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Selina solutions for Concise Mathematics Class 10 ICSE chapter 7 - Ratio and Proportion (Including Properties and Uses) [Latest edition]

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Concise Mathematics Class 10 ICSE - Shaalaa.com

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

Exercise 7(A)Exercise 7(B)Exercise 7(C)Exercise 7(D)

Exercise 7(A) [Pages 0 - 88]

Selina solutions for Concise Mathematics Class 10 ICSE Chapter 7 Ratio and Proportion (Including Properties and Uses) Exercise 7(A) [Pages 0 - 88]

Exercise 7(A) | Q 1 | Page 87

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

Exercise 7(A) | Q 2 | Page 87

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

Exercise 7(A) | Q 3 | Page 87

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

Exercise 7(A) | Q 4 | Page 87

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

Exercise 7(A) | Q 5 | Page 87

Find the number which bears the same ratio to `7/33` that `8/21` does to `4/9`.

Exercise 7(A) | Q 6 | Page 87

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

Exercise 7(A) | Q 7 | Page 87

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

Exercise 7(A) | Q 8 | Page 87

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

Exercise 7(A) | Q 9 | Page 87

Divide Rs 1,290 into A,B and C such that A is `2/5` of B and B : C = 4 : 3.

Exercise 7(A) | Q 10 | Page 87

A school has 630 students. The ratio of the number of boys to the number of girls is 3 : 2. This ratio changes to 7 : 5 after the admission of 90 new students. Find the number of newly admitted boys.

Exercise 7(A) | Q 11 | Page 87

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

Exercise 7(A) | Q 12 | Page 87

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.

Exercise 7(A) | Q 13 | Page 87

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.

Exercise 7(A) | Q 14 | Page 87

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

(i) the original fare is Rs 245;

(ii) the increased fare is Rs 207.

Exercise 7(A) | Q 15 | Page 87

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?

Exercise 7(A) | Q 16

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.

Exercise 7(A) | Q 17 | Page 88

In a mixture of 126 kg of milk and water, milk and water are in ratio 5 : 2. How much water must be added to the mixture to make this ratio 3 : 2?

Exercise 7(A) | Q 18.1 | Page 88

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

find:

(i) A: B: C (ii) A: C

Exercise 7(A) | Q 18.2 | Page 88

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

Exercise 7(A) | Q 19.1 | Page 88

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

Exercise 7(A) | Q 19.2 | Page 88

If 2a = 3b and 4b = 5c, find: a : c.

Exercise 7(A) | Q 20.1 | Page 88

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

Exercise 7(A) | Q 20.2 | Page 88

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

Exercise 7(A) | Q 20.3 | Page 88

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

Exercise 7(A) | Q 21.1 | Page 88

Find the duplicate ratio of 3: 4

Exercise 7(A) | Q 21.2 | Page 88

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

Exercise 7(A) | Q 22.1 | Page 88

Find the triplicate ratio of 1 : 3

Exercise 7(A) | Q 22.2 | Page 88

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

Exercise 7(A) | Q 23.1 | Page 88

Find the sub-duplicate ratio of 9: 16

Exercise 7(A) | Q 23.2 | Page 88

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

Exercise 7(A) | Q 24.1 | Page 88

Find the sub-triplicate ratio of 64: 27

Exercise 7(A) | Q 24.2 | Page 88

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

Exercise 7(A) | Q 25.1 | Page 88

Find the reciprocal ratio of:
5 : 8

Exercise 7(A) | Q 25.2 | Page 88

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

Exercise 7(A) | Q 26 | Page 88

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

Exercise 7(A) | Q 27 | Page 88

If m: n is the duplicate ratio of m + x: n + x; show that x2 = mn.

Exercise 7(A) | Q 28 | Page 88

If (3x - 9) : (5x + 4) is the triplicate ratio of 3 : 4, find the value of x. 

Exercise 7(A) | Q 29 | Page 88

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.

Exercise 7(A) | Q 30.1 | Page 88

If r=pq, show that p : q is the duplicate ratio of (p + r) : (q + r).

Exercise 7(A) | Q 30.2 | Page 88

If (p - x) : (q - x) be the duplicate ratio of p : q then show that : `1/"p" + 1/"q" = 1/"x"`

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Exercise 7(B) [Page 94]

Selina solutions for Concise Mathematics Class 10 ICSE Chapter 7 Ratio and Proportion (Including Properties and Uses) Exercise 7(B) [Page 94]

Exercise 7(B) | Q 1.1 | Page 94

Find the fourth proportional to 1.5, 4.5 and 3.5

Exercise 7(B) | Q 1.2 | Page 94

Find the fourth proportional to 3a, 6a2 and 2ab2

Exercise 7(B) | Q 2.1 | Page 94

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

Exercise 7(B) | Q 2.2 | Page 94

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

Exercise 7(B) | Q 3.1 | Page 94

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

Exercise 7(B) | Q 3.2 | Page 94

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

Exercise 7(B) | Q 4 | Page 94

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

Exercise 7(B) | Q 5 | Page 94

If x2, 4 and 9 are in continued proportion, find x.

Exercise 7(B) | Q 6 | Page 94

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

Exercise 7(B) | Q 7.1 | Page 94

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

Exercise 7(B) | Q 7.2 | Page 94

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

Exercise 7(B) | Q 7.3 | Page 94

If `"a"/"b" = "c"/"d"`, show that: `(a^3c + ac^3)/(b^3d + bd^3) = ("a" + "c")^4/("b" + "d")^4`

Exercise 7(B) | Q 8 | Page 94

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

Exercise 7(B) | Q 9 | Page 94

If y is the mean proportional between x and z; show that xy + yz is the mean proportional between x2+y2 and y2+ z2.

Exercise 7(B) | Q 10 | Page 94

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

Exercise 7(B) | Q 11 | Page 94

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

Exercise 7(B) | Q 12 | Page 94

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`

Exercise 7(B) | Q 13 | Page 94

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)`

Exercise 7(B) | Q 14 | Page 94

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

Exercise 7(B) | Q 15 | Page 94

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

Exercise 7(B) | Q 16 | Page 94

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

Exercise 7(B) | Q 17 | Page 94

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

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Exercise 7(C) [Pages 101 - 102]

Selina solutions for Concise Mathematics Class 10 ICSE Chapter 7 Ratio and Proportion (Including Properties and Uses) Exercise 7(C) [Pages 101 - 102]

Exercise 7(C) | Q 1.1 | Page 101

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

Exercise 7(C) | Q 1.2 | Page 101

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

Exercise 7(C) | Q 1.3 | Page 101

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

Exercise 7(C) | Q 2 | Page 101

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

Exercise 7(C) | Q 3 | Page 101

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

Exercise 7(C) | Q 4 | Page 101

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

Exercise 7(C) | Q 5 | Page 101

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

Exercise 7(C) | Q 6.1 | Page 101

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

Exercise 7(C) | Q 6.2 | Page 101

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

Exercise 7(C) | Q 7 | Page 101

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

Exercise 7(C) | Q 8 | Page 101

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

Exercise 7(C) | Q 9 | Page 101

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

Exercise 7(C) | Q 10.1 | Page 101

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

Exercise 7(C) | Q 10.2 | Page 101

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)`

Exercise 7(C) | Q 11.1 | Page 102

Using properties of proportion solve for x 

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

Exercise 7(C) | Q 11.2 | Page 102

Using properties of proportion solve for x:

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

Exercise 7(C) | Q 11.3 | Page 102

Using properties of proportion solve for x:

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

Exercise 7(C) | Q 12 | Page 102

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

Exercise 7(C) | Q 13 | Page 102

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

Exercise 7(C) | Q 14 | Page 102

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

Exercise 7(C) | Q 15 | Page 102

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

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Exercise 7(D) [Pages 102 - 103]

Selina solutions for Concise Mathematics Class 10 ICSE Chapter 7 Ratio and Proportion (Including Properties and Uses) Exercise 7(D) [Pages 102 - 103]

Exercise 7(D) | Q 1 | Page 102

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

Exercise 7(D) | Q 2 | Page 102

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

Exercise 7(D) | Q 3 | Page 102

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

Exercise 7(D) | Q 4.1 | Page 102

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

Exercise 7(D) | Q 4.2 | Page 102

Find the triplicate ratio of 2a: 3b

Exercise 7(D) | Q 4.3 | Page 102

Find the sub-duplicate ratio of 9x2a: 25y6b2

Exercise 7(D) | Q 4.4 | Page 102

Find the sub-triplicate ratio of 216: 343

Exercise 7(D) | Q 4.5 | Page 102

Find the reciprocal ratio of 3: 5

Exercise 7(D) | Q 4.6 | Page 102

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.

Exercise 7(D) | Q 5.1 | Page 102

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

Exercise 7(D) | Q 5.2 | Page 102

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

Exercise 7(D) | Q 5.3 | Page 102

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

Exercise 7(D) | Q 6 | Page 102

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

Exercise 7(D) | Q 7 | Page 102

A woman reduces her weight in the ratio 7 : 5. What does her weight become if originally it was 84 kg?

Exercise 7(D) | Q 8 | Page 102

If 15(2x2 – y2) = 7xy, find x: y; if x and y both are positive.

Exercise 7(D) | Q 9.1 | Page 102

Find the fourth proportional to 2xy, x2 and y2.

Exercise 7(D) | Q 9.2 | Page 102

Find the third proportional to a2 – b2 and a + b.

Exercise 7(D) | Q 9.3 | Page 102

Find the mean proportional to (x – y) and (x3 – x2y).

Exercise 7(D) | Q 10 | Page 102

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

Exercise 7(D) | Q 11 | Page 102

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.

Exercise 7(D) | Q 12 | Page 102

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

Exercise 7(D) | Q 13 | Page 102

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

Exercise 7(D) | Q 14 | Page 102

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

Exercise 7(D) | Q 15 | Page 102

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?

Exercise 7(D) | Q 16.1 | Page 102

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)`

Exercise 7(D) | Q 16.2 | Page 102

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)`

Exercise 7(D) | Q 17.1 | Page 102

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

Exercise 7(D) | Q 17.2 | Page 102

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

Exercise 7(D) | Q 18 | Page 102

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

Exercise 7(D) | Q 19 | Page 102

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)

Exercise 7(D) | Q 20.1 | Page 102

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

Exercise 7(D) | Q 20.2 | Page 102

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

Exercise 7(D) | Q 21 | Page 102

Using componendo and dividendo find the value of x

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

Exercise 7(D) | Q 22 | Page 102

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`

Exercise 7(D) | Q 23 | Page 103

Given `("x"^3 + 12"x")/(6"x"^2 + 8) = ("y"^3 + 27"y")/(9"y"^2 + 27)`

using componendo and dividendo, find x : y

Exercise 7(D) | Q 24 | Page 103

If `"x"/"a" = "y"/"b" = "z"/"c"` show that:

`"x"^3/"a"^3 + "y"^3/"b"^3 + "z"^3/"c"^3 = (3"xyz")/("abc")`

Exercise 7(D) | Q 25 | Page 103

If b is the mean proportion between a and c, show that: 

`("a"^4 + "a"^2"b"^2   + "b"^4)/("b"^4 + "b"^2"c"^2   +"c"^4) = "a"^2/"c"^2`

Exercise 7(D) | Q 26 | Page 103

If `(7"m" + 2"n")/(7"m" - 2"n") = 5/3` use properties of proportion to find:

(i) m : n

(ii) `("m"^2 + "n"^2)/("m"^2 - "n"^2)`

Exercise 7(D) | Q 27.1 | Page 103

If x and y both are positive and (2x2- 5y2): xy = 1: 3, find x: y.

Exercise 7(D) | Q 27.2 | Page 103

Find x, if `16(("a" - "x")/("a" + "x"))^3 = ("a" + "x")/("a" - "x")`

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

Exercise 7(A)Exercise 7(B)Exercise 7(C)Exercise 7(D)
Concise Mathematics Class 10 ICSE - Shaalaa.com

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

Selina solutions for Concise Mathematics Class 10 ICSE chapter 7 (Ratio and Proportion (Including Properties and Uses)) 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 Concise Mathematics Class 10 ICSE solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Selina textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

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