#### Chapters

Chapter 2: Sales Tax and Value Added Tax

Chapter 3: Banking

Chapter 4: Shares and Dividends

Chapter 5: Linear Inequations

Chapter 6: Quadratic Equations

Chapter 7: Problems Based On Quadratic Equations

Chapter 8: Reflection

Chapter 9: Ratio and Proportion

Chapter 10: Remainder And Factor Theorems

Chapter 11: Matrices

Chapter 12: Distance and Section Formulae

Chapter 13: Equation of A Straight Line

Chapter 14: Symmetry

Chapter 15: Similarity

Chapter 16: Loci

Chapter 17: Circles

Chapter 18: Constructions

Chapter 19: Mensuration I

Chapter 20: Mensuration II

Chapter 21: Trigonometric Identities

Chapter 22: Heights and Distances

Chapter 23: Graphical Representations

Chapter 24: Measures Of Central Tendency

Chapter 25: Probability

## Chapter 9: Ratio and Proportion

#### Exercise 9.1

### Frank solutions for ICSE Class 10 Mathematics Part 2 Chapter 9 Ratio and Proportion Exercise 9.1

Find the ratio of the following in the simplest fcrm:

5.60 and 2.40

Find the ratio of the following in the simplest fcrm:

432 and 120

Find the ratio of the following in the simplest fcrm:

Rs 5.40 and 180 paise

Find the ratio of the following in the simplest fcrm:

a^{4} + b^{4} and a^{3} - b^{3}

Find the ratio of the following in the simplest fcrm:

x^{2} + 4x +4 and x^{2} - x - 6

If a: b = 4: 7, find the following:

`(5"a" + 2"b")/(5"a" - 2"b")`

If a : b = 4 : 7, find the following:

`(6"a" - "b")/("a" + 3"b")`

If a: b = 4: 7, find the following:

`(5"a" - 4"b")/(2"a" - 3"b")`

If m : n = 3 : 8, find the value of (3m+2n) : (5m+n)

A man's monthly inoome is Rs 5,000. He saves every month a minimum of R~ 800. Find the ratio of his:

Annual expenses to annual Income.

A man's monthly inoome is Rs 5,000. He saves every month a minimum of R~ 800. Find the ratio of his:

monthly savings to monthly expenses.

If a + b : a - b = 11 : 8 ; find the vaIue of a : b

If p : q = 2 : 5 , q : r = 4 : 3, then find p : r

If a : e = 5 : 12, e : i = 8 : 3 and i : u = 9 : 16, then find a : u

Find the compounded ratio of the following:

15: 16 and 8 : 5

Find the compounded ratio of the following:

(a^{2} - b^{2} ): (a^{2} + b^{2}) and (a^{4} - b^{4} ): (a+ b)^{4}

Find the compounded ratio of the following:

3 : 5, 7 : 9 and 15 : 28

Find the compounded ratio of the following:

`sqrt 8 : 4 , 3 : sqrt 5` and `sqrt 20 : sqrt 27`

Find the compounded ratio of the following:

(m-n):(m+n), (m+n)^{2} : (m^{2}+n^{2}) and (m^{4} - n^{4}): (m^{2}-n^{2}) ^{2}

Find the duplicate ratio of the following :

`sqrt 10 : sqrt 14`

Find the duplicate ratio of the following :

`3 sqrt (2"a") : 2 sqrt (3"a")`

Find the duplicate ratio of the following :

`2/3 : 4/9`

Find the duplicate ratio of the following :

(a+b): (a^{2}- b^{2})

Find the triplicate ratio of the following :

3 : 5

Find the triplicate ratio of the following :

`2 sqrt 5 : 5 sqrt 2`

Find the triplicate ratio of the following :

`sqrt 15 : sqrt 18`

Find the triplicate ratio of the following :

`root (3)("ab"^2) : root (3)("a"^2"b")`

Find the sub-duplicate ratio of the following :

x^{6} : y^{4}

Find the sub-duplicate ratio of the following :

c

Find the sub-duplicate ratio of the following :

`1/16 : 1/36`

Find the sub-triplicate ratio of the following :

125a^{3} : 343b^{6}

Find the sub-duplicate ratio of the following :

`(9"a"^2)/5 : (25"a"^2)/3`

Find the sub-triplicate ratio of the following :

512: 216

Find the sub-triplicate ratio of the following :

m^{3}n^{6} : m^{6}n^{3}

Find the sub-triplicate ratio of the following :

125a^{3} : 343b^{6}

Find the sub-triplicate ratio of the following :

`(64 "m"^3)/(729 "n"^3) : (216 "m"^3)/(27"n"^3)`

Find the reciprocal ratio of e following :

`17/45 : 51/27`

Find the reciprocal ratio of e following :

`1/45 : 1/54`

Find the reciprocal ratio of e following :

a^{3}b^{2} : a^{2}b^{3}

Find the reciprocal ratio of e following :

81pq^{2} : 54p^{2}q

Which of the Following Ratio is Greater ?

3 : 5 and 2 : 11

which of the following ratio is greater?

2 : 3 and 13 : 19

Which of the following ratio is greater ?

5 : 8 and 7 : 10

Which of the following ratio is greater ?

`5/2 : 15/4` and `5/3 : 11/6`

Two numbers are in the ratio 7 : 10. If 8 is added to each number, the ratio becomes 3 : 4. Find the nLimbers.

Two positive numbers are in the ratio 3: 4 and the sum of their squares is 1225. Find the numbers.

Two numbers are in the ratio 5 : 7 and the difference of their squares is 600. Find the numbers.

What quantity must be subtracted from each term of the ratio 39 : 89 to make it equal to 2 : 5 ?

What quantity must be added to each term of the ratio 19 : 51 to make it equal to 3 : 7 ?

What quantity must be added to each term of the ratio (p + q) : (p - q) to make it equal to (p + q)^{2} : (p - q)^{2} ?

If (3x-4) : (2x+5) is the duplicate ratio of 3 : 4 , find x.

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

If r^{2} = pq , show that p : q is the duplicate ratio of (p+r) : ( q+r)

A sum of money Is divided in the ratio 2 : 3. If the larger portion is Rs 7,47,300. find the sum distributed.

Divide a number into two parts in the ratio 5:7 so that smaller part is 60. Find the number.

A bag contains Rs 1800 in the form of Rs 1 , Rs 2 and Rs 5 ooins. The ratio of the number of the respective coins is 3: 7: 11. Find the total number of ooi ns in the bag.

The present age of two persons are in the ratio 4: 3. Nine years hence their ages will be in the ratio 23: 18. Find their present ages.

The ratio of the pocket money saved by a boy and his sister is 5:4. If the brother saves Rs 100 more, how much more should the sister save in order to keep the ratio of their savings unchanged ?

A cistern of milk caitains a mixture of milk and water in the ratio of 11 :4. If the cistern contains 60 litres of milk, how much more water must be added to make the

ratio of milk to water as 11:6 ?

In an examination the ratio of the number of suo:essful candidates to unsuccessful candidates is 7:5. Had 30 more appeared and 10 more passed, the ratio of successful candidates to unsuccessful candidates would have been 4:3. Find the number of candidates who appeared in the examination originally.

#### Exercise 9.2

### Frank solutions for ICSE Class 10 Mathematics Part 2 Chapter 9 Ratio and Proportion Exercise 9.2

Find the value of the unknown in the following proportion :

5 : 12 :: 15 : x

Find the value of the unknown in the following proportion :

3 : 4 : : p : 12

Find the value of the unknown in the following proportion :

`1/2 : "m" :: 14/9 : 4/3`

Find the value of the unknown in the following proportion :

c

Find the fourth proportion to the following:

3,5 and 15

Find the fourth proportion to the following:

0.7, 4.9 and 1.6

Find the fourth proportion to the following:

(p^{2}q - qr^{2} ), (pqr - pr^{2} ) and (pq^{2} - pr^{2})

Find the fourth proportion to the following :

(x^{2} - y^{2}),(x^{3} + y^{3})anc(x^{3} - xy^{2} + x^{2}y- y^{3})

Find the third proportion to the following :

3 and 15

Find the third proportion to the following :

16x^{2} and 24x

Find the third proportion to the following :

(x - y) and m (x - y)

Find the third proportion to the following :

`9/25` and `18/25`

Find the mean proportion of the following :

24 and 6

Find the mean proportion of the following :

0.09 and 0.25

Find the mean proportion of the following :

ab^{3} and a^{3}b

Find the mean proportion of the following :

`28/3` and `175/27`

If x, 12 and 16 are in continued proportion! find x.

If `1/12` , x and `1/75` are in continued proportion , find x.

If y is the mean proportional between x and z, show that :

xyz (x+y+z)^{3} =(xy+yz+xz)^{3}

If y is the mean proportional between x and y; show that y(x+z) is the mean p roporti ona I between x^{2}+ y^{2} and y^{2}+ z^{2 }

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.

Given four quantities p, q, r and s are in proportion, show that

q^{2}(p - r) : rs (q - s) =(p^{2}- q^{2}- pq): ( r^{2}-s^{2}-rs).

If a : b = c : d; then show that (ax+ by) : b = (cx+ dy) : d

If ( a+c) : b = 5 : 1 and (bc + cd) : bd = 5 : 1, then prove that a : b = c : d

Find the smallest number that must be subtracted from each of the numbers 20, 29, 84 and 129 so that they are in proportion.

Find two nurnbers whose mean proportional is 12 and the third proportional is 324.

Find two numbers whose mean proportional is 18 and the third proportional is 486.

#### Exercise 9.3, Exercise 9.3

### Frank solutions for ICSE Class 10 Mathematics Part 2 Chapter 9 Ratio and Proportion Exercise 9.3, Exercise 9.3

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

2a+7b : 2a-7b = 2c+7d : 2c-7d

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

7a+11b : 7a -11b = 7c +11d : 7c - 11d

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

`(4"a" + 9"b")/(4"c" + 9"d") = (4"a" - 9"b")/(4"c" - 9"d")`

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

(ax+ by): (cx + dy)=(ax - by) : (cx - dy)

If `(7"a" + 12"b")/(7"c" + 12"d")` then prove that `"a"/"b"="c"/"d"`

If (7m +8n)(7p - 8q) = (7m - 8n)(7p + 8q), then prove that m: n = p: q

If a : b =c : d, then prove that `("a"^2 + "c"^2)/("b"^2 + "d"^2) = ("ac")/("bc")`

If a : b = c : d , then 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 : d :: e : f, then prove that `("ae" + "bf")/("ae" - "bf")` = `("ce" + "df")/("ce" - "df")`

If p, q, r ands are In continued proportion, then prove that (p^{3}+q^{3}+r^{3}) ( q^{3}+r^{3}+s^{3}) : : P : s

If u, v, w, and x are in continued proportion, then prove that (2u+3x) : (3u+4x) : : (2u^{3}+3v^{3}) : (3u^{3}+4v^{3})

If p, q and r in continued proportion, then prove the following:

(p^{2} - q^{2})(q^{2} + r^{2}) = (q^{2} - r^{2})(p^{2} + q^{2})

If p, q and r in continued proportion, then prove the following:

(p + q + r )(p - q + r) = p^{2} + q^{2} + r^{2}

If u, v, w, and x are in continued proportion, then prove that (2u+3x) : (3u+4x) : : (2u^{3}+3v^{3}) : (3u^{3}+4v^{3})

`("pqr")^2 (1/"p"^4 + 1/"q"^4 + 1/"r"^4) = ("p"^4 + "q"^4 + "r"^4)/"q"^2`

If p, q and r in continued proportion, then prove the following :

`"p"^2 - "q"^2 + "r"^2 = "q"^4 (1/"p"^2 - 1/"q"^2 - 1/"r"^2)`

If a, b, c and dare in continued proportion, then prove that

ad (c^{2} + d^{2}) = c^{3} (b + d)

If a, b, c and dare in continued proportion, then prove that

`sqrt (("a + b + c")("b + c + d")) = sqrt "ab" + sqrt "bc" + sqrt "cd"`

If a, b, c and dare in continued proportion, then prove that

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

If `"x"/("b + c - a") =" y" /("c + a - b") = "z"/("a + b - c")` , then prove that each ratio is equal to the ratio of `("x + y+z")/("a + b + c")`

If `"a"/("b + c") = "b"/("c + a") = "c"/("a + b")` , then prove that a(b - c) + b(c-a) + c (a - b) = 0

If x = `(root (3)("m + 1") + root (3)("m - 1"))/(root (3)("m + 1") + root (3)("m - 1")` then prove that x^{3} - 3mx^{2} + 3x = m

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

Show that the value of x is 11, when `("x"^3 + "3x")/(3"x"^2 + 1) = 341/91`

## Chapter 9: Ratio and Proportion

## Frank solutions for ICSE Class 10 Mathematics Part 2 chapter 9 - Ratio and Proportion

Frank solutions for ICSE Class 10 Mathematics Part 2 chapter 9 (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 ICSE Class 10 Mathematics Part 2 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in ICSE Class 10 Mathematics Part 2 chapter 9 Ratio and Proportion are Ratios, Concept of Proportion, Componendo and Dividendo Properties, Alternendo and Invertendo Properties, Direct Applications, Ratio and Proportion Example.

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