Nootan solutions for Mathematics [English] Class 10 ICSE chapter 7 - Ratio and proportion [Latest edition]

Find the ratio compounded at 2 : 3, 4 : 5, 5 : 6

Find the ratio compounded at a : b, b : c, c : d

Find the ratio compounded at x + y : x − y, x³ − y³ : x² − y²

Find the duplicate ratio of 4 : 7

Find the duplicate ratio of m : n

Find the duplicate ratio of 5x : 6y

Find the sub-duplicate ratio of 49 : 81.

Find the sub-duplicate ratio of 16x⁴ : 25y⁴.

Find the sub-duplicate ratio of 169m⁶ : 196n⁶.

Find the triplicate ratio of 3 : 5.

Find the triplicate ratio of x : y.

Find the triplicate ratio of 2x : 3y.

Find the sub-triplicate ratio of a⁹ : b¹².

Find the sub-triplicate ratio of 64a³ : 343b⁶.

Find the reciprocal ratio of 3 : 4.

Find the reciprocal ratio of 4² : 5².

Find the reciprocal ratio of `1/3 : 4`

If 4x + 3 : 7x − 1 be the duplicate ratio of 3 : 4, find the value of x.

If a : b be the triplicate ratio of (a + x) : (b + x), prove that x³ − 3abx − ab(a + b) = 0.

Find which is greater?

`3/4 or 15/16`

Find which is greater?

2 : 3 or 3 : 4

If x : y = 7 : 12, find the value of 3x + 5y : 6x + 7y.

If x : y = 3 : 4, find the ratio of 3y − x : 2x + y.

If 3x + 2y : 5x + 6y = 1 : 2, find the ratio of x : y.

If 5a + 3b : 3a + 5b = 3 : 4, find a : b.

What number must be added to each terms of the ratio 14 : 23 so that it may becomes equal to 8 : 11?

Two numbers are in the ratio of 3 : 4. If 4 be added to each, the ratio becomes 4 : 5, find the numbers.

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

If 6A = 3B = 5C, find A : B : C.

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

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

If `(5x + 4y)/(8x + 5y) = 2/3`, find the value of 4x + 7y : 9x + 2y.

Two numbers are in the ratio of 2 : 3, and if 6 subtracted from each, the remainders are in the ratio of 4 : 9. Find them.

If `x/y = 3(1)/3`, find the value of `(x - 3y)/(2x - 5y)`.

A’s present age is to B’s present age as 8 : 7. 27 years ago their ages were 5 : 4. Find their present ages.

If `a/b = 2/3` and `x/y = 3/4`, find the value of `(4ay - 3bx)/(5ax - 2by)`.

Two numbers are in the ratio of 3 : 5 and the sum of their squares is 1666. Find them.

The ratio between A and B’s share is 7 : 11. If B’s share is ₹ 22.55, find A’s share.

A bag contains ₹ 102 in the form of ₹ 1,50 paise and 25 paise coins in the ratio of 16 : 5 : 28. Find the number of coins of each kind.

In a mixture of 28 litres, the ratio of milk and water is 5 : 2. Another 2 litres of water is added to the mixture. Find the ratio of milk and water in the new mixture.

State whether the following are in proportion:

6 : 18 :: 5 : 15

State whether the following are in proportion:

3 : 7.5 :: 2 : 7

State whether the following are in proportion:

10 : 21 :: 4 : 8.4

State whether the following are in proportion:

9 : 12 :: 12 : 16

State whether the following are in proportion:

11 : 13 :: 4 : 5

Find the fourth proportional to the following:

11, 15, 33

Find the fourth proportional to the following:

1.5, 4.5, 3.5

Find the fourth proportional to the following:

`1/2, 1/3, 1/4`

Find the fourth proportional to the following:

`1/7, 1/5, 1(1)/7`

Find the value of x in the following:

₹ 10 : ₹ 25 = 14m : x

Find the value of x in the following:

75 : 15 = x : 7

Find the value of x in the following:

x : 7.5 = 7 : 17.5

Find the third proportional to the following:

20, 15

Find the third proportional to the following:

2.4, 3.6

Find the third proportional to the following:

`2(2)/3 m`, 4m

Find the third proportional to the following:

3, 12

Find the mean proportional of the following:

9, 25

Find the mean proportional of the following:

`1/16, 4/25`

Find the mean proportional of the following:

17.5, 0.007

Find the mean proportional of the following:

0.25, 40000

Find the mean proportional of the following:

₹ 5, ₹ 125

Find the mean proportional of the following:

2x, 8x

Find the mean proportional of the following:

`(a - b)/(a + b) and (a^2b^2)/(a^2 - b^2)`

What number must be added to each of the numbers 4, 6, 8, 11 in order to get the four numbers in proportion?

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

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

The following numbers, K + 3, K + 2, 3K – 7 and 2K – 3 are in proportion. Find k. 

k + 6 is the mean proportion between k − 2 and k². Find the value of k.

What number must be added to each of the numbers 6, 14 and 30 so that the resulting numbers may be in continued proportion?

Find two numbers such that their mean proportional is 28 and third proportional in 224.

Find two numbers such that their mean proportional is 12 and third proportional in 96.

If `a/b = c/d` then prove that each of the given ratio is equal to `(3a + 2c)/(3b + 2d)`.

If `a/b = c/d` then prove that each of the given ratio is equal to `sqrt((4a^2 - 3c^2)/(4b^2 - 3d^2))`.

If `a/b = c/d` then prove that each of the given ratio is equal to `root3 ((a^3 + 4c^3)/(b^3 + 4d^3))`.

If `a/b = c/d` then prove that `(5a + 3b)/(5c + 3d) = ((4a^3 - 3b^3)/(4c^3 - 3d^3))^(1/3)`

If `a/b = c/d` then prove that `(6a + 7b)/(6c + 7d) = sqrt((5a^2 + 3b^2)/(5c^2 + 3d^2))`

If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are proportional.

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^2a^2 + y^2b^2 + z^2c^2)/(a^3x + b^3y + c^3z) = ((xyz)/(abc))^(1/3)`

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

If `a/b = c/d = e/f`, prove that: (a² + c² + e²) (b² + d² + f²) = (ab + cd + ef)²

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

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

If b is the mean proportional between a and c, show that `(a^4 + a^2b^2 + b^4)/(b^4 + b^2c^2 + c^4) = a^2/c^2`.

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

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

If a, b, c, d are in proportion, prove that (a² + c²) : (b² + d²) = ac : bd

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

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

 If a, b, c, d are in proportion, prove that abcd(a⁻² + b⁻² + c⁻² + d⁻²) = a² + b² + c² + d².

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

If x, y, z are in continued proportion, prove that: `(px^2 + qxy + ry^2)/(py^2 + qyz + rz^2) = x/z`.

If x, y and z are in continued proportion, Prove that:

`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`

If x, y, z are in continued proportion, prove that: x²y²z²(x⁻⁴ + y⁻⁴ + z⁻⁴) = y⁻²(x⁴ + y⁴ + z⁴)

If x, y, z are in continued proportion, prove that: xyz(x + y + z)³ = (xy + yz + zx)³.

If a, b, c, d are in continued proportion, prove that: `sqrt(ab) + sqrt(bc) - sqrt(cd) = sqrt((a + b - c)(b + c - d))`.

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: (a² + b²) (c² + d²) = (b² + c²)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 - b)/c + (a - c)/b)^2 - ((d - b)/c + (d - c)/b)^2 = (a - d)^2 (1/c^2 - 1/b^2)`.

If `x/y = p/q`, prove that `(5x + 7y)/(5x - 7y) = (5p + 7q)/(5p - 7q)`.

If `a/b = c/d`, prove that `(4a + 9b)/(4a - 9b) = (4c + 9d)/(4c - 9d)`.

If a : b :: c : d, show that (2a + 3b) : (2c + 3d) :: (2a − 3b) : (2c − 3d).

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

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

If (6a + 5b) (6c − 5d) = (6a − 5b) (6c + 5d), prove that a, b, c, d are in proportion.

If (9a² + 8b²) (9c² − 8d²) = (9a² − 8b²) (9c² + 8d²), prove that a : b :: c : d.

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

If x = `(2ab)/(a + b)`, prove that `(x + a)/(x - a) + (x + b)/(x - b)` = 2

If x = `(8ab)/(a + b)`, prove that `(x + 4a)/(x - 4a) + (x + 4b)/(x - 4b)` = 2

Using properties of proportion solve for x, given

`(sqrt(5x)+sqrt(2x -6))/(sqrt(5x)- sqrt(2x -6)) = 4`

Using properties of proportion, find the value of x from the following:

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

Using properties of proportion, find the value of x from the following:

`(sqrt(12x + 1) + sqrt(2x - 3))/(sqrt(12x + 1) - sqrt(2x - 3)) = 3/2`

Using properties of proportion, solve for x. Given that x is positive:

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

Using properties of proportion, find the value of x from the following:

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

If x = `(sqrt(2a + 3b) + sqrt(2a - 3b))/(sqrt(2a + 3b) - sqrt(2a - 3b))` then show that 3bx² − 4ax + 3b = 0.

If x = `(sqrt(b^2 + ab) + sqrt(b^2 - ab))/(sqrt(b^2 + ab) - sqrt(b^2 - ab))` show that ax² − 2bx + a = 0.

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

If x = `(sqrt(2a + 1) + sqrt(2a - 1))/(sqrt(2a + 1) - sqrt(2a - 1))` prove that x² − 4ax + 1 = 0

Using properties of proportion, find x : y, if `(x^2 + 2x)/(2x + 4) = (y^2 + 3y)/(3y + 9)`

If `(x^2 + y^2)/(x^2 - y^2) = 17/8`, then find the value of:

1. x : y
2. `(x^3 + y^3)/(x^3 - y^3)`

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

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 + 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.

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

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

If `(a + b)^3/(a - b)^3 = 64/27`

1. Find `(a + b)/(a - b)`
2. Hence using properties of proportion, find a : b.

x, 8, 16 are in continued proportion. The value of x is ______.

The mean proportional of two numbers is 16 and the third proportional is 128. The larger number is ______.

If b is the mean proportional of a and c then the mean proportional of a² + b² and b² + c² is ______.

If 3x + 4y : 3x − 4y = 17 : 1 then x : y is equal to ______.

The sub-triplicate ratio of 125 : 27 is ______.

The quantity added to each term of the ratio 6 : 11 so that it becomes equal to 3 : 4 is ______.

If x, y, z are in continued proportion then `(x^2 + y^2)/(y^2 + z^2)` is equal to ______.

If A : B = 2 : 3, В : C = 4 : 5 then A : B : C is equal to ______.

If a : b = 3 : 11 then (15a – 3b) : (9a + 5b) is equal to ______.

If `(1 + x + x^2)/(1 - x + x^2) = (62(1 + x))/(63(1 - x))` then the value of x is ______.

The given table shows the distance covered and the time taken by a train moving at a uniform speed along a straight track:

The values of x and y are:

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. The duplicate ratio of `sqrt2 : sqrt3` is 2 : 3
2. 7 : 8 > 4 : 5

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If (4a + 3b) : (6a +7b) = 16 : 25 then a : b = 31 : 4.
2. If (1 + x) : (3x + 1) is the duplicate ratio of 3 : 5 then x = 8.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. The third proportional of 5, 10 is 40.
2. If three quantities are in continued proportion, then first is to the third is the sub-duplicate ratio of first to the second.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If x : y = 2 : 3, y : z = 4 : 5, then x : y : z = 8 : 12 : 15.
2. lf x, 2, 8, y are in continued proportion, then x = 1.

Using componendo and dividendo solve for x:

`(sqrt(2x + 2) + sqrt(2x - 1))/(sqrt(2x + 2) - sqrt(2x - 1))` = 3

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

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

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

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`

If x = `(root3 (a + 1) + root3 (a - 1))/(root3 (a + 1) - root(3)(a - 1)`, prove that x³ – 3ax² + 3x – a = 0.

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

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

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

If (3x² + 2y²) : (3x² − 2y²) = 11 : 9, find the value of `(3x^4 + 25y^4)/(3x^4 - 25y^4)`.

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