#### Chapters

Chapter 2: Banking (Recurring Deposit Account)

Chapter 3: Shares and Dividend

Chapter 4: Linear Inequations (In one variable)

Chapter 5: Quadratic Equations

Chapter 6: Solving (simple) Problems (Based on Quadratic Equations)

Chapter 7: Ratio and Proportion (Including Properties and Uses)

Chapter 8: Remainder and Factor Theorems

Chapter 9: Matrices

Chapter 10: Arithmetic Progression

Chapter 11: Geometric Progression

Chapter 12: Reflection

Chapter 13: Section and Mid-Point Formula

Chapter 14: Equation of a Line

Chapter 15: Similarity (With Applications to Maps and Models)

Chapter 16: Loci (Locus and Its Constructions)

Chapter 17: Circles

Chapter 18: Tangents and Intersecting Chords

Chapter 19: Constructions (Circles)

Chapter 20: Cylinder, Cone and Sphere

Chapter 21: Trigonometrical Identities

Chapter 22: Height and Distances

Chapter 23: Graphical Representation

Chapter 24: Measure of Central Tendency(Mean, Median, Quartiles and Mode)

Chapter 25: Probability

#### Selina Selina ICSE Concise Mathematics Class 10 (2018-2019)

## Chapter 7: Ratio and Proportion (Including Properties and Uses)

#### Chapter 7: Ratio and Proportion (Including Properties and Uses) Exercise 7A solutions [Page 0]

if a : b = 5:3 find `(5a - 3b)/(5xa + 3b)`

If x: y = 4: 7, find the value of (3x + 2y): (5x + y).

If a : b = 3 : 8. Find the value of `(4a + 3b)/(6a - b)`

If (a – b): (a + b) = 1: 11, find the ratio (5a + 4b + 15): (5a – 4b + 3).

If `(y- x)/x = 3/8` , find the value of `y /x`

If `(m + n)/(m + 3n) = 2/3` find `(2n^2)/(3m^2 + mn)`

Find x/y when `x^2 + 6y^2 =5xy`

If the ratio between 8 and 11 is the same as the ratio of 2x - y to x + 2y. Find the value of `(7x)/(9y)`

Two numbers are in the ratio 2 : 3. If 5 is added to each number, the ratio becomes 5 : 7. Find the numbers.

Two positive numbers are in the ratio 3 : 5 and the difference between their squares is 400. Find the numbers.

What quantity must be subtracted from each term of the ratio 9: 17 to make it equal to 1: 3?

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.

The work done by (x – 2) men in (4x + 1) days and the work done by (4x + 1) men in (2x – 3) days are in the ratio 3: 8. Find the value of x.

The bus fare between two cities is increased in the ratio 7: 9. Find the increase in the fare, if the original fare is Rs 245;

The bus fare between two cities is increased in the ratio 7: 9. Find the increase in the fare, if the increased fare is Rs 207.

By increasing the cost of entry ticket to a fair in the ratio 10: 13, the number of visitors to the fair has decreased in the ratio 6: 5. In what ratio has the total collection increased or decreased?

In a basket, the ratio between the number of oranges and the number of apples is 7: 13. If 8 oranges and 11 apples are eaten, the ratio between the number of oranges and the number of apples becomes 1: 2. Find the original number of oranges and the original number of apples in the basket.

The ratio between the number of boys and the number of girls in a class is 4:3 . If there were 20 more boys and 12 less girls, the ratio would have been 2:1. Find the total number of students in

the class.

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

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

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

If 3A = 4B = 6C; find A: B: C.

Find the compound ratio of :

3 : 5 and 8 : 15

Find the compound ratio of 2 : 3, 9: 14 and 14: 27

Find the compound ration of `2a: 3b, mn: x^2 and x: n`

Find the compound ratio of `sqrt2: 1, 3: sqrt5` and `sqrt20: 9`

Find the duplicate ratio of 3: 4

Find the duplicate ratio of `3sqrt3: 2sqrt5`

Find the triplicate ratio of 1 : 3

Find the triplicate ration of `m/2: n/3`

Find the sub-duplicate ratio of 9: 16

Find the sub-duplicate ratio of `(x - y)^4 : (x + y)^6`

Find the sub-triplicate ratio of 64: 27

Find the sub-triplicate ratio of `x^3: 125y^3`

Find the reciprocal ratio of:

5 : 8

Find the reciprocal ratio of `x/3 : y/7`

If 3x + 4 : x + 5 is the duplicate ratio of 8 : 15, find x.

If m: n is the duplicate ratio of m + x: n + x; show that x^{2} = mn.

If 4x + 4 : 9x – 10 is the triplicate ratio of 4 : 5, find x.

Find the ratio compounded of the reciprocal ratio of 15: 28, the sub-duplicate ratio of 36: 49 and the triplicate ratio of 5: 4.

If `(a + b)/(am + bn)= (b + c)/(mb + nc )= (c + a)/(mc + na)` prove that each of theseccc `2/(m+n)`provided a + b + c ≠ 0

#### Chapter 7: Ratio and Proportion (Including Properties and Uses) Exercise 7B solutions [Page 0]

Find the fourth proportional to 1.5, 4.5 and 3.5

Find the fourth proportional to 3a, 6a^{2} and 2ab^{2}

Find the third proportional to `2 2/3 and 4`

Find the third proportional to a - b and `a^2 - b^2`

Find the mean proportional between :

17.5 and 0.007

Find the mean proportional between `6 + 3sqrt3 and 8 - 4sqrt3`

Find the mean proportional between `a - b and a^3 - a^2b`

If x + 5 is the mean proportional between x + 2 and x + 9; find the value of x.

What least number must be added to each of the numbers 16, 7, 79 and 43 so that the resulting

numbers are in proportion?

What number must be added to each of the numbers 6, 15, 20 and 43 to make them

proportional?

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

What least number be subtracted from each of the numbers 7, 17 and 47 so that the remainders are in continued proportion?

If y is the mean proportional between x and z; show that xy + yz is the mean proportional between x^{2}+y^{2} and y^{2}+ z^{2}.

If q is the mean proportional between p and r, show that: pqr (p + q + r)^{3} = (pq + qr + rp)^{3}

If three quantities are in continued proportion; show that the ratio of the first to the third is the duplicate ratio of the first to the second

If y is the mean proportional between x and z. prove that `(x^2 - y^2 + z^2)/(x^(-2) - y^(-2) + z^(-2)) = y^4`

Given four quantities a, b, c and d are in proportion. Show that `(a - c)b^2 : (b - d)cd = (a^2 - b^2 - ab) : (c^2 - d^2 - cd)`

Find two numbers such that the mean proportional between them is 12 and the third proportional to them is 96.

Find the third proportional to `x/y + y/x` and `sqrt(x^2 + y^2)`

If p: q = r: s; then show that: mp + nq : q = mr + ns : s.

If p + r = mq and `1/q + 1/s = m/r` then prove that p : q = r : s

If `a/b = c/d` prove that each of the given ratios is equal to

`(5a + 4c)/(5b + 4d)`

If a/b = c/d prove that each of the given ratio is equal to `(13a - 8c)/(13b - 8d)`

If a/b = c/d prove that each of the given ratio is equal to `sqrt((3a^2 - 10c^2)/(3b^2 - 10d^2))`

if `a/b = c/d` prove that each of the given ratio is equal to: `((8a^3 + 15c^3)/(8b^3 + 15d^3))^(1/3)`

If a, b, c and d are in proportion prove that `(13a + 17b)/(13c + 17d) = sqrt((2ma^2 - 3nb^2)/(2mc^2 - 3nd^2)`

If a, b, c and d are in proportion prove that `sqrt((4a^2 + 9b^2)/(4c^2 + 9d^2)) = ((xa^3 - 4yb^3)/(xc^3 - 5yd^3))^(1/3)`

If `x/a = y/b = z/c` prove that `(2x^3 - 3y^3 + 4z^3)/(2a^3 - 3b^3 + 4c^3) = ((2x - 3y + 4z)/(2a - 3b + 4c))^3`

#### Chapter 7: Ratio and Proportion (Including Properties and Uses) Exercise 7C solutions [Page 0]

If a : b = c : d, prove that: 5a + 7b : 5a – 7b = 5c + 7d : 5c – 7d.

If a : b = c : d, prove that: (9a + 13b) (9c – 13d) = (9c + 13d) (9a – 13b).

If a : b = c : d, prove that: xa + yb : xc + yd = b : d.

If a : b = c : d, prove that: (6a + 7b) (3c – 4d) = (6c + 7d) (3a – 4b).

Given `a/b = c/d` prove that `(3a - 5b)/(3a + 5b) = (3c - 5d)/(3c + 5d)`

If `(5x + 6y)/(5u + 6v) = (5x - 6y)/(5u - 6v)` then prove that x : y = u : v

If (7a + 8b) (7c – 8d) = (7a – 8b) (7c + 8d), prove that a: b = c: d.

if x = `(6ab)/(a + b)` find the value of `(x + 3a)/(x - 3a) = (x + 3b)/(x - 3b)`

If a = `(4sqrt6)/(sqrt2 + sqrt3)` find the value of `(a + 2sqrt2)/(a - 2sqrt2) + (a + 2sqrt3)/(a - 2sqrt3)`

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

If `(a - 2b - 3c + 4d)/(a + 2b - 3c - 4d) = (a - 2b + 3c - 4d)/(a + 2b + 3c + 4d)` show that 2ad = 3bc

If `(a^2 + b^2)(x^2 + y^2)` = (ax + by)^2 ; prove that `a/x = b/y`

If a, b, and c are in continued proportion, prove that `(a^2 + ab + b^2)/(b^2 + bc + c^2) = a/c`

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

Using properties of proportion solve for x

`(sqrt(x + 5) + sqrt(x - 16))/(sqrt(x + 5) - sqrt(x - 16)) = 7/3`

Using properties of proportion solve for x:

`(sqrt(x + 1) + sqrt(x - 1))/(sqrt(x + 1) - sqrt(x - 1)) = (4x - 1)/2`

Using properties of proportion solve for x:

`(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5)) = 5`

If x = `(sqrt(a + 3b) + sqrt(a - 3b))/(sqrt(a + 3b) - sqrt(a - 3b))` prove that `3bx^2 - 2ax + 3b = 0`

Using the properties of proportion solve for x given `(x^4 + 1)/(2x^2) = 17/8`

If `x = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n))` express n in terms of x and m

If `(x^2 + 3xy^2)/(3x^2y + y^3) = (m^2 + 3mn^2)/(3m^2n + n^3)` show that nx = my

#### Chapter 7: Ratio and Proportion (Including Properties and Uses) Exercise 7D solutions [Page 0]

If a: b = 3: 5, find: (10a + 3b): (5a + 2b)

If 5x + 6y: 8x + 5y = 8: 9, find x: y.

If (3x – 4y): (2x – 3y) = (5x – 6y): (4x – 5y), find x: y.

Find the duplicate ratio of `2sqrt2 : 3sqrt5`

Find the triplicate ratio of 2a: 3b

Find the sub-duplicate ratio of 9x^{2}a^{4 }: 25y^{6}b^{2}

Find the sub-triplicate ratio of 216: 343

Find the reciprocal ratio of 3: 5

Find the ratio compounded of the duplicate ratio of 5: 6, the reciprocal ratio of 25: 42 and the sub-duplicate ratio of 36: 49.

Find the value of x, if: (2x + 3): (5x – 38) is the duplicate ratio of `sqrt5: sqrt6`

Find the value of x, if: (2x + 1): (3x + 13) is the sub-duplicate ratio of 9: 25.

Find the value of x, if: (3x – 7): (4x + 3) is the sub-triplicate ratio of 8: 27.

What quantity must be added to each term of the ratio x: y so that it may become equal to c: d?

Two numbers are in the ratio 5:7. If 3 is subtracted from each of them, the ratio between them becomes 2:3. Find the numbers.

If 15(2x^{2} – y^{2}) = 7xy, find x: y; if x and y both are positive.

Find the fourth proportional to 2xy, x^{2} and y^{2}.

Find the third proportional to a^{2} – b^{2} and a + b.

Find the mean proportional to (x – y) and (x^{3} – x^{2}y).

Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.

If x and y be unequal and x: y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y.

If q is the mean proportional between p and r prove that `(p^3 + q^3 + r^3)/(p^2q^2r^2) = 1/p^3 + 1/q^3 = 1/r^3`

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

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

If (4a + 9b) (4c – 9d) = (4a – 9b) (4c + 9d), prove that: a: b = c: d.

if `a/b = c/d` show that `(a + b) : (c + d) = sqrt(a^2 + b^2) : sqrt(c^2 + d^2)`

if `x/a = y/b = z/c` prove that `(ax - by)/((a + b)(x - y)) + (by - cz)/((b + c)(y - z)) + (cz - az)/((c + a)(z - x)) = 3`

There are 36 members in a student council in a school and the ratio of the number of boys to the number of girls is 3: 1. How any more girls should be added to the council so that the ratio of the number of boys to the number of girls maybe 9: 5?

Given, `x/(b - c ) = y/(c - a ) = z/(a - b)` , Prove that

ax+ by + cz = 0

If 7x – 15y = 4x + y, find the value of x: y. Hence, use componendo and dividend to find the values of:

`(9x + 5y)/(9x - 5y)`

If 7x – 15y = 4x + y, find the value of x: y. Hence, use componendo and dividend to find the values of:

`(3x^2 + 2y^2)/(3x^2 - 2y^2)`

If `(4m + 3n)/(4m - 3n) = 7/4` use properties of proportion to find m : n

If `(4m + 3n)/(4m - 3n) = 7/4` use properties of proportion to find `(2m^2 - 11n^2)/(2m^2 + 11n^2)`

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

Given x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))`

Use componendo and dividendo to prove that b^2 = (2a^2x)/(x^2 + 1)

If `(x^2 + y^2)/(x^2 - y^2) = 2 1/8` find `x/y`

if `(x^2 + y^2)/(x^2 - y^2) = 2 1/8` find `(x^3 + y^3)/(x^3 - y^3)`

Using componendo and dividendo find the value of x

`(sqrt(3x + 4) + sqrt(3x - 5))/(sqrt(3x + 5) - sqrt(3x - 5)) = 9`

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

## Chapter 7: Ratio and Proportion (Including Properties and Uses)

#### Selina Selina ICSE Concise Mathematics Class 10 (2018-2019)

#### Textbook solutions for Class 10

## Selina solutions for Class 10 Mathematics chapter 7 - Ratio and Proportion (Including Properties and Uses)

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Concepts covered in Class 10 Mathematics chapter 7 Ratio and Proportion (Including Properties and Uses) are Ratio and Proportion Example, Direct Applications, Alternendo and Invertendo Properties, Componendo and Dividendo Properties, Proportions, Ratios.

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