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

If a : b = 5 : 3, find : `(5a - 3b)/(5a + 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).

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

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

Find `x/y`; when x² + 6y² = 5xy. 

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")`

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

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.

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:

1. the original fare is Rs. 245;
2. 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.

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?

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

1. A : B : C
2. A : C

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

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

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

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

Find the compound ratio of 2a : 3b, mn : x² and x : n

Find the compound ratio of `sqrt(2) : 1, 3: sqrt(5)` and `sqrt(20) : 9`

Find duplicate ratio of 3 : 4

Find duplicate ratio of `3sqrt(3) : 2sqrt(5)`

Find triplicate ratio of 1 : 3

Find triplicate ratio of `m/2 : n/3`

Find sub-duplicate ratio of 9 : 16

Find sub-duplicate ratio of (x – y)⁴ : (x + y)⁶

Find sub-triplicate ratio of 64 : 27

Find sub-triplicate ratio of x³ : 125y³

Find the reciprocal ratio of 5 : 8

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

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

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

If (3x - 9) : (5x + 4) is the triplicate ratio of 3 : 4, find the value of 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 r² ₌pq, show that p : q is the duplicate ratio of (p + r) : (q + r).

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

Find the fourth proportional to 1.5, 4.5 and 3.5

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

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

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

Find the mean proportional between `6 + 3sqrt(3)` and `8 - 4sqrt(3)`

Find the mean proportional between a – b and a³ – a²b

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

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

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

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

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

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

What least number must 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 x² + y² and y² + z².

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)b² : (b – d)cd = (a² – b² – ab) : (c² – d² – 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: 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 (a² + b²)(x² + y²) = (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 and 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: 3bx² – 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^3 + 3xy^2)/(3x^2y + y^3) = (m^3 + 3mn^2)/(3m^2n + n^3)`, show that nx = my.

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 `2sqrt(2) : 3sqrt(5)`

Find the triplicate ratio of 2a : 3b

Find the sub-duplicate ratio of 9x²a⁴ : 25y⁶b²

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 `sqrt(5) : sqrt(6)`.

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?

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

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

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

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

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

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 `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)`

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 may be 9 : 5?

If 7x – 15y = 4x + y, find the value of x : y. Hence, use componendo and dividendo 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 + 4) - sqrt(3x - 5)) = 9`

If `x = (sqrt(a + 1) + sqrt(a - 1))/(sqrt(a + 1) - sqrt(a - 1))`, using properties of proportion show that: x² – 2ax + 1 = 0.

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

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

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`

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)`

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

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