ICSE solutions for Mathematics [English] Class 10 chapter 8 - Ratio and Proportion [Latest edition]

Which is greater 4: 5 or 19: 25.

Arrange 5: 6, 8: 9, 13: 18 and 7: 12 in ascending order of magnitude.

Find:
The sub-triplicate ratio of the sub-duplicate ratio of 729x¹⁸: 64y⁶.

Find:
The sub-duplicate ratio of the sub-triplicate ratio of 4096x⁶: 729y¹².

If 3x – 2y – 7z = 0 and 2x + 3y – 5z = 0 find x : y : z.

If b is the mean proportional between a and c, prove that `(a^2 - b^2 + c^2)/(a^-2 -b^-2 + c^-2)` = b⁴.

Two numbers are in the ratio of 3: 5. If 8 is added to each number, the ratio becomes 2 : 3. Find the numbers.

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

Divide Rs. 720 between Sunil, Sbhil and Akhil. So that Sunil gets 4/5 of Sohil’s and Akhil’s share together and Sohil gets 2/3 of Akhil’s share.

The ratio between two numbers is 3: 4. If their L.C.M., is 180. Find the numbers.

If (x - 9) : (3x + 6) is the duplicate ratio of 4 : 9, find the value of x.

Find the compound ratio of the following:
If A : B = 4 : 5, B : C = 6 : 7 and C : D = 14 : 15. Find A : D.

Find the compound ratio of the following:
If P : Q = 6 : 7, Q : R = 8 : 9 find P : Q : R.

Find: The duplicate ratio of 7: 9

Find: The triplicate ratio of 3: 7

Find: The sub-duplicate ratio of 256: 625

Find: The sub-triplicate ratio of 216: 343

Find: The reciprocal ratio of 8: 15.

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)`

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

What least number must be added to each of the numbers 5, 11, 19 and 37, so that they are in proportion?

What quantity must be added to each term of the ratio a + b: a - b to make it equal to (a + b)² : (a - b)² ?

The work done by (x – 3) men in (2x + 1) days and the work done by (2x + 1) men in (x + 4) days are in the ratio of 3: 10. Find the value of x.

Find the third proportional to:
x - y, x² - y²

Find the third proportional to:
`a/b + b/c, sqrt(a^2 + b^2)`.

Find the fourth proportional to:
2xy, x², y²

Find the fourth proportional to:
x³ - y², x⁴ + x²y² + y⁴, x - y.

Find the two numbers such that their mean proprtional is 24 and the third proportinal is 1,536.

Using componendo and idendo, find the value of x
`(sqrt(3x + 4) + sqrt(3x - 5))/(sqrt(3x + 4) - sqrt(3x - 5)` = 9

Solve for x: `(3x^2 + 5x + 18)/(5x^2 + 6x + 12) = (3x + 5)/(5x - 6)`.

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

Given that `(a^3 + 3ab^2)/(b^2 + 3a^2b) = (63)/(62)`.
Using Componendo and Dividendo find a : b.

If `(3x + 5y)/(3x - 5y) = (7)/(3)`, find x : y.

Solve for x : `(1 - px)/(1 + px) = sqrt((1 + qx)/(1 - qx)`

Find the value of
`(x + sqrt(3))/(x - sqrt(3)) + (x + sqrt(2))/(x - sqrt(2)), if x = (2sqrt(6))/(sqrt(3) + sqrt(2)`.

If (a – x) : (b – x) be the duplicate ratio of a: b show that:
`(1)/x = (1)/a + (1)/b.`

If a: b with a ≠ b is the duplicate ratio of a + c: b + c, show that c² = ab.

If a : b = 5 : 3, show that (5a + 8b) : (6a – 7b) = 49 : 9.

If `a/b = c/d = e/f`, Prove that each of these ratios is equal to `(a + c + e)/(b + d + f)`.

If `a/(b + c) = b/(c + a) = c/(a + b)`, show that each ratio is equal to `(1)/(2) or -1`.

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

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.

Show, that a, b, c, d are in proportion if:
(6a + 7b) : (6c + 7d) : : (6a - 7b) : (6c - 7d)

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

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

If `(by + cz )/(b^2 + c^2) = (cz + ax)/(c^2 + a^2) = (ax + by)/(a^2 + b^2)` then show that each ratio is equal to `x/a = y/b = z/c`.

If y = `(sqrt(a + 3b) + sqrt(a - 3b))/(sqrt(a + 3b) - sqrt(a - 3b))`, show that 3by² - 2ay + 3b = 0.

If a, b, c are in continued proportion, prove that a : c = (a² + b²) : (b² + c²).

If y = `((p + 1)^(1/3) + (p - 1)^(1/3))/((p + 1)^(1/3) - (p - 1)^(1/3)` find that y³ - 3py² + 3y - p = 0.

If `x/(b + c - a) = y/(c + a - b) = z/(a + b - c)` then show that (b - c)x + (c - a)y + (a - b) z = 0.

If p + r = 2q and `(1)/q + (1)/s = (2)/r`, then prove that p : q = r : s.

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`

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 `a/b = c/d,` show that (9a + 13b) (9c - 13d) = (9c + 13b) (9a - 13d).

If `p/q = r/s`, prove that `(2p + 3q)/(2p - 3q) = (2r + 3s)/(2r - 3s)`.

If a, b, c, d are in continued proportion, prove that (b - c)² + (c - a)² + (d - b)² = (d - a)².

If q is the mean proportional between p and r, prove that
`p^2 - q^2 + r^2 = q^4[(1)/p^2 - (1)/q^2 + (1)/r^2]`.

If `a/b = c/d` Show that a + b : c + d = `sqrt(a^2 + b^2) : sqrt(c^2 + d^2)`.

If a : b = c : d, show that (a - c) b² : (b - d) cd = (a² - b² - ab) : (c² - d² - cd).

If `a/b = c/d = e/f`, prove that `(ab + cd + ef)^2 = (a^2 + c^2 + e^2) (b^2 + d^2 + f^2)`.

If `a/b = c/d = r/f`, prove that `((a^2b^2 + c^2d^2 + e^2f^2)/(ab^3 + cd^3 + ef^3))^(3/2) = sqrt((ace)/(bdf)`

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:
(a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)².

If `x/a = y/b = z/c`, show that `x^3/a^3 - y^3/b^3 = z^3/c^3 = (xyz)/(zbc).`

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

If `a = (b + c)/(2), c = (a + b)/(2)` and b is mean proportional between a and c, prove that `(1)/a + (1)/c = (1)/b`.

if `(3a + 4b)/(3c + 4d) = (3a - 4b)/(3c - 4d)` Prove that `a/b = c/d`.

If a : b = c : d, show that (2a - 7b) (2c + 7d) = (2c - 7d) (2a + 7b).

If `(3x + 4y)/(3u + 4v) = (3x - 4y)/(3u - 4v)`, then show that `x/y = u/v`.

The hypotenuse of a right angled triangle is 3`sqrt(5)`. If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15 cm. Find the length of each side.