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

If A : B = 7 : 5 and B : C = 7 : 5, then A : C is ______.

If `(x^2 - 1)/(x^2 + 1) = 3/5`, the value of x is ______.

If (2x + 3y) : (3x + 2y) = 4 : 3; then x : y is ______.

Two numbers are in the ratio 5 : 8. If 10 is subtracted from each number, the ratio becomes 4 : 7. The numbers are ______.

The compound ratio of x – y : x + y and (x + y)² : x² – y² is ______.

If the sub-duplicate ratio of 6x² : 24y² is ______.

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

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

Divide Rs. 1290 in to 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 work done by (3x − 1) men in (2x + 3) days and the work done by (3x − 4) men in (2x + 1) days are in the ratio 4 : 3. Find the value of x.

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

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.

If x, 2, 10 and y are in continued proportion, the values of x and y are respectively:

If x : y = y : z, then x² : y² is ______.

The mean proportion between `3 + 2sqrt(2)` and `3 - 2sqrt(2)` is ______.

If 2x, 9 and 18 are in continued proportion, the value of x is ______.

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.

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.

(That is, if the three quantities are x, y, and z such that x : y = y : z, prove that \[x : z = x^2 : y^2\].)

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 x + y + z = 0, the value of `x/(y + z)` is ______.

If a, b, c and d are in proportion, the value of `(8a^2 - 5b^2)/(8c^2 - 5d^2)` is equal to ______.

`(x + y)/z = (y + z)/x = (z + x)/y` is equal to ______.

If `(x^2 - 4)/(x^2 + 4) = 3/5`, the value of x is ______.

If `x/(a + b - c) = y/(b + c - a) = z/(c + a - b) = 5` and a + b + c = 7; the value of x + y + z is ______.

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

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

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

`(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.

13. From the following tables, find the values of 𝑎, 𝑏 and 𝑐:

For the following table, find the values of 𝑎, 𝑏 and 𝑐:

The mean proportional to `sqrt(3) + sqrt(2)` and `sqrt(3) - sqrt(2)` is ______.

If (a + b) : (a – b) = 13 : 3 ; a : b is ______.

The table, given below, shows the values of x and y, where x is proportional (directly proportional) to y.

The values of A and B are:

If (m + n) : (n − m) = 5 : 2, m : n is ______.

If x = y, the value of (3x + y) : (5x – 3y) is ______.

𝑥 : 𝑦 = 3 : 2 and \[(𝑥^2 + 𝑦^2) : (𝑥^2 − 𝑦^2)\].

Assertion (A): The value of ratio \[(𝑥^2 + 𝑦^2) : (𝑥^2 − 𝑦^2)\] = 13 : 5.

Reason (R): 𝑥 : 𝑦 = 3 : 2 \[\Rightarrow \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \frac{(3k)^{2} + (2k)^{2}}{(3k)^{2} - (2k)^{2}} = \frac{13}{5};\ k \neq 0.\]

\[\frac{(𝑥+𝑦)^4}{(𝑥−𝑦)^4} =\frac{16}1\].

Assertion (A): 𝑥 : 𝑦 = 3 : 1.

Reason (R): \[\frac{𝑥 + 𝑦}{𝑥−𝑦} = 21\] and \[\frac{𝑥+𝑦+𝑥−𝑦} {𝑥+𝑦−𝑥+𝑦} = \frac{2 + 1}{2−1}.\]

Two irrational numbers \[\sqrt6\] and \[\sqrt5\].

Statement (1): The mean proportion of \[\sqrt6\] and \[\sqrt5\] is \[\frac{\sqrt6 +\sqrt5}2\].

Statement (2): The mean proportion of two positive real numbers 𝑥 and 𝑦 is \[\sqrt{𝑥×𝑦}\].

Numbers 𝑎,𝑏 and 𝑐 are in continued proportion.

Statement (1): (𝑎 + 𝑏 + 𝑐)⁢(𝑎 − 𝑏 + 𝑐) = \[𝑎^2+𝑏^2+𝑐^2\]

Statement (2): \[𝑏^2 = 𝑎⁢𝑐\] and (𝑎 +𝑏 + 𝑐) ⁢(𝑎 − 𝑏 + 𝑐) = \[(𝑎+𝑐)^2 −𝑏^2\]

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

If 5x + 6y : 8x + 5y = 8 : 9, 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 `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)`

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

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

If 𝑎 : 𝑏 : : 𝑐 : 𝑑 then prove that: 𝑎 : (𝑎 − 𝑏) : : 𝑐 : (𝑐 − 𝑑).

Using properties of proportion solve: 

`(k^5 + x^5)/(x^5 − k^5) = 122/121`

Using properties of proportion solve:

\[ \Rightarrow \frac{x^2-x+1}{x^2+x+1}=\frac{112(1-x)}{104(1+x)} \]

If 𝑥 ≠ 𝑦 and \[\frac{𝑥^2−𝑥+1}{𝑦^2−𝑦+1} = \frac{𝑥^2+𝑥+1}{𝑦^2+𝑦+1}\], prove that 𝑥⁢𝑦 = 1. Use properties of proportion.

Amit runs a grocery store in a busy neighbourhood market. He stocks two varieties of lentils.

• The premium quality sells at ₹ 280 per kg.
• The standard quality sells at ₹ 240 per kg.

Customers in his area are price-conscious, so Amit wants to create a mixture of the two varieties that appears high quality but is also affordable. He decides to sell the mixture at ₹ 265 per kg.

In what ratio should the ₹ 240/kg and ₹ 280/kg varieties be mixed to obtain the mixture costing ₹ 265 per kg?

In a school chemistry lab, the lab assistant has two bottles of acid solution on his shelf:

• Solution A: 55% acid concentration
• Solution B: 80% acid concentration

The science teacher wants 60% a acid solution for tomorrow's experiment that is to be used by the students.

In what ratio should the 55% solution and the 80% solution be mixed to obtain a 60% acid solution?