R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE chapter 7 - Ratio and Proportion [Latest edition]

Find the ratio between 60 paise and ₹ 1.35

Find the ratio between 1.8 m and 75 cm

Find the ratio between 1.5 kg and 600 g

Find the ratio between 35 min and $$1\frac{3}{4}$$ hrs

If A : B = 5 : 6 and B : C = 9 : 11, find A : C.

If A : B = 5 : 6 and B : C = 9 : 11, find A : B : C.

If $$\mathrm{P}:\mathrm{Q} = 7 : 4$$ and $$\mathrm{Q}:\mathrm{R} = 5 : 14$$, find $$\mathrm{R}:\mathrm{P}$$.

If $$\mathrm{P}:\mathrm{Q} = 7 : 4$$ and $$\mathrm{Q}:\mathrm{R} = 5 : 14$$, find $$\mathrm{P}:\mathrm{Q}:\mathrm{R}$$.

If $$3\mathrm{A} = 5\mathrm{B} = 6\mathrm{C}$$, find $$\mathrm{A}:\mathrm{B}:\mathrm{C}$$.

If $$\frac{\mathrm{A}}{2} = \frac{\mathrm{B}}{3} = \frac{\mathrm{C}}{6}$$, find A : B : C.

If $$a : b = 8 : 5$$, find $$(7a + 5b) : (8a - 9b)$$.

If $$x : y = 3 : 2$$, find $$(5x - 3y) : (7x + 2y)$$.

If $$x : y = 10 : 3$$, find $$(3x^{2} + 2y^{2}) : (3x^{2} - 2y^{2})$$.

If $$a : b = 2 : 5$$, find $$(3a^2 - 2ab + 5b^2) : (a^2 + 7ab - 2b^2)$$.

If $$(5x + 2y) : (7x + 4y) = 13 : 20$$, find $$x : y$$.

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

If $$(6x^2 - xy) : (2xy - y^2) = 6 : 1$$, find $$x : y$$.

If $$(4x^2 - 3y^2) : (2x^2 + 5y^2) = 12 : 19$$, find $$x : y$$.

If $$x^2 + 4y^2 = 4xy$$, find $$x : y$$.

If $$10x^2 - 23xy + 9y^2 = 0$$, find $$x : y$$.

A ratio is equal to $$3 : 4$$. If its consequent is 144, what is its antecedent?

Two numbers are in the ratio $$8 : 13$$. If 14 is added to each of the numbers, the ratio becomes $$2 : 3$$. Find the numbers.

Two numbers are in the ratio $$5 : 7$$. If 8 is subtracted from each, the ratio becomes $$3 : 5$$. Find the numbers.

What least number must be added to each term of the ratio $$5 : 7$$ to make it $$8 : 9$$?

Out of the monthly income of ₹45000, Rahul spends ₹31500 and the rest he saves. Find the ratio of his

1. income to expenditure 
2. income to savings 
3. savings to expenditure.

The cost of making an umbrella is divided between material, labour and overheads in the ratio $$6 : 4 : 1$$. If the material costs ₹132, find the cost of production of an umbrella.

Divide ₹6720 in the ratio $$5 : 3$$.

Divide ₹11620 among A, B and C in the ratio $$35 : 28 : 20$$.

Divide ₹782 among P, Q and R in the ratio $$\frac{1}{2} : \frac{2}{3} : \frac{3}{4}$$.

[Hint : Given ratio = 6 : 8 : 9.]

If ₹5100 be divided among A, B, C in such a way that A gets $$\frac{2}{3}$$ of what B gets and B gets $$\frac{1}{4}$$ of what C gets, find their respective shares.

Divide ₹8300 among A, B and C such that 4 times A's share, 5 times B's share and 7 times C's share may all be equal.

A sum of money is divided between A and B in the ratio $$6 : 11$$. If B's share is ₹7315, find (i) A's share (ii) the total amount of money.

The ages of Tanvy and Divya are in the ratio $$5 : 7$$. Five years hence, their ages will be in the ratio $$3 : 4$$. Find their present ages.

One year ago, the ratio of Amit’s and Arun’s ages was $$6 : 7$$ respectively. Four years hence, their ages will be in the ratio $$7 : 8$$. How old is Amit?

Reena reduces her weight in the ratio $$5 : 4$$. What is her weight now, if originally it was $$70 \text{ kg}$$?

$$68 \text{ kg}$$ of a mixture contains milk and water in the ratio $$27 : 7$$. How much more water is to be added to this mixture to get a new mixture containing milk and water in the ratio $$3 : 1$$?

A mixture contains milk and water in the ratio $$5 : 1$$. On adding 5 litres of water, the ratio of milk to water becomes $$5 : 2$$. Find the quantity of milk in the original mixture.

In an examination, the ratio of passes to failures was $$4 : 1$$. Had 30 less appeared and 20 less passed, the ratio of passes to failures would have been $$5 : 1$$. How many students appeared for the examination?

Find the angles of a triangle which are in the ratio $$5 : 4 : 3$$.

The sides of a triangle are in the ratio $$\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$$ and its perimeter is $$91 \text{ cm}$$. Find the lengths of the sides of the triangle.

In a school, the boys and girls are in the ratio $$9 : 5$$. If there are 425 girls, what is the total number of students in the school?

Compare the following ratios: 

(7 : 9) and (11 : 16) 

Compare the following ratios: 

(19 : 25) and (17 : 20)

Compare the following ratios: 

$$\left(\frac{1}{2} : \frac{1}{5}\right)$$ and (5 : 2)

Arrange the following ratios in descending order of magnitude: 

(5 : 6), (8 : 9), (13 : 18) and (19 : 24)

Arrange the following ratios in descending order of magnitude: 

(6 : 7), (13 : 14), (19 : 21) and (23 : 28)

Arrange the following ratios in descending order of magnitude: 

(7 : 12), (9 : 16), (13 : 20) and (5 : 8)

Arrange the following ratios in ascending order of magnitude:

(4 : 9), (6 : 11), (7 : 13) and (27 : 50)

Arrange the following ratios in ascending order of magnitude: 

(2 : 3), (8 : 15), (11 : 12) and (7 : 16)

Arrange the following ratios in ascending order of magnitude:

(3 : 5), (4 : 9), (5 : 11) and (10 : 17)

If (3a + 2b) : (5a + 3b) = 18 : 29. Find a : b

Find $$x$$, when $$3 : 4 :: 2.4 : x$$.

Find $$x$$, when $$1:3::x:7$$.

Find $$x$$, when $$x : 1.5 :: 3 : 5$$.

Find the fourth proportional to 3, 8 and 21.

Find the fourth proportional to 1.4, 3.2 and 7.

Find the fourth proportional to 1.5, 4.5 and 3.6.

Find the fourth proportional to $$a^2$$, $$ab$$ and $$b^2$$.

Find the fourth proportional to $$(a^2 - ab + b^2)$$, $$(a^3 + b^3)$$ and $$(a - b)$$.

Find the third proportional to 9 and 6.

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

Find the third proportional to 1.6 and 2.4.

Find the third proportional to $$(2 + \sqrt{3})$$ and $$(5 + 4\sqrt{3})$$.

Find the third proportional to $$\left(\frac{a}{b} +\frac{b}{a}\right)$$ and $$\sqrt{a^2 + b^2}$$.

Find the mean proportion between 28 and 63.

Find the mean proportion between 2.5 and 0.9.

Find the mean proportion between 6.25 and 1.6.

Find the mean proportion between $$(\sqrt{26} - \sqrt{17})$$ and $$(\sqrt{26} + \sqrt{17})$$.

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

6 is the mean proportion between two numbers x and y and 48 is the third proportional of x and y. Find the numbers.

What least number must be added to each of the numbers 5, 11, 19 and 37, so that the resulting numbers are proportional.

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

What least number must be subtracted from each of the numbers 23, 30, 57 and 78, so that the remainders are in proportion?

If $$(x - 2)$$, $$(x + 2)$$, $$(2x + 1)$$ and $$(2x + 19)$$ are in proportion, find the value of $$x$$.

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

If $$(x + 5)$$ is the geometric mean between $$(x + 2)$$ and $$(x + 9)$$, find the value of $$x$$.

Find two numbers whose mean proportion is 36 and the third proportional is 288.

If $$a : b :: c : d$$, prove that $$(a^2 + ab) : (c^2 + cd) = (b^2 - 2ab) : (d^2 - 2cd)$$.

If $$a : b :: c : d$$, prove that $$(a^2 + b^2) : (c^2 + d^2) = (ab + ad - bc) : (cd - ad + bc)$$.

If $$a : b :: c : d$$, prove that $$(a^2 + ac + c^2) : (a^2 - ac + c^2) = (b^2 + bd + d^2) : (b^2 - bd + d^2)$$.

If $$a : b :: c : d$$, show that $$\frac{a + b}{c + d} = \frac{\sqrt{2a^2 + 7b^2}}{\sqrt{2c^2 + 7d^2}}$$.

If $$a : b :: c : d$$, show that $$\frac{ma^2 + nc^2}{mb^2 + nd^2} = \frac{\sqrt{a^4 + c^4}}{\sqrt{b^4 + d^4}}$$.

If $$a : b :: c : d$$, show that $$\frac{a^2 + ab + b^2}{a^2 - ab + b^2} = \frac{c^2 + cd + d^2}{c^2 - cd + d^2}$$.

If $$a : b :: c : d$$, show that $$\frac{(a + c)^3}{(b + d)^3} = \frac{a(a - c)^2}{b(b - d)^2}$$.

[Hint : Use k-method.]

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{x^2 + y^2 + z^2}{a^2 + b^2 + c^2} = \left(\frac{px + qy + rz}{pa + qb + rc}\right)^2$$.

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}$$.

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{x^3}{a^2} + \frac{y^3}{b^2} + \frac{z^3}{c^2} = \frac{(x + y + z)^3}{(a + b + c)^2}$$.

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{ax - by}{(a + b)(x - y)} + \frac{by - cz}{(b + c)(y - z)} + \frac{cz - ax}{(c + a)(z - x)} = 3$$.

[Hint : Use k-method in each.]

If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$, prove that $$(b^2 + d^2 + f^2)(a^2 + c^2 + e^2) = (ab + cd + ef)^2$$.

If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$ prove that $$\frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{ace}{bdf}$$.

If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$ prove that $$\left(\frac{a^2}{b^2} + \frac{c^2}{d^2} + \frac{e^2}{f^2}\right) = \left(\frac{ac}{bd} + \frac{ce}{df} + \frac{ae}{bf}\right)$$

If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$ prove that $$(bdf) \cdot \left(\frac{a + b}{b} + \frac{c + d}{d} + \frac{e + f}{f}\right)^3 = 27(a + b)(c + d)(e + f)$$.

[Hint : Use k-method in each.]

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

If $$a, b, c$$ are in continued proportion, prove that $$\frac{a + b + c}{a - b + c} = \frac{(a + b + c)^2}{(a^2 + b^2 + c^2)}$$.

If $$a, b, c$$ are in continued proportion, prove that $$\frac{a^2 + ab + b^2}{b^2 + bc + c^2} = \frac{a}{c}$$.

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

If $$a, b, c$$ are in continued proportion, prove that $$a^2b^2c^2(a^{-3} + b^{-3} + c^{-3}) = (a^3 + b^3 + c^3)$$.

If $$a, b, c$$ are in continued proportion, prove that $$ad(c^2 + d^2) = c^3(b + d)$$.

\[ [\textbf{Hint :}\ \dfrac{a}{b} = \dfrac{b}{c} = k \Rightarrow b = ck \ \textit{and} \ a = ck^{2}\,] \]

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

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

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

If $$a, b, c, d$$ are in continued proportion, prove that $$\left(\frac{a - b}{c} + \frac{a - c}{b}\right)^2 - \left(\frac{d - b}{c} + \frac{d - c}{b}\right)^2 = (a - d)^2 \left(\frac{1}{c^2} + \frac{1}{b^2}\right)$$.

\[ [\textbf{Hint :}\ \dfrac{a}{b} = \dfrac{b}{c} = \dfrac{c}{d} = k \Rightarrow c = dk,\ b = dk^{2} \ \text{and} \ a = dk^{3}.\,] \]

If $$ax = by = cz$$, prove that $$\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} = \frac{bc}{a^2} + \frac{ca}{b^2} + \frac{ab}{c^2}$$.

\[ [\textbf{Hint :}\ ax = by = cz,\ \Rightarrow \ \dfrac{x}{bc} = \dfrac{y}{ca} = \dfrac{z}{ab} = k \Rightarrow x = kbc,\ y = kca \ \textit{and} \ z = kab.] \]

If $$\frac{x}{b + c - a} = \frac{y}{c + a - b} = \frac{z}{a + b - c}$$, prove that each ratio is equal to $$\left(\frac{x + y + z}{a + b + c}\right)$$. Also, show that $$(b - c)x + (c - a)y + (a - b)z = 0$$.

\[ \begin{array}{l} [\textbf{Hint :}\ \text{Each ratio} = \dfrac{x + y + z}{(b + c - a) + (c + a - b) + (a + b - c)} = \dfrac{x + y + z}{a + b + c}. \\[16pt] \text{Let each ratio be equal to } k\text{. Then,} \\[8pt] x = (b + c - a)k,\ y = (c + a - b)k \ \text{and} \ z = (a + b - c)k.] \end{array} \]

If $$b$$ is the mean proportion between $$a$$ and $$c$$, show that $$\frac{a^4 + a^2b^2 + b^4}{b^4 + b^2c^2 + c^4} = \frac{a^2}{c^2}$$.

[Hint : Substitute \[b^{2} = ac\] in L.H.S. to get R.H.S.]

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

If $$a : b = c : d$$, prove that $$(3a + 2b) : (3a - 2b) = (3c + 2d) : (3c - 2d)$$.

If $$(3a + 5b) : (3a - 5b) = (3c + 5d) : (3c - 5d)$$, prove that $$a : b = c : d$$.

If $$(4a^2 + 7b^2) : (4a^2 - 7b^2) = (4c^2 + 7d^2) : (4c^2 - 7d^2)$$, prove that $$a : b = c : d$$.

If $$(ma + nb) : (mc + nd) = (ma - nb) : (mc - nd)$$, prove that $$a : b = c : d$$.

If $$\frac{5x + 6y}{5x - 6y} = \frac{5u + 6v}{5u - 6v}$$ show that $$\frac{x}{y} = \frac{u}{v}$$.

[Hint : We have $$\frac{5x + 6y}{5x - 6y} = \frac{5u + 6v}{5u - 6v}$$. Apply Componendo & Dividendo.]

If $$x = \frac{6ab}{a + b}$$, prove that $$\left(\frac{x + 3a}{x - 3a} + \frac{x + 3b}{x - 3b}\right) = 2$$.

[Hint : We have $$\frac{x}{3a} = \frac{2b}{a + b}$$ and $$\frac{x}{3b} = \frac{2a}{a + b}$$. Apply Componendo & Dividendo on each.]

If $$\frac{x^3 + 3x}{3x^2 + 1} = \frac{341}{91}$$, prove that $$x = 11$$.

[Hint : By componendo and dividendo, $$\frac{(x + 1)^3}{(x - 1)^3} = \frac{432}{250} = \frac{216}{125} = \left(\frac{6}{5}\right)^3 \Rightarrow \frac{x + 1}{x - 1} = \frac{6}{5}$$]

If $$\frac{\sqrt{x + 2} + \sqrt{x - 3}}{\sqrt{x + 2} - \sqrt{x - 3}} = 5$$, prove that $$x = 7$$.

If $$\frac{\sqrt{x + 5} + \sqrt{x - 16}}{\sqrt{x + 5} - \sqrt{x - 16}} = \frac{7}{3}$$, prove that $$x = 20$$.

If $$\frac{\sqrt{3x} + \sqrt{2x - 1}}{\sqrt{3x} - \sqrt{2x - 1}} = 5$$, prove that $$x = \frac{3}{2}$$.

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

$$\frac{2x + \sqrt{4x^2 - 1}}{2x - \sqrt{4x^2 - 1}} = 4$$

Using properties of proportion solve for $$x$$, given $$\frac{\sqrt{5x} + \sqrt{2x - 6}}{\sqrt{5x} - \sqrt{2x - 6}} = 4$$.

Using componendo and dividendo solve for x:

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

If $$16\left(\frac{a - x}{a + x}\right)^3 = \left(\frac{a + x}{a - x}\right)$$, prove that $$x = \frac{a}{3}$$.

\[ [\textbf{Hint :}\ \text{We have}\ \left( \dfrac{a + x}{a - x} \right)^{4} = 16 = 2^{4} \Rightarrow \dfrac{a + x}{a - x} = 2.\,] \]

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

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

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

If $$x = \frac{\sqrt{b+3a} + \sqrt{b-3a}}{\sqrt{b+3a} - \sqrt{b-3a}}$$, prove that $$3ax^2 - 2bx + 3a = 0$$.

If $$\frac{\sqrt{2a+3b} + \sqrt{2a-3b}}{\sqrt{2a+3b} - \sqrt{2a-3b}}$$ prove that $$3bx^2 - 4ax + 3b = 0$$.

If $$x = \frac{\sqrt[3]{m+1} + \sqrt[3]{m-1}}{\sqrt[3]{m+1} - \sqrt[3]{m-1}}$$, prove that $$x^3 - 3x^2m + 3x - m = 0$$.

\[ \begin{array}{l} [\textbf{Hint :}\ \text{By componendo and dividendo, we have}\ \dfrac{x + 1}{x - 1} = \dfrac{\sqrt[3]{m + 1}}{\sqrt[3]{m - 1}} \Leftrightarrow \dfrac{(x + 1)^{3}}{(x - 1)^{3}} = \dfrac{m + 1}{m - 1}. \\[16pt] \text{Now, apply componendo and dividendo.}] \end{array} \]

What quantity must be added to each term of the ratio $$a : b$$ to make it $$(c : d)$$?

If $$\frac{a+3b+2c+6d}{a-3b+2c-6d} = \frac{a+3b-2c-6d}{a-3b-2c+6d}$$, prove that $$\frac{a}{b} = \frac{c}{d}$$.

\[ \begin{array}{l} \left[ \textbf{Hint :}\ \text{We have}\ \dfrac{(a + 3b) + (2c + 6d)}{(a + 3b) - (2c + 6d)} = \dfrac{(a - 3b) + (2c - 6d)}{(a - 3b) - (2c - 6d)} \right. \\[16pt] \text{Now apply componendo and dividendo.}] \end{array} \]

If $$\frac{2a+2b-3c-3d}{2a-2b-3c+3d} = \frac{a+b-4c-4d}{a-b-4c+4d}$$, prove that $$\frac{a}{b} = \frac{c}{d}$$.

\[ \begin{array}{l} \left[ \textbf{Hint :}\ \text{We have}\ \dfrac{(2a - 3c) + (2b - 3d)}{(2a - 3c) - (2b - 3d)} = \dfrac{(a - 4c) + (b - 4d)}{(a - 4c) - (b - 4d)} \right. \\[16pt] \text{Now apply componendo and dividendo.}] \end{array} \]

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

\[ \begin{array}{l} [\textbf{Hint :}\ \text{We have}\ \dfrac{(a + b) + (c + d)}{(a + b) - (c + d)} = \dfrac{(a - b) + (c - d)}{(a - b) - (c - d)}. \\[16pt] \text{Now apply componendo and dividendo.}] \end{array} \]

Given : $$\frac{x^3+12x}{6x^2+8} = \frac{y^3+27y}{9y^2+27}$$. Using componendo, find $$x : y$$. 

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

Using properties of proportion, find the value of ‘x’:

`(6x^2 + 3x − 5)/(3x − 5) = (9x^2 + 2x + 5)/(2x + 5); x ≠ 0`

Which of the following is the ratio between a number and the number obtained by adding one-fifth of that number to it?

Govind spends ₹8100 in buying some jeans at ₹1200 each and some shirts at ₹300 each. The ratio of the number of shirts to that of jeans, when the maximum possible number of jeans is purchased is ______. 

Which of the following represents xy = 64?

If a carton, containing a dozen mirrors is dropped, then which of the following cannot be the ratio of broken mirrors to unbroken mirrors?

A mixture of paint is prepared by mixing 2 parts of red pigments with 5 parts of the base. Using the given information in the following table, find the values of a, b and c to get the required mixture of paint.

If (x + 1) : 8 = 3.75 : 7, then the value of x is ______.

If `sqrt(2) : (1 + sqrt(3)) :: sqrt(6) : x`, then x is equal to ______.

0.5 of a number is equal to 0.07 of another. The ratio of the numbers is ______.

25% of x is equal to 35% of y, then x : y equals.

If `A = 1/3 B` and `B = 1/2 C`, then A : B : C is equal to ______.

If 2p = 3q = 4r then p : q : r is equal to ______.

If `a/3 = b/4 = c/7`, then `(a + b + c)/c` = ?

If x, 5.4, 5, 9 are in proportion, then x is equal to ______.

The fourth proportional to 5, 8, 15 is ______.

The fourth proportional to 0.12, 0.21 and 8 is ______.

The third proportional to 38 and 15 is ______.

The third proportional to (x² – y²) and (x – y) is ______.

The mean proportion between 9 and 16 is ______.

The mean proportion between 0.02 and 0.32 is ______.

The mean proportion between `(3 + sqrt(2))` and `(12 - sqrt(32))` is ______.

The mean proportion between x and y is 6. The third proportional to x and y is 48. Then x : y equals ______.

The ratio between the third proportional to 12 and 30 and mean proportion of 9 and 25, is ______.

When 30% of one number is subtracted from another number, the second number reduces to its four-fifths. What is the ratio of the first to the second number?

Five mangoes and four guavas cost as much as three mangoes and seven guavas. What is the ratio of the cost of one mango to the cost of one guava?

Of 132 examinees of a certain class, the ratio of passed to failed students is 9 : 2. If 4 more students passed, what would have been the ratio of passed to failed students?

There are three boxes – P, Q and R, containing marbles in the ratio 1 : 2 : 3. Total number of marbles is 60. The above ratio can be changed to 3 : 4 : 5 by transferring:

If x² + y² = 4xy, then x : y is ______.

If 3A = 5B and 4B = 6C, then A : C is equal to ______.

If A : B = 7 : 9 and B : C = 5 : 4, then A : B : C is ______.

If 8a = 9b, then the ratio of `a/9` to `b/8` is ______.

If a, b, c and d are proportional, then `(a + b)/(a − b)` is equal to ______.

If x : y = 3 : 2, then the ratio (2x² + 3y²) : (3x² – 2y²) is equal to ______.

If (5a + 3b) : (2a – 3b) = 23 : 5, then the value of a : b is ______.

If (4x² – 3y²) : (2x² + 5y²) = 12 : 19 then x : y is ______.

If a : b = b : c then a⁴ : b⁴ would be equal to ______.

If a : b : c = 2 : 3 : 4, then `1/a : 1/b : 1/c` is equal to ______.

If `1/x : 1/y : 1/z = 2 : 3 : 5`, then x : y : z is equal to ______.

If (x + y) : (x – y) = 4 : 1, then (x² + y²) : (x² – y²) is ______.

If a : b : c = 2 : 3 : 4 and 2a – 3b + 4c = 33, then the value of c is ______.

If x, y, z are in continued proportion, then (y² + z²) : (x² + y²) is equal to ______.

If `a/b = b/c = c/d`, then `(b^3 + c^3 + d^3)/(a^3 + b^3 + c^3)` is equal to ______.

If a : b = c : d, then `(ma + nc)/(mb + nd)` is equal to ______.

In an alloy, the ratio of copper and zinc is 5 : 2. If 1.250 kg of zinc is mixed in 17 kg 500 g of alloy, then the ratio of copper and zinc in the alloy will be ______.

A sum of ₹6400 is divided among three workers in the ratio `3/5 : 2 : 5/3`. The share of the second worker is ______.

If a sum of ₹x is divided between A and B in the ratio `a/b : c/d`, then A gets :

The ratio of the number of boys and girls in a school of 720 students is 7 : 5. How many more girls should be admitted to make the ratio 1 : 1?

Two numbers are in the ratio 7 : 11. If 7 is added to each of the numbers, the ratio becomes 2 : 3. The smaller number is ______.

What must be subtracted from each of 7, 9, 11 and 15, so that the resulting numbers are in proportion?

If p, q and r are in continued proportion, then:

The ratio of milk to water in 80 litres of a mixture is 7 : 3. The quantity of water (in litres) to be added to it to make the ratio 2 : 1, is ______.

What number must be added to each of the numbers 7, 11 and 19 so that the resulting numbers may be in continued proportion?

Two numbers are in the ratio of 3 : 5. If 9 is subtracted from each, they are in the ratio of 12 : 23. The larger number is ______.

What number must be added to the terms of 3 : 5 to make the ratio 5 : 6?

Six numbers a, b, c, d, e, f are such that ab = 1, `bc = 1/2`, cd = 6, de = 2 and `ef = 1/2`. What is the value of (ad : be : cf)?

If `a = (4xy)/(x + y)`, then `((a + 2x)/(a - 2x) + (a + 2y)/(a - 2y))` equals ______.

If `(sqrt(1 + x) + sqrt(1 - x))/(sqrt(1 + x) - sqrt(1 - x)) = a/b`, then x equals ______.

If `x = (sqrt(2a + 1) + sqrt(2a - 1))/(sqrt(2a + 1) - sqrt(2a - 1))`, then which of the following is true?

If `a/(b + c) = b/(c + a) = c/(a + b)`, then each ratio is equal to ______.

If b is the mean proportion between a and c, then `(a^2 - b^2 + c^2)/(a^-2 - b^-2 + c^-2)` is equal to ______.

If b is the mean proportion between a and c, then the mean proportion between (a² + b²) and (b² + c²) is ______.

Assertion (A): If $$\frac{a}{b} = \frac{c}{d}$$, then $$\frac{a + c}{b + d} = \frac{a}{b}$$.

Reason (R): If two or more than two ratios are equal, then, each ratio = $$\frac{\text{sum of antecedents}}{\text{sum of consequents}}$$

Assertion (A): The mean proportion between $$a^2b$$ and $$\frac{1}{b}$$ is $$\frac{a}{b}$$.

Reason (R): The mean proportion between x and y is given by $$\sqrt{xy}$$.

Assertion (A): If $$2x = 3y$$ and $$4y = 5z$$, then $$x : y : z$$ is equal to $$15 : 10 : 8$$.

Reason (R): If $$x = y$$ and $$y = z$$, then we cannot find the ratio $$x : y : z$$.

The mean proportion between two numbers is 6 and their third proportion is 48. Find the two numbers.

The profit in rupees in a local restaurant and the number of customers who visited the restaurant are tabulated below for each week for one month.

Find:

1. if the number of customers and profit per week in continued proportion or not? Justify your answer.
2. the value of x and y.