#### Chapters

Chapter 2: Banking

Chapter 3: Shares and Dividends

Chapter 4: Linear Inequations

Chapter 5: Quadratic Equations in One Variable

Chapter 6: Factorization

Chapter 7: Ratio and Proportion

Chapter 8: Matrices

Chapter 9: Arithmetic and Geometric Progressions

Chapter 10: Reflection

Chapter 11: Section Formula

Chapter 12: Equation of a Straight Line

Chapter 13: Similarity

Chapter 14: Locus

Chapter 15: Circles

Chapter 16: Constructions

Chapter 17: Mensuration

Chapter 18: Trigonometric Identities

Chapter 19: Trigonometric Tables

Chapter 20: Heights and Distances

Chapter 21: Measures of Central Tendency

Chapter 22: Probability

## Chapter 7: Ratio and Proportion

### ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved Chapter 7 Ratio and Proportion Exercise 7.1

An alloy consists of `27(1)/(2)` kg of copper and `2(3)/(4)` kg of tin. Find the ratio by weight of tin to the alloy

Find the compounded ratio of: 2 : 3 and 4 : 9

Find the compounded ratio of: 4 : 5, 5 : 7 and 9 : 11

Find the compounded ratio of: (a – b) : (a + b), (a + b)^{2} : (a^{2} + b^{2}) and (a^{4} – b^{4}) : (a^{2} – b^{2})^{2}

Find the duplicate ratio of 2 : 3

Find the duplicate ratio of `sqrt(5) : 7`

Find the duplicate ratio of 5a : 6b

Find the triplicate ratio of 3 : 4

Find the triplicate ratio of `(1)/(2) : (1)/(3)`

Find the triplicate ratio of 1^{3} : 2^{3}

Find the sub-duplicate ratio of 9 : 16

Find the sub-duplicate ratio of `(1)/(4) : (1)/(9)`

Find the sub-duplicate ratio of 9a^{2} : 49b^{2}

Find the sub-triplicate ratio of 1 : 216

Find the sub-triplicate ratio of `(1)/(8) : (1)/(125)`

Find the sub-triplicate ratio of 27a^{3} : 64b^{3}

Find the reciprocal ratio of 4 : 7

Find the reciprocal ratio of 3^{2} : 4^{2}

Find the reciprocal ratio of `(1)/(9) : 2`

Arrange the following ratios in ascending order of magnitude: 2 : 3, 17 : 21, 11 : 14 and 5 : 7

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

If x : y = 2 : 3, and y : z = 4 : 7, find x : y : z

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

If 3A = 4B = 6C, find A : B : C

If `(3x + 5y)/(3x - 5y) = (7)/(3)` , Find x : y

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

If (4x² + xy) : (3xy – y²) = 12 : 5, find (x + 2y) : (2x + y).

If y (3x – y) : x (4x + y) = 5 : 12. Find (x² + y²) : (x + y)².

If (x – 9) : (3x + 6) is the duplicate ratio of 4 : 9, find the value of x.

If (3x + 1) : (5x + 3) is the triplicate ratio of 3 : 4, find the value of x.

If (x + 2y) : (2x – y) is equal to the duplicate ratio of 3 : 2, find x : y.

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

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.

A woman reduces her weight in the ratio 7 : 5. What does her weight become if originally it was 91 kg.

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?

The sides of a triangle are in the ratio 7 : 5 : 3 and its perimeter is 30 cm. Find the lengths of sides.

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) : (1)/(4)` If the sum of their squares is 244, find the numbers.

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.

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.

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 ?

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.

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 every month, find their monthly pocket money.

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?

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

### ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved Chapter 7 Ratio and Proportion Exercise 7.2

Find the value of x in the following proportions : 10 : 35 = x : 42

Find the value of x in the following proportions : 3 : x = 24 : 2

Find the value of x in the following proportions : 2.5 : 1.5 = x : 3

Find the value of x in the following proportions : x : 50 :: 3 : 2

Find the fourth proportional to 3, 12, 15

Find the fourth proportional to `(1)/(3), (1)/(4), (1)/(5)`

Find the fourth proportional to 1.5, 2.5, 4.5

Find the fourth proportional to 9.6 kg, 7.2 kg, 28.8 kg

Find the third proportional to 5, 10

Find the third proportional to 0.24, 0.6

Find the third proportional to Rs. 3, Rs. 12

Find the third proportional to `5(1)/(4) and 7.`

Find the mean proportion of: 5 and 80

Find the mean proportion of: `(1)/(12) and (1)/(75)`

Find the mean proportion of: 8.1 and 2.5

Find the mean proportion of: (a – b) and (a³ – a²b), 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, a² + b², and b² + c² are proportional.

If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between (a² + b²) and (b² + c²).

If y is mean proportional between x and z, prove that

xyz (x + y + z)³ = (xy + yz + zx)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 `x^3/a^2 + y^3/b^2 + z^3/c^2 = (x+ y+ z)^3/(a + b+ c)^2`

If `x/a = y/b = z/c`, prove that `[(a^2x^2 + b^2y^2 + c^2z^2)/(a^2x + b^3y +c^3z)]^3 = "xyz"/"abc"`

If `x/a = y/b = z/c`, prove that `"ax - by"/((a + b)(x- y)) + "by - cz"/((b + c)(y - z)) + "cz - ax"/((c + a)(z - x)` = 3

If `a/c = c/d = c/f` prove that : (b^{2} + d^{2} + f^{2}) (a^{2} + c^{2} + e^{2}) = (ab + cd + ef)^{2 }

If `a/c = c/d = c/f` prove that : `(a^3 + c^3)^2/(b^3 + d^3)^2 = e^6/f^6`

If `a/c = c/d = c/f` prove that : `(a^2)/(b^2) + (c^2)/(d^2) + (e^2)/(f^2) = "ac"/"bd" + "ce"/"df" + "ae"/"df"`

If `a/c = c/d = c/f` prove that : `bd f[(a + b)/b + (c + d)/d + (c + f)/f]^3` = 27(a + b)(c + d)(e + f)

If ax = by = cz; prove that `x^2/"yz" + y^2/"zx" + z^2/"xy" = "bc"/a^2 + "ca"/b^2 + "ab"/c^2`

If a, b, c and d are in proportion, prove that: (5a + 7b) (2c – 3d) = (5c + 7d) (2a – 3b)

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

If a, b, c and d are in proportion, prove that: (a^{4} + c^{4}) : (b^{4} + d^{4}) = a^{2} c^{2} : b^{2} d^{2}.

If a, b, c and d are in proportion, prove that: `(a^2 + ab)/(c^2 + cd) = (b^2 - 2ab)/(d^2 - 2cd)`

If a, b, c and d are in proportion, prove that: `(a + c)^3/(b + d)^3 = (a(a - c)^2)/(b(b - d)^2)`

If a, b, c and d are in proportion, prove that: `(a^2 + ab + b^2)/(a^2 - ab + b^2) = (c^2 + cd + d^2)/(c^2 - cd + d^2)`

If a, b, c and d are in proportion, prove that: `(a^2 + b^2)/(c^2 + d^2) = "ab + ad - bc"/"bc + cd - ad"`

If a, b, c and d are in proportion, prove that: `abcd [(1/a^2 + 1/b^2 + 1/c^2 + 1/d^2]` = a^{2} + b^{2} + c^{2} + d^{2 }

If x, y, z are in continued proportion, prove that: `(x + y)^2/(y + z)^2 = x/z`.

If a, b, c are in continued proportion, prove that: `(pa^2+ qab+ rb^2)/(pb^2+qbc+rc^2) = a/c`

If a, b, c are in continued proportion, prove that: `"a + b"/"b + c" = (a^2(b - c))/(b^2(a - b)`

If a, b, c are in continued proportion, prove that: `(1)/a^3 + (1)/b^3 + (1)/c^3 = a/(b^2c^2) + b/(c^2a^2) + c/(a^2b^2)`

If a, b, c are in continued proportion, prove that: a : c = (a^{2} + b^{2}) : (b^{2} + c^{2})

If a, b, c are in continued proportion, prove that: a^{2} b^{2} c^{2} (a^{-4} + b^{-4} + c^{-4})

If a, b, c are in continued proportion, prove that: abc(a + b + c)^{3} = (ab + bc + ca)^{3}

If a, b, c are in continued proportion, prove that: (a + b + c) (a – b + c) = a^{2} + b^{2} + c^{2}

If a, b, c, d are in continued proportion, prove that: `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3) = a/d`

If a, b, c, d are in continued proportion, prove that: (a^{2} – b^{2}) (c^{2} – d^{2}) = (b^{2} – c^{2})^{2}

If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)^{2}

If a, b, c, d are in continued proportion, prove that: a : d = triplicate ratio of (a – b) : (b – c)

If a, b, c, d are in continued proportion, prove that: `((a -b)/c + (a - c)/b)^2 - ((d - b)/c + (d - c)/b)^2 = (a - d)^2 (1/c^2 - 1/b^2)`

### ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved Chapter 7 Ratio and Proportion Exercise 7.3

If a : b : : c : d, prove that `(2a +5b)/(2a - 5b) = (2c + 5d)/(2c - 5d)`

If a : b : : c : d, prove that `(5a + 11b)/(5c + 11d) = (5a - 11b)/(5c - 11d)`

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

If a : b : : c : d, prove that (la + mb) : (lc + mb) :: (la – mb) : (lc – mb)

If `(5x + 7y)/(5u + 7v) = (5x - 7y)/(5u - 7v)`, show that `x/y = u/v`

If `(8a - 5b)/(8c - 5a) = (8a + 5b)/(8c + 5d)`, prove that `a/b = c/d`

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

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

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

If (11a² + 13b²) (11c² – 13d²) = (11a² – 13b²)(11c² + 13d²), prove that a : b :: c : d.

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

If x = `(2a + b)/(a + b)` find the value of `(x + a)/(x - a) + (x + b)/(x - b)`

If x = `(8ab)/"a + b"` find the value of `(x + 4a)/(x - 4a) + (x + 4b)/(x - 4b)`

If x = `(4sqrt(6))/(sqrt(2) + sqrt(3)` find the value of `(x + 2sqrt(2))/(x - 2sqrt(2)) + (x + 2sqrt(3))/(x - 2sqrt(3)`

Solve x : `(sqrt(36x + 1) + 6sqrt(x))/(sqrt(36x + 1) -6sqrt(x))` = 9

Find x from the following equations : `(sqrt(2 - x) + sqrt(2 + x))/(sqrt(2 - x) - sqrt(2 + x)` = 3

Find x from the following equations : `(sqrt(x + 4) + sqrt(x - 10))/(sqrt(x + 4) - sqrt(x - 10)) = (5)/(2)`

Find x from the following equations : `(sqrt(1 + x) + sqrt(1 - x))/(sqrt(1 + x) - sqrt(1 - x)) = a/b`

Find x from the following equations : `(sqrt(12x + 1) + sqrt(2x - 3))/(sqrt(12x + 1) - sqrt(2x - 3)) = (3)/(2)`

Find x from the following equations : `(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5)) = (5)/(1)`

Find x from the following equations : `(sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) = c/d`

Solve `(1 + x + x^2)/(1 - x + x^2) = (62(1 +x))/(63(1 + x)`

Solve for `x : 16((a - x)/(a + x))^3 = (a + x)/(a - x)`

If x = `(sqrt(a + x) + sqrt(a - 1))/(sqrt(a + 1 - sqrt(a - 1)`, using properties of proportion , show that x^{2} – 2ax + 1 = 0

Given x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(^2 + b^2) - sqrt(a^2 - b^2)` Use componendo and dividendo to prove that b^{2} = `(2a^2x)/(x^2 + 1)`

Given that `(a^3 + 3ab^2)/(b^3 + 3a^2b) = (63)/(62)`. Using componendo and dividendo find a: b.

Give `(x^3 + 12x)/(6x^2 + 8) = (y^3 + 27y)/(9y^2 + 27)` Using componendo and dividendo find x : y.

Using the properties of proportion, solve the following equation for x; given `(x^3 + 3x)/(3x^2 + 1) = (341)/(91)`

If `(x + y)/(ax + by) = (y + z)/(ay + bz) = (z + x)/(az + bx)`, prove that each of these ratio is equal to `(2)/(a + b)` unless x + y + z = 0

### ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved Chapter 7 Ratio and Proportion Multiple Choice Question

**Choose the correct answer from the given options :**

The ratio of 45 minutes to `5(3)/(4)` hours is

180:23

3:23

23:3

6:23

**Choose the correct answer from the given options :**

The ratio of 4 litres to 900 mL is

4 : 9

40 : 9

9 : 40

20 : 9

**Choose the correct answer from the given options :**

When the number 210 is increased in the ratio 5 : 7, the new number is

150

180

294

420

**Choose the correct answer from the given options :**

Two numbers are in the ratio 7 : 9. If the sum of the numbers is 288, then the smaller number is

126

162

112

144

**Choose the correct answer from the given options :**

A ratio equivalent to the ratio `(2)/(3) : (5)/(7)` is

4:6

5:7

15:14

14:15

**Choose the correct answer from the given options :**

The ratio of number of edges of a cube to the number of its faces is

2 : 1

1 : 2

3 : 8

8 : 3

**Choose the correct answer from the given options :**

If x, 12, 8 and 32 are in proportion, then the value of x is

6

4

3

2

**Choose the correct answer from the given options :**

The fourth proportional to 3, 4, 5 is

6

`(20)/(3)`

`(15)/(4)`

`(12)/(5)`

**Choose the correct answer from the given options :**

The third proportional to `6(1)/(4)` and 5 is

4

`8(1)/(2)`

3

none of these

**Choose the correct answer from the given options :**

The mean proportional between `(1)/(2)` and 128 is

64

32

16

8

### ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved Chapter 7 Ratio and Proportion Chapter Test

Find the compound ratio of:

(a + b)^{2} : (a – b )^{2} ,

(a^{2} – b^{2}) : (a^{2} + b^{2}),

(a^{4} – b^{4}) : (a + b)^{4}

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

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

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?

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.

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

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.

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}.

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: p^{3} – 3q^{2} + r^{2} = `q^4(1/p^2 - 3/q^2 + 1/r^2)`

If `a/b = c/d = c/f`, prove that each ratio is `sqrt((3a^2 - 5c^2 + 7e^2)/(3b^2 - 5d^2 + 7f^2)`

If `a/b = c/d = e/f`, prove that each ratio is `[(2a^3 + 5c^3 + 7e^3)/(2b^3 + 5d^3 + 7f^3)]^(1/3)`

If `x/a = y/b = z/c`, prove that `(3x^3 - 5y^3 + 4c^3)/(3a^3 - 5b^3 + 4c^3) = ((3x - 5y + 4c)/(3a - 5b + ac))^3`

If x : a = y : b, prove that `(x^4 + a^4)/(x^3 + a^3) + (y^4 + b^4)/(y^3 + b^3) = ((x + y)^4 + (a + b)^4)/((x+ y)^3 + (a + b)^3`

If `x/(b + c - a) = y/(c + a - b) = z/(a + b - c)` prove that each ratio’s equal to : `(x + y + z)/(a + b + c)`

If a : b = 9 : 10, find the value of `(5a + 3b)/(5a - 3b)`

If a : b = 9 : 10, find the value of `(2a^2 - 3b^2)/(2a^2 + 3b^2)`

If (3x² + 2y²) : (3x² – 2y²) = 11 : 9, find the value of `(3x^4 + 5y^4)/(3x^4 - 5y^4)`

If x = `(2mab)/(a + b)`, find the value of `(x + ma)/(x - ma) + (x + mb)/(x - mb)`

If x = `"pab"/(a + b)`, provee that `(x + pa)/(x - pa) - (x + pb)/(x - pb) = (2(a^2 - b^2))/(ab)`

Find x from the equation `(a+ x + sqrt(a^2 x^2))/(a + x - sqrt(a^2 - x^2)) = b/x`

If x = `(root(3)(a + 1) + root(3)(a - 1))/(root(3)(a + 1) - root(3)(a - 1)`,prove that :

x³ – 3ax² + 3x – a = 0

If `(by + cz)/(b^2 + c^2) = (cz + ax)/(c^2 + a^2) = (ax + by)/(a^2 + b^2)`, prove that each of these ratio is equal to `x/a = y/b = z/c`

## Chapter 7: Ratio and Proportion

## ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved chapter 7 - Ratio and Proportion

ML Aggarwal solutions for Understanding ICSE Mathematics Class 10 Solved chapter 7 (Ratio and Proportion) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Understanding ICSE Mathematics Class 10 Solved solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Understanding ICSE Mathematics Class 10 Solved chapter 7 Ratio and Proportion are Concept of Ratio, Concept of Proportion, Componendo and Dividendo Properties, Alternendo and Invertendo Properties, Direct Applications, Concept of Ratio.

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