Mathematics · Textbook solutions

Ratio and Proportion

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 147 questions

Ratio, Proportion and Properties of Ratio

6 q

Solved Examples

Worked · 6
  1. 4.1 SolvedEx.1
    The proportion of compounds of nitrogen, phosphorous and potassium in certain fertilizer is 18 : 18 : 10. Here compound of nitrogen is 18%, compound of phosphorous is 18% and that of potassium is 10%. Remaining part is of other substances. Find the weight of each of the above compounds in 20 kg of fertilizer.
  2. 4.1 SolvedEx.2
    The ratio of ages of Seema and Rajashree is 3 : 1. The ratio of ages of Rajashree and Atul is 2 : 3. Then find the ratio of ages of Seema, Rajashree and Atul.
  3. 4.1 SolvedEx.3
    The length of a rectangular field is 1.2 km and its breadth is 400 metre. Find the ratio of length to breadth.
  4. 4.1 SolvedEx.4
    The ratio of expenditure and income of Mahesh is 3 : 5. Find the percentage of expenses to his income.
  5. 4.1 SolvedEx.5
    The ratio of number of mango trees to chikoo trees in an orchard is 2 : 3. If 5 more trees of each type are planted the ratio of trees would be 5 : 7. Then find the number of mango and chickoo trees in the orchard.
  6. 4.1 SolvedEx.6
    The ratio of two numbers is 5 : 7. If 40 is added in each number, then the ratio becomes 25 : 31. Find the numbers.

Practice Set 4.1

23 q
  1. From the following pairs of numbers, find the reduced form of ratio of first number to second number.
    Ex 4.1 Q1(i)
    72, 60
  2. Ex 4.1 Q1(ii)
    38, 57
  3. Ex 4.1 Q1(iii)
    52, 78
  4. Find the reduced form of the ratio of the first quantity to second quantity.
    Ex 4.1 Q2(i)
    700 ₹, 308 ₹
  5. Ex 4.1 Q2(ii)
    14 ₹, 12 ₹ 40 paise.
  6. Ex 4.1 Q2(iii)
    5 litre, 2500 ml
  7. Ex 4.1 Q2(iv)
    3 years 4 months, 5 years 8 months
  8. Ex 4.1 Q2(v)
    3.8 kg, 1900 gm
  9. Ex 4.1 Q2(vi)
    7 minutes 20 seconds, 5 minutes 6 seconds.
  10. Express the following percentages as ratios in the reduced form.
    Ex 4.1 Q3(i)
    75:10075 : 100
  11. Ex 4.1 Q3(ii)
    44:10044 : 100
  12. Ex 4.1 Q3(iii)
    6.25%6.25\%
  13. Ex 4.1 Q3(iv)
    52:10052 : 100
  14. Ex 4.1 Q3(v)
    0.64%0.64\%
  15. Ex 4.1 Q4
    Three persons can build a small house in 8 days. To build the same house in 6 days, how many persons are required?
  16. Convert the following ratios into percentage.
    Ex 4.1 Q5(i)
    15:2515 : 25
  17. Ex 4.1 Q5(ii)
    47:5047 : 50
  18. Ex 4.1 Q5(iii)
    710\dfrac{7}{10}
  19. Ex 4.1 Q5(iv)
    546600\dfrac{546}{600}
  20. Ex 4.1 Q5(v)
    716\dfrac{7}{16}
  21. Ex 4.1 Q6
    The ratio of ages of Abha and her mother is 2 : 5. At the time of Abha's birth her mothers age was 27 year. Find the present ages of Abha and her mother.
  22. Ex 4.1 Q7
    Present ages of Vatsala and Sara are 14 years and 10 years respectively. After how many years the ratio of their ages will become 5 : 4?
  23. Ex 4.1 Q8
    The ratio of present ages of Rehana and her mother is 2 : 7. After 2 years, the ratio of their ages will be 1 : 3. What is Rehana's present age ?

Comparison of Ratios

4 q

Solved Examples

Worked · 4
  1. Compare the following pairs of ratios.
    4.2 SolvedEx.1
    49\dfrac{4}{9}, 78\dfrac{7}{8}
  2. 4.2 SolvedEx.2
    138\dfrac{\sqrt{13}}{\sqrt{8}}, 75\dfrac{\sqrt{7}}{\sqrt{5}}
  3. 4.2 SolvedEx.3
    If aa and bb are integers and a<ba < b, b>1b > 1 then compare a−1b−1\dfrac{a-1}{b-1}, a+1b+1\dfrac{a+1}{b+1}.
  4. 4.2 SolvedEx.4
    If a:b=2:1a : b = 2 : 1 and b:c=4:1b : c = 4 : 1 then find the value of (a432b2c2)3\left(\dfrac{a^4}{32b^2c^2}\right)^3.

Practice Set 4.2

20 q
  1. Using the property ab=akbk\dfrac{a}{b} = \dfrac{ak}{bk}, fill in the blanks substituting proper numbers in the following.
    Ex 4.2 Q1(i)
    57=⋯28=35⋯=⋯3.5\dfrac{5}{7} = \dfrac{\cdots}{28} = \dfrac{35}{\cdots} = \dfrac{\cdots}{3.5}
  2. Ex 4.2 Q1(ii)
    914=4.5⋯=⋯42=⋯3.5\dfrac{9}{14} = \dfrac{4.5}{\cdots} = \dfrac{\cdots}{42} = \dfrac{\cdots}{3.5}
  3. Find the following ratios.
    Ex 4.2 Q2(i)
    The ratio of radius to circumference of the circle.
  4. Ex 4.2 Q2(ii)
    The ratio of circumference of circle with radius rr to its area.
  5. Ex 4.2 Q2(iii)
    The ratio of diagonal of a square to its side, if the length of side is 7 cm.
  6. Ex 4.2 Q2(iv)
    The lengths of sides of a rectangle are 5 cm and 3.5 cm. Find the ratio of its perimeter to area.
  7. Compare the following pairs of ratios.
    Ex 4.2 Q3(i)
    53\dfrac{\sqrt{5}}{3}, 37\dfrac{3}{\sqrt{7}}
  8. Ex 4.2 Q3(ii)
    3557\dfrac{3\sqrt{5}}{5\sqrt{7}}, 63125\dfrac{\sqrt{63}}{\sqrt{125}}
  9. Ex 4.2 Q3(iii)
    518\dfrac{5}{18}, 17121\dfrac{17}{121}
  10. Ex 4.2 Q3(iv)
    8048\dfrac{\sqrt{80}}{\sqrt{48}}, 4527\dfrac{\sqrt{45}}{\sqrt{27}}
  11. Ex 4.2 Q3(v)
    9.25.1\dfrac{9.2}{5.1}, 3.47.1\dfrac{3.4}{7.1}
  12. Solve the following.
    Ex 4.2 Q4(i)
    □\squareABCD is a parallelogram. The ratio of ∠A\angle A and ∠B\angle B of this parallelogram is 5 : 4. Find the measure of ∠B\angle B.
  13. Ex 4.2 Q4(ii)
    The ratio of present ages of Albert and Salim is 5 : 9. Five years hence ratio of their ages will be 3 : 5. Find their present ages.
  14. Ex 4.2 Q4(iii)
    The ratio of length and breadth of a rectangle is 3 : 1, and its perimeter is 36 cm. Find the length and breadth of the rectangle.
  15. Ex 4.2 Q4(iv)
    The ratio of two numbers is 31 : 23 and their sum is 216. Find these numbers.
  16. Ex 4.2 Q4(v)
    If the product of two numbers is 360 and their ratio is 10 : 9, then find the numbers.
  17. If a:b=3:1a : b = 3 : 1 and b:c=5:1b : c = 5 : 1 then find the value of the following.
    Ex 4.2 Q5(i)
    (a315b2c)3\left(\dfrac{a^3}{15b^2c}\right)^3
  18. Ex 4.2 Q5(ii)
    a27bc\dfrac{a^2}{7bc}
  19. Ex 4.2 Q6
    If 0.04×0.4×a=0.4×0.04×b\sqrt{0.04 \times 0.4 \times a} = 0.4 \times 0.04 \times \sqrt{b} then find the ratio ab\dfrac{a}{b}.
  20. Ex 4.2 Q7
    (x+3):(x+11)=(x−2):(x+1)(x + 3) : (x + 11) = (x - 2) : (x + 1) then find the value of xx.

Operations on Equal Ratios

8 q

Solved Examples

Worked · 8
  1. 4.3 SolvedEx.1
    If ab=53\dfrac{a}{b} = \dfrac{5}{3} then find the ratio a+7b7b\dfrac{a + 7b}{7b}.
  2. 4.3 SolvedEx.2
    If ab=74\dfrac{a}{b} = \dfrac{7}{4} then find the ratio 5a−bb\dfrac{5a - b}{b}.
  3. 4.3 SolvedEx.3
    If ab=73\dfrac{a}{b} = \dfrac{7}{3} then find the value of the ratio a+2ba−2b\dfrac{a + 2b}{a - 2b}.
  4. 4.3 SolvedEx.4
    If a3=b2\dfrac{a}{3} = \dfrac{b}{2} then find the value of the ratio 5a+3b7a−2b\dfrac{5a + 3b}{7a - 2b}.
  5. 4.3 SolvedEx.5
    If xy=45\dfrac{x}{y} = \dfrac{4}{5} then find the value of the ratio 4x−y4x+y\dfrac{4x - y}{4x + y}.
  6. 4.3 SolvedEx.6
    If 5x=4y5x = 4y then find the value of the ratio 3x2+y23x2−y2\dfrac{3x^2 + y^2}{3x^2 - y^2}.
  7. 4.3 SolvedEx.7
    Solve the equation. 3x2+5x+710x+14=3x2+4x+38x+6\dfrac{3x^2 + 5x + 7}{10x + 14} = \dfrac{3x^2 + 4x + 3}{8x + 6}
  8. 4.3 SolvedEx.8
    Solve. x+7+x−2x+7−x−2=51\dfrac{\sqrt{x + 7} + \sqrt{x - 2}}{\sqrt{x + 7} - \sqrt{x - 2}} = \dfrac{5}{1}

Practice Set 4.3

15 q
  1. If ab=73\dfrac{a}{b} = \dfrac{7}{3} then find the values of the following ratios.
    Ex 4.3 Q1(i)
    5a+3b5a−3b\dfrac{5a + 3b}{5a - 3b}
  2. Ex 4.3 Q1(ii)
    2a2+3b22a2−3b2\dfrac{2a^2 + 3b^2}{2a^2 - 3b^2}
  3. Ex 4.3 Q1(iii)
    a3−b3b3\dfrac{a^3 - b^3}{b^3}
  4. Ex 4.3 Q1(iv)
    7a+9b7a−9b\dfrac{7a + 9b}{7a - 9b}
  5. If 15a2+4b215a2−4b2=477\dfrac{15a^2 + 4b^2}{15a^2 - 4b^2} = \dfrac{47}{7} then find the values of the following ratios.
    Ex 4.3 Q2(i)
    ab\dfrac{a}{b}
  6. Ex 4.3 Q2(ii)
    7a−3b7a+3b\dfrac{7a - 3b}{7a + 3b}
  7. Ex 4.3 Q2(iii)
    b2−2a2b2+2a2\dfrac{b^2 - 2a^2}{b^2 + 2a^2}
  8. Ex 4.3 Q2(iv)
    b3−2a3b3+2a3\dfrac{b^3 - 2a^3}{b^3 + 2a^3}
  9. Ex 4.3 Q3
    If 3a+7b3a−7b=43\dfrac{3a + 7b}{3a - 7b} = \dfrac{4}{3} then find the value of the ratio 3a2−7b23a2+7b2\dfrac{3a^2 - 7b^2}{3a^2 + 7b^2}.
  10. Solve the following equations.
    Ex 4.3 Q4(i)
    x2+12x−203x−5=x2+8x+122x+3\dfrac{x^2 + 12x - 20}{3x - 5} = \dfrac{x^2 + 8x + 12}{2x + 3}
  11. Ex 4.3 Q4(ii)
    10x2+15x+635x2−25x+12=2x+3x−5\dfrac{10x^2 + 15x + 63}{5x^2 - 25x + 12} = \dfrac{2x + 3}{x - 5}
  12. Ex 4.3 Q4(iii)
    (2x+1)2+(2x−1)2(2x+1)2−(2x−1)2=178\dfrac{(2x+1)^2 + (2x-1)^2}{(2x+1)^2 - (2x-1)^2} = \dfrac{17}{8}
  13. Ex 4.3 Q4(iv)
    4x+1+x+34x+1−x+3=41\dfrac{\sqrt{4x+1} + \sqrt{x+3}}{\sqrt{4x+1} - \sqrt{x+3}} = \dfrac{4}{1}
  14. Ex 4.3 Q4(v)
    (4x+1)2+(2x+3)24x2+12x+9=6136\dfrac{(4x+1)^2 + (2x+3)^2}{4x^2 + 12x + 9} = \dfrac{61}{36}
  15. Ex 4.3 Q4(vi)
    (3x−4)3−(x+1)3(3x−4)3+(x+1)3=61189\dfrac{(3x-4)^3 - (x+1)^3}{(3x-4)^3 + (x+1)^3} = \dfrac{61}{189}

Theorem on Equal Ratios

5 q

Solved Examples

Worked · 5
  1. Fill in the blanks in the following statements.
    4.4 SolvedEx.1(i)
    a3=b7=4a+9b⋯\dfrac{a}{3} = \dfrac{b}{7} = \dfrac{4a + 9b}{\cdots}
  2. 4.4 SolvedEx.1(ii)
    x3=y5=z4=5x−3y+4z⋯\dfrac{x}{3} = \dfrac{y}{5} = \dfrac{z}{4} = \dfrac{5x - 3y + 4z}{\cdots}
  3. 4.4 SolvedEx.2
    If ax−2y+3z=by−2z+3x=cz−2x+3y\dfrac{a}{x - 2y + 3z} = \dfrac{b}{y - 2z + 3x} = \dfrac{c}{z - 2x + 3y} and x+y+z≠0x + y + z \neq 0 then prove that each ratio =a+b+c2(x+y+z)= \dfrac{a + b + c}{2(x + y + z)}.
  4. 4.4 SolvedEx.3
    If yb+c−a=zc+a−b=xa+b−c\dfrac{y}{b + c - a} = \dfrac{z}{c + a - b} = \dfrac{x}{a + b - c} then prove that az+x=bx+y=cy+z\dfrac{a}{z + x} = \dfrac{b}{x + y} = \dfrac{c}{y + z}.
  5. 4.4 SolvedEx.4
    Solve : 14x2−6x+810x2+4x+7=7x−35x+2\dfrac{14x^2 - 6x + 8}{10x^2 + 4x + 7} = \dfrac{7x - 3}{5x + 2}

Practice Set 4.4

11 q
  1. Fill in the blanks of the following.
    Ex 4.4 Q1(i)
    x7=y3=3x+5y⋯=7x−9y⋯\dfrac{x}{7} = \dfrac{y}{3} = \dfrac{3x + 5y}{\cdots} = \dfrac{7x - 9y}{\cdots}
  2. Ex 4.4 Q1(ii)
    a3=b4=c7=a−2b+3c⋯=⋯6−8+14\dfrac{a}{3} = \dfrac{b}{4} = \dfrac{c}{7} = \dfrac{a - 2b + 3c}{\cdots} = \dfrac{\cdots}{6 - 8 + 14}
  3. 5m−n=3m+4n5m - n = 3m + 4n then find the values of the following expressions.
    Ex 4.4 Q2(i)
    m2+n2m2−n2\dfrac{m^2 + n^2}{m^2 - n^2}
  4. Ex 4.4 Q2(ii)
    3m+4n3m−4n\dfrac{3m + 4n}{3m - 4n}
  5. Prove the following.
    Ex 4.4 Q3(i)
    If a(y+z)=b(z+x)=c(x+y)a(y + z) = b(z + x) = c(x + y) and out of aa, bb, cc no two of them are equal then show that, y−za(b−c)=z−xb(c−a)=x−yc(a−b)\dfrac{y - z}{a(b - c)} = \dfrac{z - x}{b(c - a)} = \dfrac{x - y}{c(a - b)}.
  6. Ex 4.4 Q3(ii)
    If x3x−y−z=y3y−z−x=z3z−x−y\dfrac{x}{3x - y - z} = \dfrac{y}{3y - z - x} = \dfrac{z}{3z - x - y} and x+y+z≠0x + y + z \neq 0 then show that the value of each ratio is equal to 1.
  7. Ex 4.4 Q3(iii)
    If ax+byx+y=bx+azx+z=ay+bzy+z\dfrac{ax + by}{x + y} = \dfrac{bx + az}{x + z} = \dfrac{ay + bz}{y + z} and x+y+z≠0x + y + z \neq 0 then show that each ratio =a+b2= \dfrac{a + b}{2}.
  8. Ex 4.4 Q3(iv)
    If y+za=z+xb=x+yc\dfrac{y + z}{a} = \dfrac{z + x}{b} = \dfrac{x + y}{c} then show that xb+c−a=yc+a−b=za+b−c\dfrac{x}{b + c - a} = \dfrac{y}{c + a - b} = \dfrac{z}{a + b - c}.
  9. Ex 4.4 Q3(v)
    If 3x−5y5z+3y=x+5zy−5x=y−zx−z\dfrac{3x - 5y}{5z + 3y} = \dfrac{x + 5z}{y - 5x} = \dfrac{y - z}{x - z} then show that every ratio =xy= \dfrac{x}{y}.
  10. Solve.
    Ex 4.4 Q4(i)
    16x2−20x+98x2+12x+21=4x−52x+3\dfrac{16x^2 - 20x + 9}{8x^2 + 12x + 21} = \dfrac{4x - 5}{2x + 3}
  11. Ex 4.4 Q4(ii)
    5y2+40y−125y+10y2−4=y+81+2y\dfrac{5y^2 + 40y - 12}{5y + 10y^2 - 4} = \dfrac{y + 8}{1 + 2y}

Continued Proportion and the k-Method

7 q

Solved Examples

Worked · 7
  1. 4.5 SolvedEx.1
    If xx is the geometric mean of 25 and 4, then find the value of xx.
  2. 4.5 SolvedEx.2
    If 4a2b4a^2b, 8ab28ab^2, pp are in continued proportion then find the value of pp.
  3. 4.5 SolvedEx.3
    Which number should be subtracted from 7, 12 and 18 such that the resultant numbers are in continued proportion?
  4. 4.5 SolvedEx.4
    If ab=cd\dfrac{a}{b} = \dfrac{c}{d} then show that 5a−3c5b−3d=7a−2c7b−2d\dfrac{5a - 3c}{5b - 3d} = \dfrac{7a - 2c}{7b - 2d}.
  5. 4.5 SolvedEx.5
    If aa, bb, cc are in continued proportion then show that (a+b)2ab=(b+c)2bc\dfrac{(a+b)^2}{ab} = \dfrac{(b+c)^2}{bc}.
  6. 4.5 SolvedEx.6
    If aa, bb, cc are in continued proportion then show that ac=a2+ab+b2b2+bc+c2\dfrac{a}{c} = \dfrac{a^2 + ab + b^2}{b^2 + bc + c^2}.
  7. 4.5 SolvedEx.7
    Five numbers are in continued proportion. The first term is 5 and the last term is 80. Find these numbers.

Practice Set 4.5

8 q
  1. Ex 4.5 Q1
    Which number should be subtracted from 12, 16 and 21 so that resultant numbers are in continued proportion?
  2. Ex 4.5 Q2
    If (28−x)(28 - x) is the mean proportional of (23−x)(23 - x) and (19−x)(19 - x) then find the value of xx.
  3. Ex 4.5 Q3
    Three numbers are in continued proportion, whose mean proportional is 12 and the sum of the remaining two numbers is 26, then find these numbers.
  4. Ex 4.5 Q4
    If (a+b+c)(a−b+c)=a2+b2+c2(a + b + c)(a - b + c) = a^2 + b^2 + c^2 show that aa, bb, cc are in continued proportion.
  5. If ab=bc\dfrac{a}{b} = \dfrac{b}{c} and aa, bb, c>0c > 0 then show that,
    Ex 4.5 Q5(i)
    (a+b+c)(b−c)=ab−c2(a + b + c)(b - c) = ab - c^2
  6. Ex 4.5 Q5(ii)
    (a2+b2)(b2+c2)=(ab+bc)2(a^2 + b^2)(b^2 + c^2) = (ab + bc)^2
  7. Ex 4.5 Q5(iii)
    a2+b2ab=a+cb\dfrac{a^2 + b^2}{ab} = \dfrac{a + c}{b}
  8. Ex 4.5 Q6
    Find mean proportional of x+yx−y\dfrac{x + y}{x - y}, x2−y2x2y2\dfrac{x^2 - y^2}{x^2y^2}.

Problem Set 4

40 q
  1. Select the appropriate alternative answer for the following questions.
    Prob Q1(i)
    If 6:5=y:206 : 5 = y : 20 then what will be the value of yy ?
    1. A.
      15
    2. B.
      24
    3. C.
      18
    4. D.
      22.5
  2. Prob Q1(ii)
    What is the ratio of 1 mm to 1 cm ?
    1. A.
      1:1001 : 100
    2. B.
      10:110 : 1
    3. C.
      1:101 : 10
    4. D.
      100:1100 : 1
  3. Prob Q1(iii)
    The ages of Jatin, Nitin and Mohasin are 16, 24 and 36 years respectively. What is the ratio of Nitin's age to Mohasin's age ?
    1. A.
      3:23 : 2
    2. B.
      2:32 : 3
    3. C.
      4:34 : 3
    4. D.
      3:43 : 4
  4. Prob Q1(iv)
    24 Bananas were distributed between Shubham and Anil in the ratio 3 : 5, then how many bananas did Shubham get ?
    1. A.
      8
    2. B.
      15
    3. C.
      12
    4. D.
      9
  5. Prob Q1(v)
    What is the mean proportional of 4 and 25 ?
    1. A.
      6
    2. B.
      8
    3. C.
      10
    4. D.
      12
  6. For the following numbers write the ratio of first number to second number in the reduced form.
    Prob Q2(i)
    21, 48
  7. Prob Q2(ii)
    36, 90
  8. Prob Q2(iii)
    65, 117
  9. Prob Q2(iv)
    138, 161
  10. Prob Q2(v)
    114, 133
  11. Write the following ratios in the reduced form.
    Prob Q3(i)
    Radius to the diameter of a circle.
  12. Prob Q3(ii)
    The ratio of diagonal to the length of a rectangle, having length 4 cm and breadth 3 cm.
  13. Prob Q3(iii)
    The ratio of perimeter to area of a square, having side 4 cm.
  14. Check whether the following numbers are in continued proportion.
    Prob Q4(i)
    2, 4, 8
  15. Prob Q4(ii)
    1, 2, 3
  16. Prob Q4(iii)
    9, 12, 16
  17. Prob Q4(iv)
    3, 5, 8
  18. Prob Q5
    aa, bb, cc are in continued proportion. If a=3a = 3 and c=27c = 27 then find bb.
  19. Convert the following ratios into percentages.
    Prob Q6(i)
    37:50037 : 500
  20. Prob Q6(ii)
    58\dfrac{5}{8}
  21. Prob Q6(iii)
    2230\dfrac{22}{30}
  22. Prob Q6(iv)
    516\dfrac{5}{16}
  23. Prob Q6(v)
    1441200\dfrac{144}{1200}
  24. Write the ratio of first quantity to second quantity in the reduced form.
    Prob Q7(i)
    1024 MB, 1.2 GB [(1024 MB = 1 GB)]
  25. Prob Q7(ii)
    17 Rupees, 25 Rupees 60 paise
  26. Prob Q7(iii)
    5 dozen, 120 units
  27. Prob Q7(iv)
    4 sq.m, 800 sq.cm
  28. Prob Q7(v)
    1.5 kg, 2500 gm
  29. If ab=23\dfrac{a}{b} = \dfrac{2}{3} then find the values of the following expressions.
    Prob Q8(i)
    4a+3b3b\dfrac{4a + 3b}{3b}
  30. Prob Q8(ii)
    5a2+2b25a2−2b2\dfrac{5a^2 + 2b^2}{5a^2 - 2b^2}
  31. Prob Q8(iii)
    a3+b3b3\dfrac{a^3 + b^3}{b^3}
  32. Prob Q8(iv)
    7b−4a7b+4a\dfrac{7b - 4a}{7b + 4a}
  33. If aa, bb, cc, dd are in proportion, then prove that
    Prob Q9(i)
    11a2+9ac11b2+9bd=a2+3acb2+3bd\dfrac{11a^2 + 9ac}{11b^2 + 9bd} = \dfrac{a^2 + 3ac}{b^2 + 3bd}
  34. Prob Q9(ii)
    a2+5c2b2+5d2=ab\sqrt{\dfrac{a^2 + 5c^2}{b^2 + 5d^2}} = \dfrac{a}{b}
  35. Prob Q9(iii)
    a2+ab+b2a2−ab+b2=c2+cd+d2c2−cd+d2\dfrac{a^2 + ab + b^2}{a^2 - ab + b^2} = \dfrac{c^2 + cd + d^2}{c^2 - cd + d^2}
  36. If aa, bb, cc are in continued proportion, then prove that
    Prob Q10(i)
    aa+2b=a−2ba−4c\dfrac{a}{a + 2b} = \dfrac{a - 2b}{a - 4c}
  37. Prob Q10(ii)
    bb+c=a−ba−c\dfrac{b}{b + c} = \dfrac{a - b}{a - c}
  38. Prob Q11
    Solve : 12x2+18x+4218x2+12x+58=2x+33x+2\dfrac{12x^2 + 18x + 42}{18x^2 + 12x + 58} = \dfrac{2x + 3}{3x + 2}
  39. Prob Q12
    If 2x−3y3z+y=z−yz−x=x+3z2y−3x\dfrac{2x - 3y}{3z + y} = \dfrac{z - y}{z - x} = \dfrac{x + 3z}{2y - 3x} then prove that every ratio =xy= \dfrac{x}{y}.
  40. Prob Q13
    If by+czb2+c2=cz+axc2+a2=ax+bya2+b2\dfrac{by + cz}{b^2 + c^2} = \dfrac{cz + ax}{c^2 + a^2} = \dfrac{ax + by}{a^2 + b^2} then prove that xa=yb=zc\dfrac{x}{a} = \dfrac{y}{b} = \dfrac{z}{c}.