:

(i) 10: 35 = x: 42

(ii) 3: x = 24: 2

(iii) 2.5: 1.5 = x: 3

(iv) x: 50 :: 3: 2

Find the fourth proportional to

(i) 3, 12, 15

(ii) 1/3, 1/4, 1/5

(iii) 1.5, 2.5, 4.5

(iv) 9.6 kg, 7.2 kg, 28.8 kg

Find the third proportional to

(i) 5, 10

(ii) 0.24, 0.6

(iii) Rs. 3, Rs. 12

(iv) 5 ¼ and 7.

Find the mean proportion of:

(i) 5 and 80

(ii) 1/12 and 1/75

(iii) 8.1 and 2.5

(iv) (a – b) and \left(a^{3}-a^{2} b\right), a ˃ b

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

What number must be added to each of the numbers 5, 11, 19, and 37 so that they are in proportion?

What number should be subtracted from each of the numbers 23, 30, 57, and 78 so that the remainders are in proportion?

If 2x – 1, 5x – 6, 6x + 2 and 15x – 9 are in proportion, find the value of x.

If x + 5 is the mean proportion between x + 2 and x + 9, find the value of x.

What number must be added to each of the numbers 16, 26, and 40 so that the resulting numbers may be in continued proportion?

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

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

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

If y is mean proportional between x and z, prove that x y z(x+y+z)^{3}=(x y+y z+z x)^{3}.

If a + c = mb and 1/b + 1/d = m/c, prove that a, b, c and d are in proportion.

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{\mathrm{a}^{2} \mathrm{x}^{2}+\mathrm{b}^{2} \mathrm{y}^{2}+\mathrm{c}^{2} \mathrm{z}^{2}}{\mathrm{a}^{3} \mathrm{x}+\mathrm{b}^{3} \mathrm{y}+\mathrm{c}^{3} \mathrm{z}}\right]^{3}=\frac{\mathrm{xyz}}{\mathrm{abc}}

(iii) \frac{a x-b y}{(a+b)(x-y)}+\frac{b y-c z}{(b+c)(y-z)}+\frac{c z-a x}{(c+a)(z-x)}=3

If a/b = c/d = e/f prove that:

(i) \left(b^{2}+d^{2}+r^{2}\right)\left(a^{2}+c^{2}+e^{2}\right)=(a b+c d+e f)^{2}

(ii) \frac{\left(\mathbf{a}^{3}+\mathbf{c}^{3}\right)^{2}}{\left(\mathbf{b}^{3}+\mathbf{d}^{3}\right)^{2}}=\frac{\mathbf{e}^{6}}{\mathbf{f}^{6}}

(iii) \frac{\mathrm{a}^{2}}{\mathrm{b}^{2}}+\frac{\mathrm{c}^{2}}{\mathrm{d}^{2}}+\frac{\mathrm{e}^{2}}{\mathrm{f}^{2}}=\frac{\mathrm{ac}}{\mathrm{bd}}+\frac{\mathrm{ce}}{\mathrm{df}}+\frac{\mathrm{ae}}{\mathrm{df}}

(iv) \mathrm{bdf}\left[\frac{\mathrm{a}+\mathrm{b}}{\mathrm{b}}+\frac{\mathrm{c}+\mathrm{d}}{\mathrm{d}}+\frac{\mathrm{c}+\mathrm{f}}{\mathrm{f}}\right]^{3}=27(\mathrm{a}+\mathrm{b})(\mathrm{c}+\mathrm{d})(\mathrm{e}+\mathrm{f})

If ax = by = cz; prove that

\frac{x^{2}}{y z}+\frac{y^{2}}{z x}+\frac{z^{2}}{x y}=\frac{b c}{a^{2}}+\frac{c a}{b^{2}}+\frac{a b}{c^{2}}

If a, b, c and d are in proportion, prove that:

(i) (5a + 7v) (2c – 3d) = (5c + 7d) (2a – 3b)

(ii) (ma + nb): b = (mc + nd): d

(iii) \left(a^{4}+c^{4}\right):\left(b^{4}+d^{4}\right)=a^{2} c^{2}: b^{2} d^{2}

(iv) \frac{a^{2}+a b}{c^{2}+c d}=\frac{b^{2}-2 a b}{d^{2}-2 c d}

(v) \frac{(\mathbf{a}+\mathbf{c})^{3}}{(\mathbf{b}+\mathbf{d})^{3}}=\frac{\mathbf{a}(\mathbf{a}-\mathbf{c})^{2}}{\mathbf{b}(\mathbf{b}-\mathbf{d})^{2}}

(vi) \frac{a^{2}+a b+b^{2}}{a^{2}-a b+b^{2}}=\frac{c^{2}+c d+d^{2}}{c^{2}-c d+d^{2}}

(vii) \frac{a^{2}+b^{2}}{c^{2}+d^{2}}=\frac{a b+a d-b c}{b c+c d-a d}

(viii) \operatorname{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}

If a, b, c are in continued proportion, prove that:

\frac{p a^{2}+q a b+r b^{2}}{p b^{2}+q b c+r c^{2}}=\frac{a}{c}

If a, b, c are in continued proportion, prove that:

If a, b, c, d are in continued proportion, prove that:

(\mathrm{i}) \frac{\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}}{\mathrm{b}^{3}+\mathrm{c}^{3}+\mathrm{d}^{3}}=\frac{\mathrm{a}}{\mathrm{d}}

(ii) \left(a^{2}-b^{2}\right)\left(c^{2}-d^{2}\right)=\left(b^{2}-c^{2}\right)^{2}

(iii) (a + d) (b + c) – (a + c) (b + d) =(b-c)^{2}

(iv) a: d = triplicate ratio of (a – b): (b – c)

(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)

Classify the following matrices:

\text { (i) }\left[\begin{array}{cc} 2 & -1 \\ 5 & 1 \end{array}\right]

\text { (ii) }\left[\begin{array}{lll} 2 & 3 & -7 \end{array}\right]

\text { (iii) }\left[\begin{array}{c} 3 \\ 0 \\ -1 \end{array}\right]

\text { (iv) }\left[\begin{array}{cc} 2 & -4 \\ 0 & 0 \\ 1 & 7 \end{array}\right]

(v)\left[\begin{array}{rrr} 2 & 7 & 8 \\ -1 & \sqrt{2} & 0 \end{array}\right]

\text { (vi) }\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]

If a: b:: c: d, prove that

(i) \frac{2 a+5 b}{2 a-5 b}=\frac{2 c+5 d}{2 c-5 d}

(ii) \frac{5 a+11 b}{5 c+11 d}=\frac{5 a-11 b}{5 c-11 d}

(iii) (2a + 3b) (2c – 3d) = (2a – 3b) (2c + 3d)

(iv) (la + mb): (lc + mb) :: (la – mb): (lc – mb)

Find the compound ratio of:

\begin{array}{l} (a+b)^{2}:(a-b)^{2} \\ \left(a^{2}-b^{2}\right):\left(a^{2}+b^{2}\right) \\ \left(a^{4}-b^{4}\right):(a+b)^{4} \end{array}

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

(i) \text { If } \frac{5 x+7 y}{5 u+7 v}=\frac{5 x-7 y}{5 u-7 v}, \text { show that } \frac{x}{y}=\frac{u}{v}

(ii) \frac{8 a-5 b}{8 c-5 d}=\frac{8 a+5 b}{8 c+5 d}, \text { prove that } \frac{a}{b}=\frac{c}{d}.

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

If (4a + 5b) (4c – 5d) = (4a – 5d) (4c + 5d), prove that a, b, c, d are in proportion.

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

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.

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?

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

If \left(11 a^{2}+13 b^{2}\right)\left(11 c^{2}-13 d^{2}\right)=\left(11 a^{2}-13 b^{2}\right)\left(11 c^{2}+13 d^{2}\right), prove that a: b :: c: d.

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.

If (a + 3b + 2c + 6d) (a – 3b – 2c + 6d) = (a + 3b – 2c – 6d) (a – 3b + 2c – 6d), prove that a: b:: c: d.

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

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

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

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

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

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

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

\text { Solve for } x: \frac{\sqrt{36 x+1}+6 \sqrt{x}}{\sqrt{36 x+1}-6 \sqrt{x}}=9

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

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

\mathrm{p}^{2}-3 \mathrm{q}^{2}+\mathrm{r}^{2}=\mathrm{q}^{4}\left(\frac{1}{\mathrm{p}^{2}}-\frac{3}{\mathrm{q}^{2}}+\frac{1}{\mathrm{r}^{2}}\right)

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}

(iii) \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}=\frac{a}{b}

(iv) \frac{\sqrt{12 x+1}+\sqrt{2 x-3}}{\sqrt{12 x+1}-\sqrt{2 x-3}}=\frac{3}{2}

(v) \frac{3 x+\sqrt{9 x^{2}+5}}{3 x-\sqrt{9 x^{2}+5}}=5

(vi) \frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}=\frac{c}{d}

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^{3}+7 e^{3}}{2 b^{3}+5 d^{3}+7 f^{3}}\right]^{\frac{1}{3}}

Using properties of proportion solve for x. Give that x is positive.

\frac{3 x+\sqrt{9 x^{2}-5}}{3 x-\sqrt{9 x^{2}-5}}=5

Solve

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

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{3 x-5 y+4 z}{3 a-5 b+4 c}\right)^{3}

Solve for x:

16\left(\frac{a-x}{a+x}\right)^{3}=\frac{a+x}{a-x}

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}}

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

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

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

(i) \frac{5 a+3 b}{5 a-3 b}

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

\text { Give } x=\frac{\sqrt{a^{2}+b^{2}}+\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}-\sqrt{a^{2}-b^{2}}} \text { use componendo and dividendo to prove that } b^{2}=\frac{2 a^{2} x}{x^{2}+1}

If \left(3 x^{2}+2 y^{2}\right):\left(3 x^{2}-2 y^{2}\right)=11:9, find the value of

\frac{3 x^{4}+25 y^{4}}{3 x^{4}-25 y^{4}}

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

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

\text { If } x=\frac{2 m a b}{a+b}, \text { find the value of } \frac{x+m a}{x-m a}+\frac{x+m b}{x-m b}.

Using the properties of proportion, solve the following equation for x; given

\frac{x^{3}+3 x}{3 x^{2}+1}=\frac{341}{91}

\text { If } x=\frac{\text { pab }}{a+b}, \text { prove that } \frac{x+p a}{x-p a}-\frac{x+p b}{x-p b}=\frac{2\left(a^{2}-b^{2}\right)}{a b}.

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

\text { If } \frac{x+y}{a x+b y}=\frac{y+z}{a y+b z}=\frac{z+x}{a z+b x},

\text { prove that each of these ratio is equal to } \frac{2}{\mathrm{a}+\mathrm{b}} \text { unless } \mathrm{x}+\mathrm{y}+\mathrm{z}=0.

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

\text { If } \frac{b y+c z}{b^{2}+c^{2}}=\frac{c z+a x}{c^{2}+a^{2}}=\frac{a x+b y}{a^{2}+b^{2}}, \text { prove that each of these ratio is equal to } \frac{x}{a}=\frac{y}{b}=\frac{z}{c}.