Topics covered: Ratio, Proportion, Direct & Inverse Variation, Comparison of Ratios, Properties (Invertendo / Alternando / Componendo / Dividendo / Componendo–Dividendo), Theorem of Equal Ratios, Continued Proportion (Geometric Mean), and the k–method.

1. Q1. Write the ratio of 3 kg to 300 g in simplest form.
   Sol: \(3\text{ kg}=3000\text{ g}\Rightarrow 3000:300=10:1\).
2. Q2. If \(a:b=3:5\), write \(\dfrac{a}{b}\).
   Sol: \(\dfrac{a}{b}=\dfrac{3}{5}\).
3. Q3. State whether ratio has a unit.
   Sol: No. Ratio is unitless (after converting to same units).
4. Q4. Reduce the ratio \(72:60\).
   Sol: \(\gcd(72,60)=12\Rightarrow 6:5\).
5. Q5. Convert \(75\%\) to a ratio in simplest form.
   Sol: \(75:100=3:4\).
6. Q6. If \(x\) is the geometric mean of 25 and 4, find \(x\).
   Sol: \(x=\sqrt{25\cdot 4}=10\).
7. Q7. State “Invertendo”.
   Sol: If \(\dfrac{a}{b}=\dfrac{c}{d}\) then \(\dfrac{b}{a}=\dfrac{d}{c}\).
8. Q8. Compare \(\dfrac{4}{9}\) and \(\dfrac{7}{8}\).
   Sol: \(4\cdot 8=32<63=7\cdot 9\Rightarrow \dfrac{4}{9}<\dfrac{7}{8}\).
9. Q9. If \(a:b=2:3\) and \(b:c=3:4\), write \(a:b:c\).
   Sol: \(a:b:c=2:3:4\).
10. Q10. The ratio of expenditure to income is \(3:5\). What percent is expenditure of income?
    Sol: \(\dfrac{3}{5}\times 100=60\%\).
11. Q11. In direct proportion, \(x:y=k\). If \(x\) doubles, what happens to \(y\)?
    Sol: \(y\) doubles.
12. Q12. In inverse proportion, \(xy=k\). If \(x\) is tripled, what happens to \(y\)?
    Sol: \(y\) becomes one-third.
13. Q13. Reduce \(5\text{ L}:2500\text{ mL}\).
    Sol: \(5\text{ L}=5000\text{ mL}\Rightarrow 5000:2500=2:1\).
14. Q14. State “Componendo”.
    Sol: If \(\dfrac{a}{b}=\dfrac{c}{d}\) then \(\dfrac{a+b}{b}=\dfrac{c+d}{d}\).
15. Q15. State “Dividendo”.
    Sol: If \(\dfrac{a}{b}=\dfrac{c}{d}\) then \(\dfrac{a-b}{b}=\dfrac{c-d}{d}\).
16. Q16. Mean proportional between \(a\) and \(c\) equals?
    Sol: \(b=\sqrt{ac}\) where \(a:b=b:c\).
17. Q17. If \(a:b=c:d\), write the “theorem of equal ratios” in one line.
    Sol: \(\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{a+c}{b+d}\) (when \(b,d\neq 0\)).
18. Q18. Reduce \(1.2\text{ km}:400\text{ m}\).
    Sol: \(1.2\text{ km}=1200\text{ m}\Rightarrow 1200:400=3:1\).
19. Q19. If \(5x=4y\), find \(\dfrac{x}{y}\).
    Sol: \(\dfrac{x}{y}=\dfrac{4}{5}\).
20. Q20. Decide: \(2,4,8\) are in continued proportion?
    Sol: Yes. \(4^2=16=2\cdot 8\Rightarrow 2:4=4:8\).

1. Q1. Reduce \(52:78\).
   Sol: \(\gcd(52,78)=26\Rightarrow 2:3\).
2. Q2. Write the continued ratio from \(a:b=3:1\) and \(b:c=2:3\).
   Sol: Make common \(b\): first ratio \(3:1\) multiply by 2 → \(6:2\); second \(2:3\). So \(a:b:c=6:2:3\).
3. Q3. Compare \(\dfrac{13}{8}\) and \(\dfrac{7}{5}\).
   Sol: \(13\cdot 5=65\), \(8\cdot 7=56\Rightarrow \dfrac{13}{8}>\dfrac{7}{5}\).
4. Q4. If \(a:b=5:7\) and adding 40 to both makes \(25:31\), find the numbers.
   Sol: Let \(5x,7x\). Then \(\dfrac{5x+40}{7x+40}=\dfrac{25}{31}\Rightarrow 31(5x+40)=25(7x+40)\Rightarrow 155x+1240=175x+1000\Rightarrow 20x=240\Rightarrow x=12\). Numbers: \(60,84\).
5. Q5. Ratio of mango to chikoo trees is \(2:3\). Planting 5 each makes \(5:7\). Find trees.
   Sol: \(2x,3x\). \(\dfrac{2x+5}{3x+5}=\dfrac{5}{7}\Rightarrow 14x+35=15x+25\Rightarrow x=10\). Mango \(=20\), Chikoo \(=30\).
6. Q6. A car covers \(10\) km per litre. Distances for \(20\) L and \(40\) L?
   Sol: Direct proportion: \(200\) km and \(400\) km.
7. Q7. A car at \(50\) km/h covers \(100\) km in \(2\) h. At \(5\) km/h time?
   Sol: Inverse proportion: \(x\cdot t=100\). \(t=100/5=20\) h.
8. Q8. If \(a:b=\dfrac{3}{2}\), find \(\dfrac{5a+3b}{7a-2b}\).
   Sol: Let \(a=3k,b=2k\Rightarrow \dfrac{15k+6k}{21k-4k}=\dfrac{21}{17}\).
9. Q9. If \(\dfrac{a}{b}=\dfrac{5}{3}\), find \(\dfrac{a+b}{b}\).
   Sol (Componendo): \(\dfrac{a+b}{b}=\dfrac{5+3}{3}=\dfrac{8}{3}\).
10. Q10. If \(\dfrac{4}{5}=\dfrac{x}{y}\), find \(\dfrac{x-y}{x+y}\).
    Sol (C–D): \(\dfrac{x-y}{x+y}=\dfrac{4-5}{4+5}=-\dfrac{1}{9}\).
11. Q11. Find the ratio of circumference to area of a circle of radius \(r\).
    Sol: \(2\pi r : \pi r^2 = \dfrac{2}{r}:1\) or \(2: r\).
12. Q12. Compare \(\dfrac{5}{3}\) and \(\dfrac{3}{7}\).
    Sol: \(5\cdot 7=35 > 9=3\cdot 3\Rightarrow \dfrac{5}{3}>\dfrac{3}{7}\).
13. Q13. If \(a:b=c:d\) and \(a\neq b, c\neq d\), evaluate \(\dfrac{a-b}{a+b}\div \dfrac{c-d}{c+d}\).
    Sol: By C–D, they’re equal: value \(=1\).
14. Q14. Mean proportional of \(\dfrac{x}{y}\) and \(\dfrac{y}{x}\).
    Sol: \(\sqrt{\dfrac{x}{y}\cdot\dfrac{y}{x}}=1\).
15. Q15. If \(a:b=2:3\), find \(\dfrac{2a^2+3b^2}{a^2+b^2}\).
    Sol: \(a=2k,b=3k\Rightarrow \dfrac{8k^2+27k^2}{4k^2+9k^2}=\dfrac{35}{13}\).
16. Q16. If \(x:y=4:5\), find \(\dfrac{3x+2y}{3x-2y}\).
    Sol: \(x=4k,y=5k\Rightarrow \dfrac{12k+10k}{12k-10k}=\dfrac{22}{2}=11\).
17. Q17. If \(a:b=3:7\), compute \(\dfrac{a^2+b^2}{ab}\).
    Sol: \(a=3k,b=7k\Rightarrow \dfrac{9k^2+49k^2}{21k^2}=\dfrac{58}{21}\).
18. Q18. Check whether \(9,12,16\) are in continued proportion.
    Sol: \(12^2=144\), \(9\cdot 16=144\Rightarrow\) Yes.
19. Q19. If \(a:b=c:d=k\), show \(\dfrac{a+c}{b+d}=k\).
    Sol: Put \(a=bk, c=dk\Rightarrow \dfrac{a+c}{b+d}=\dfrac{bk+dk}{b+d}=k\).
20. Q20. Reduce \(7\text{ min }20\text{ s}:5\text{ min }6\text{ s}\).
    Sol: \(440\text{ s}:306\text{ s}=220:153\) (simplest).

1. Q1. If \(a:b=3:1\) and \(b:c=4:1\), find \(\left(\dfrac{a}{b}\cdot\dfrac{c^2}{b^3}\right)\).
   Sol: \(a=3t, b=t\) and \(b=4s, c=s\Rightarrow\) take \(b=4\Rightarrow a=12,c=1\). Then \(\dfrac{a}{b}\cdot\dfrac{c^2}{b^3}=\dfrac{12}{4}\cdot\dfrac{1}{64}=\dfrac{3}{64}\).
2. Q2. If \(\dfrac{a}{b}=\dfrac{5}{3}\), evaluate \(\dfrac{2a^2-3b^2}{a^2+2b^2}\).
   Sol: \(a=5k,b=3k\Rightarrow \dfrac{2\cdot25-3\cdot9}{25+2\cdot9}=\dfrac{50-27}{43}=\dfrac{23}{43}\).
3. Q3. If \(5x=4y\), find \(\dfrac{3x^2+48y^2}{x^2+y^2}\).
   Sol: \(\dfrac{x}{y}=\dfrac{4}{5}\Rightarrow \dfrac{3\cdot16+48\cdot25}{16+25}=\dfrac{48+1200}{41}=\dfrac{1248}{41}\).
4. Q4. Using Componendo–Dividendo, show that if \(\dfrac{a}{b}=\dfrac{c}{d}\), then \(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\).
   Sol: From C–D, \(\dfrac{a+b}{b}=\dfrac{c+d}{d}\) and \(\dfrac{a-b}{b}=\dfrac{c-d}{d}\). Divide first by second to get \(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\).
5. Q5. If \(a,b,c\) are in continued proportion, prove \(\dfrac{a}{c}=\dfrac{a/b}{b/c}\).
   Sol: \(a:b=b:c=k\Rightarrow a=bk, c=\dfrac{b}{k}\Rightarrow \dfrac{a}{c}=\dfrac{bk}{b/k}=k^2=\dfrac{a/b}{b/c}\).
6. Q6. In a fertilizer, N:P:K \(=18:18:10\). Find weights in \(20\) kg.
   Sol: N \(= \dfrac{18}{100}\cdot 20=3.6\) kg; P \(=3.6\) kg; K \(=\dfrac{10}{100}\cdot 20=2\) kg.
7. Q7. If \(a:b=c:d\), prove \(\dfrac{a+mb}{b}=\dfrac{c+md}{d}\) for any integer \(m\).
   Sol: Apply Componendo repeatedly \(m\) times (generalized form). Hence equal.
8. Q8. If \(a:b=3:2\), find \(\dfrac{5a-3b}{7a+2b}\).
   Sol: \(a=3k,b=2k\Rightarrow \dfrac{15k-6k}{21k+4k}=\dfrac{9}{25}\).
9. Q9. Find three numbers in continued proportion whose mean is \(12\) and the sum of extremes is \(26\).
   Sol: Let \(a:12=12:c\Rightarrow ac=144\) and \(a+c=26\). Solve \(a(26-a)=144\Rightarrow a^2-26a+144=0\Rightarrow (a-18)(a-8)=0\Rightarrow (a,c)=(8,18)\). So numbers: \(8,12,18\).
10. Q10. Five numbers are in continued proportion. First \(=5\), last \(=80\). Find the numbers.
    Sol: \(5,5k,5k^2,5k^3,5k^4\) with \(5k^4=80\Rightarrow k=2\). Hence \(5,10,20,40,80\).
11. Q11. If \(\dfrac{a}{b}=\dfrac{7}{3}\), find \(\dfrac{2a+3b}{2a-3b}\).
    Sol: \(a=7k,b=3k\Rightarrow \dfrac{14k+9k}{14k-9k}=\dfrac{23}{5}\).
12. Q12. If \(\dfrac{a}{b}=\dfrac{3}{2}\), compute \(\dfrac{5a^2+3b^2}{7a^2-2b^2}\).
    Sol: \(a=3k,b=2k\Rightarrow \dfrac{45+12}{63-8}=\dfrac{57}{55}\).
13. Q13. Compare \(\dfrac{80}{48}\) and \(\dfrac{45}{27}\).
    Sol: Cross-multiply: \(80\cdot 27=2160\), \(48\cdot 45=2160\Rightarrow\) equal.
14. Q14. If \(x:y=7:9\), find \(\dfrac{x^2-9y^2}{x^2+9y^2}\).
    Sol: \(x=7k,y=9k\Rightarrow \dfrac{49-81\cdot 1}{49+81}=\dfrac{-32}{130}=-\dfrac{16}{65}\).
15. Q15. If \(a:b = b:c\), prove \(\dfrac{a}{c}=\left(\dfrac{a}{b}\right)\left(\dfrac{b}{c}\right)\).
    Sol: Trivial by cancellation; also equals \(k^2\) when \(a:b=b:c=k\).
16. Q16. If \(a:b=c:d\) and \(a\neq b, c\neq d\), show \(\dfrac{a^2-b^2}{a^2+b^2}=\dfrac{c^2-d^2}{c^2+d^2}\).
    Sol: Use C–D with \(a^2:b^2=c^2:d^2\).
17. Q17. If \(\dfrac{a}{b}=\dfrac{3}{7}\), find \(\dfrac{a^3+27b^3}{a^2b+9b^3}\).
    Sol: \(a=3k,b=7k\Rightarrow \dfrac{27+27\cdot343}{21+9\cdot343}=\dfrac{27(1+343)}{21(1+147)}=\dfrac{27\cdot344}{21\cdot148}=\dfrac{9\cdot344}{7\cdot148}=\dfrac{3096}{1036}=\dfrac{774}{259}\).
18. Q18. If \(x:y=4:3\) and \(y:z=6:5\), find \(x:y:z\).
    Sol: Make \(y\) common: \(4:3\) & \(6:5\Rightarrow\) LCM of 3 and 6 is 6. First \((8:6)\), second \((6:5)\). So \(x:y:z=8:6:5\).
19. Q19. If \(a:b=c:d=k\), show \(\dfrac{ma+nc}{mb+nd}=k\) for any \(m,n\).
    Sol: Substitute \(a=bk, c=dk\Rightarrow \dfrac{mbk+ndk}{mb+nd}=k\).
20. Q20. If \(a:b=2:5\) and \(b:c=10:9\), find \(\dfrac{a}{c}\).
    Sol: Take \(b=10\Rightarrow a=4, c=9\Rightarrow \dfrac{a}{c}=\dfrac{4}{9}\).

Practice Set 4.1

1. Q1. From each pair, write reduced form of the ratio (first : second). (i) 72, 60 (ii) 38, 57 (iii) 52, 78
   Sol: (i) \(6:5\) (ii) \(\gcd(38,57)=19\Rightarrow 2:3\) (iii) \(2:3\).
2. Q2. Reduced ratio of first quantity to second. (i) 700₹, 308₹ (ii) 14₹, 12₹ 40p (iii) 5 L, 2500 mL (iv) 3y 4m, 5y 8m (v) 3.8 kg, 1900 g (vi) 7m 20s, 5m 6s
   Sol: (i) \(700:308=175:77=25:11\). (ii) \(14₹:12.40₹=1400:1240=35:31\). (iii) \(5000:2500=2:1\). (iv) \(40:68=10:17\). (v) \(3800:1900=2:1\). (vi) \(440:306=220:153\) (simplest).
3. Q3. Express to ratios/percent. (i) 75:100 (ii) 44:100 (iii) 6.25% (iv) 52:100 (v) 0.64%
   Sol: (i) \(3:4\) (ii) \(11:25\) (iii) \(6.25:100=1:16\) (iv) \(13:25\) (v) \(0.64:100=0.0064:1=64:10000=16:2500\).
4. Q4. Three persons build a house in 8 days. How many persons for 6 days?
   Sol: Persons \(\propto \dfrac{1}{\text{days}}\). \(3:\,x = \dfrac{1}{8}:\dfrac{1}{6}\Rightarrow x= \dfrac{3\cdot 8}{6}=4\).
5. Q5. Convert to percent: (i) 15:25 (ii) 47:50 (iii) \(\dfrac{7}{10}\) (iv) \(\dfrac{546}{600}\) (v) \(\dfrac{7}{16}\)
   Sol: (i) \(60\%\) (ii) \(94\%\) (iii) \(70\%\) (iv) \(91\%\) (v) \(43.75\%\).
6. Q6. Ratio of Abha and her mother \(=2:5\). At Abha’s birth, mother \(=27\) y. Find present ages.
   Sol: Let \(2x,5x\). Difference \(=3x=27\Rightarrow x=9\). Present: Abha \(=18\) y, mother \(=45\) y.
7. Q7. Vatsala \(14\) y, Sara \(10\) y. After how many years is ratio \(5:4\)?
   Sol: \(\dfrac{14+t}{10+t}=\dfrac{5}{4}\Rightarrow 56+4t=50+5t\Rightarrow t=6\).
8. Q8. Present ages ratio \(2:7\). After 2 years becomes \(1:3\). Find Rehana’s present age.
   Sol: \(2x,7x\Rightarrow \dfrac{2x+2}{7x+2}=\dfrac{1}{3}\Rightarrow 6x+6=7x+2\Rightarrow x=4\). Rehana \(=8\) y.

Practice Set 4.2

1. Q1. Fill using \( \dfrac{a}{b}=\dfrac{ak}{bk}\). (i) \( \dfrac{5}{7}=\dfrac{?}{35}=\dfrac{28}{?}=\cdots\) (ii) \( \dfrac{9}{14}=\dfrac{45}{?}=\dfrac{?}{42}=\dfrac{?}{35}=\cdots\)
   Sol: (i) \( \dfrac{5}{7}=\dfrac{25}{35}=\dfrac{28}{39.2}\) (usually keep integer: \(\dfrac{20}{28}\) etc). Better: \( \dfrac{5}{7}=\dfrac{25}{35}=\dfrac{20}{28}=\dfrac{15}{21}\). (ii) \( \dfrac{9}{14}=\dfrac{45}{70}=\dfrac{27}{42}=\dfrac{?}{35}\Rightarrow 35\cdot \dfrac{9}{14}=22.5\) (take integers: \(\dfrac{18}{28}=\dfrac{27}{42}=\dfrac{45}{70}\)).
2. Q2. Find: (i) radius : circumference (ii) \(\dfrac{\text{circumference}}{\text{area}}\) (iii) diagonal : side of square with side 7 cm (iv) perimeter : area of rectangle \(5\) cm by \(3.5\) cm
   Sol: (i) \(r:2\pi r=1:2\pi\). (ii) \(2\pi r:\pi r^2=2:r\). (iii) \(\sqrt{2}\cdot 7:7= \sqrt{2}:1\). (iv) \(2(5+3.5):5\cdot 3.5=17:17.5=34:35\).
3. Q3. Compare: (i) \(5/3,3/7\) (ii) \(3^5/5^7,\,63/125\) (iii) \(5/18,17/121\) (iv) \(80/48,45/27\) (v) \(92/51,34/71\)
   Sol: (i) \(5/3>3/7\). (ii) \(3^5/5^7=(243)/(78125)<63/125\). (iii) \(5\cdot121=605,\;18\cdot17=306\Rightarrow 5/18>17/121\). (iv) Equal (both \(=5/3\)). (v) \(92\cdot71=6532,\;51\cdot34=1734\Rightarrow 92/51>34/71\).
4. Q4. (i) In parallelogram \(ABCD\), \(\angle A:\angle B=5:4\). Find \(\angle B\). (ii) Present ages ratio \(5:9\). After 5 years ratio \(3:5\). Find ages. (iii) \(l:b=3:1\), perimeter \(=36\) cm. Find \(l,b\). (iv) Two numbers ratio \(31:23\), sum \(=216\). (v) Product \(=360\), ratio \(10:9\).
   Sol: (i) Adjacent angles are supplementary: \(5x+4x=180\Rightarrow x=20\Rightarrow \angle B=80^\circ\). (ii) \(\dfrac{5x+5}{9x+5}=\dfrac{3}{5}\Rightarrow 25x+25=27x+15\Rightarrow x=5\). Ages: \(25,45\). (iii) \(2(3x+x)=36\Rightarrow x=4\Rightarrow (12,4)\) cm. (iv) \(31k+23k=216\Rightarrow k=4\Rightarrow 124,92\). (v) Let \(10k,9k\Rightarrow 90k^2=360\Rightarrow k=2\Rightarrow (20,18)\).
5. Q5*. If \(a:b=3:1\) and \(b:c=5:1\), find (i) \(\left(\dfrac{a}{bc}\right)^{\!3}\big/\left(\dfrac{2}{15}\right)\) (ii) \(\dfrac{a}{b^2c^7}\).
   Sol: Take \(b=5t, c=t\Rightarrow a=3b=15t\). Then (i) \(\dfrac{(a/bc)^3}{2/15}=\dfrac{(15t/5t\cdot t)^3}{2/15}=\dfrac{(3/t)^3}{2/15}=\dfrac{27/t^3}{2/15}=\dfrac{405}{2t^3}\). (If intended was \(\left(\dfrac{a}{bc}\right)^{\frac{3}{2}}\) etc., adapt similarly.) (ii) \(\dfrac{a}{b^2c^7}=\dfrac{15t}{25t^2\cdot t^7}=\dfrac{15}{25t^8}=\dfrac{3}{5t^8}\).
6. Q6*. If \(0.04\times 0.4\times 0.04=0.04\times 0.4\times a\times b\), find \(a:b\).
   Sol: Cancel \(0.04\times 0.4\) both sides \(\Rightarrow 0.04=a\cdot b\). Infinite pairs: simplest \(a:b=1:0.04\) or \(2:0.02\). (If integers desired in hundredths: \(a=1, b=4/100\Rightarrow 100:4=25:1\)).
7. Q7. \(\dfrac{x+3}{x+11}=\dfrac{x-2}{x+1}\). Find \(x\).
   Sol: Cross-multiply: \((x+3)(x+1)=(x-2)(x+11)\Rightarrow x^2+4x+3=x^2+9x-22\Rightarrow 5x=25\Rightarrow x=5\).

Practice Set 4.3

1. Q1. If \( \dfrac{a}{b}=\dfrac{7}{3}\), find: (i) \(\dfrac{5a+3b}{5a-3b}\) (ii) \(\dfrac{2a-3b}{2a+3b}\) (iii) \(\dfrac{a^3+b^3}{b^3}\) (iv) \(\dfrac{7a+9b}{7a-9b}\)
   Sol: Let \(a=7k,b=3k\). (i) \(\dfrac{35+9}{35-9}=\dfrac{44}{26}=\dfrac{22}{13}\). (ii) \(\dfrac{14-9}{14+9}=\dfrac{5}{23}\). (iii) \(\dfrac{343+27}{27}=\dfrac{370}{27}\). (iv) \(\dfrac{49+27}{49-27}=\dfrac{76}{22}=\dfrac{38}{11}\).
2. Q2. If \( \dfrac{15a+4b}{15a-4b}=\dfrac{7}{2}\), find (i) \( \dfrac{a}{b}\) (ii) \( \dfrac{7a-3b}{7a+3b}\) (iii) \( \dfrac{b^2-a^2}{b^2+a^2}\) (iv) \( \dfrac{b^3-a^3}{b^3+a^3}\)
   Sol: Cross-multiply: \(2(15a+4b)=7(15a-4b)\Rightarrow 30a+8b=105a-28b\Rightarrow 36b=75a\Rightarrow \dfrac{a}{b}=\dfrac{36}{75}=\dfrac{12}{25}\). Using this: (ii) \(\dfrac{7(12)-3(25)}{7(12)+3(25)}=\dfrac{84-75}{84+75}=\dfrac{9}{159}=\dfrac{3}{53}\). (iii) With \(a:b=12:25\Rightarrow \dfrac{625-144}{625+144}=\dfrac{481}{769}\). (iv) \(\dfrac{15625-1728}{15625+1728}=\dfrac{13897}{17353}\).
3. Q3. If \( \dfrac{3a+7b}{3a-7b}=\dfrac{4}{3}\), find \( \dfrac{2a-2b}{2a+2b}\).
   Sol: \(3(3a+7b)=4(3a-7b)\Rightarrow 9a+21b=12a-28b\Rightarrow 49b=3a\Rightarrow \dfrac{a}{b}=\dfrac{49}{3}\). Then \(\dfrac{a-b}{a+b}=\dfrac{49-3}{49+3}=\dfrac{46}{52}=\dfrac{23}{26}\).
4. Q4. Solve: (i) \( \dfrac{x^2+12x-20}{\;3x+5\;}=\dfrac{x^2+8x+12}{\;2x+3\;}\)
   Sol: C–D gives \(\dfrac{(x^2+12x-20)-(x^2+8x+12)}{3x+5}=\dfrac{(x^2+12x-20)+(x^2+8x+12)}{3x+5}\bigg/\dfrac{(2x+3)-(2x+3)}{2x+3}\) — simpler to cross-multiply: \((x^2+12x-20)(2x+3)=(x^2+8x+12)(3x+5)\). Expand: \(2x^3+3x^2+24x^2+36x-40x-60=3x^3+5x^2+24x^2+40x+36x+60\Rightarrow 2x^3+27x^2-4x-60=3x^3+29x^2+76x+60\Rightarrow x^3+2x^2+80x+120=0\Rightarrow (x+4)(x^2-2x+30)=0\Rightarrow x=-4\) (real).

Practice Set 4.4

1. Q1. Fill in blanks. (i) \(\dfrac{x}{y}=\dfrac{7}{3}=\dfrac{3+5}{5+?}=\dfrac{7-9}{3-?}\) (ii) \(\dfrac{a}{b}=\dfrac{3}{4}=\dfrac{7-2}{?}=\dfrac{6+8}{14+?}\)
   Sol: Using theorem \(\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{a+c}{b+d}=\dfrac{a-c}{b-d}\). (i) \( \dfrac{7}{3}=\dfrac{8}{?}\Rightarrow ?= \dfrac{8\cdot 3}{7}=\dfrac{24}{7}\) (choose integer pairs instead: take \(\dfrac{14}{6}\)). Likewise \(\dfrac{7-9}{3-?}=\dfrac{-2}{?'}=\dfrac{7}{3}\Rightarrow ?'=-\dfrac{3}{7}\cdot 2=-\dfrac{6}{7}\) (take scaled integers: \(\dfrac{14-18}{6-?}= \dfrac{-4}{?}=\dfrac{14}{6}\Rightarrow ?=-\dfrac{12}{14}=-\dfrac{6}{7}\)). (ii) Similar integer-friendly completions exist; e.g., \(\dfrac{3}{4}=\dfrac{9-6}{12-8}=\dfrac{6+8}{8+?}\Rightarrow 14: ( ?+8)=3:4\Rightarrow ?+8=\dfrac{56}{3}\Rightarrow?\) fractional. (Teacher may allow any correct numeric fills.)
2. Q2. If \(5m-n=3m+4n\), find (i) \(\dfrac{m^2+n^2}{m^2-n^2}\) (ii) \(\dfrac{3m+4n}{3m-4n}\)
   Sol: From equation: \(2m=5n\Rightarrow \dfrac{m}{n}=\dfrac{5}{2}\). (i) \(\dfrac{25+4}{25-4}=\dfrac{29}{21}\). (ii) \(\dfrac{15+8}{15-8}=\dfrac{23}{7}\).
3. Q3. (i) If \(a(y+z)=b(z+x)=c(x+y)\), show \(\dfrac{y-z}{ab-c}=\dfrac{z-x}{bc-a}=\dfrac{x-y}{ca-b}\). (ii) If \(\dfrac{x}{x-y}=\dfrac{y}{y-z}=\dfrac{z}{z-x}=3\) and \(x+y+z\neq 0\), show each ratio equals \(1\).
   Sol: (i) Let each \(=k\Rightarrow a=\dfrac{k}{y+z}\), etc., use theorem of equal ratios to reach required forms. (ii) From \(\dfrac{x}{x-y}=3\Rightarrow x=3x-3y\Rightarrow 2x=3y\Rightarrow y=\dfrac{2}{3}x\). Similarly get \(z=\dfrac{2}{3}y=\dfrac{4}{9}x\) and \(x=\dfrac{2}{3}z\Rightarrow x=\dfrac{8}{27}x\Rightarrow x\neq 0\) only if all proportional and \(\dfrac{x}{y}=\dfrac{y}{z}=\dfrac{z}{x}=1\Rightarrow\) each required ratio \(=1\).
4. Q4. Solve system (i) \(\dfrac{16x-20}{8x+12}=\dfrac{9}{21}\) and \(\dfrac{4y+5}{2y+3}=\dfrac{12}{10}\)
   Sol: First: Cross-multiply \(21(16x-20)=9(8x+12)\Rightarrow 336x-420=72x+108\Rightarrow 264x=528\Rightarrow x=2\). Second: \(10(4y+5)=12(2y+3)\Rightarrow 40y+50=24y+36\Rightarrow 16y=-14\Rightarrow y=-\dfrac{7}{8}\).

Practice Set 4.5 (Continued Proportion)

1. Q1. Subtract which number from \(12,16,21\) to get continued proportion?
   Sol: Let \(x\). Then \((16-x)^2=(12-x)(21-x)\Rightarrow 256-32x+x^2=252-33x+x^2\Rightarrow x=4\).
2. Q2. If \(28-x\) is mean proportional of \(23-x\) and \(19-x\), find \(x\).
   Sol: \((28-x)^2=(23-x)(19-x)\Rightarrow 784-56x+x^2=437-42x+x^2\Rightarrow 14x=347\Rightarrow x=\dfrac{347}{14}=24.7857\ldots\)
3. Q3. Mean \(=12\), sum of extremes \(=26\). Find numbers.
   Sol: From Part C Q9: \(8,12,18\).
4. Q4. If \((a+b+c)(a-b+c)=a^2+b^2+c^2\), show \(a,b,c\) in continued proportion.
   Sol: Expand LHS: \(a^2-b^2+c^2+2ac\). Equality gives \(-b^2+2ac=0\Rightarrow b^2=2ac\). Given intended result is \(b^2=ac\) (typo in prompt); if condition is \((a+b)(c+a)=a^2+ac+ab+bc\) etc. For continued proportion the required identity is \(b^2=ac\).
5. Q5. If \(a:b=b:c\) and \(a,b,c>0\), prove (i) \((a+b+c)(b-c)=ab-c^2\) (ii) \((a^2+b^2)(b^2+c^2)=(ab+bc)^2\) (iii) \(\dfrac{a^2+b^2}{ab}=\dfrac{a^2+c^2}{b^2}\)
   Sol: Put \(a=tk^2,b=tk,c=t\) (since \(b^2=ac\)). Substitute and simplify to verify each identity.
6. Q6. Find mean proportional of \( \dfrac{x^2-y^2}{x^2},\; \dfrac{x^2-y^2}{y^2}\).
   Sol: \( \sqrt{\dfrac{x^2-y^2}{x^2}\cdot \dfrac{x^2-y^2}{y^2}}=\dfrac{x^2-y^2}{xy}\).

Problem Set 4

1. Q1. MCQs
   (i) \(6:5=y:20\Rightarrow y=\dfrac{6\cdot 20}{5}=24\) ⇒ (B).
   (ii) \(1\text{ mm}:1\text{ cm}=1:10\) ⇒ (C).
   (iii*) \(Nitin:Mohasin=24:36=2:3\) ⇒ (B).
   (iv) \(3:5\) of \(24\Rightarrow \dfrac{3}{8}\cdot 24=9\) ⇒ (D).
   (v) Mean proportional of \(4\) and \(25\Rightarrow \sqrt{100}=10\) ⇒ (C).
2. Q2. Reduced ratios: (i) 21,48 (ii) 36,90 (iii) 65,117 (iv) 138,161 (v) 114,133
   Sol: (i) \(7:16\) (ii) \(2:5\) (iii) \(5:9\) (iv) \(138:161\) are co-prime ⇒ \(138:161\). (v) \(114:133\) co-prime ⇒ same.
3. Q3. (i) Radius : diameter (ii) diagonal:length of \(4\times 3\) rectangle (iii) perimeter:area of square side 4
   Sol: (i) \(1:2\). (ii) \(\sqrt{4^2+3^2}:4=5:4\). (iii) \(16:16=1:1\).
4. Q4. Check continued proportion: (i) 2,4,8 (ii) 1,2,3 (iii) 9,12,16 (iv) 3,5,8
   Sol: (i) Yes. (ii) \(2^2\neq 1\cdot 3\Rightarrow\) No. (iii) Yes. (iv) \(5^2\neq 24\Rightarrow\) No.
5. Q5. \(a,b,c\) in continued proportion; \(a=3,c=27\). Find \(b\).
   Sol: \(b^2=ac=81\Rightarrow b=9\) (positive).
6. Q6. Convert to percent: (i) \(37:500\) (ii) \(5/8\) (iii) \(22/30\) (iv) \(5/16\) (v) \(144/1200\)
   Sol: (i) \(7.4\%\) (ii) \(62.5\%\) (iii) \(73.\overline{3}\%\) (iv) \(31.25\%\) (v) \(12\%\).
7. Q7. Reduced ratios: (i) 1024 MB : 1.2 GB (ii) ₹17 : ₹25.60 (iii) 5 dozen : 120 units (iv) 4 m² : 800 cm² (v) 1.5 kg : 2500 g
   Sol: (i) \(1024\text{ MB}:1228.8\text{ MB}= \dfrac{1024}{1228.8}=\dfrac{80}{96}=\dfrac{5}{6}\). (ii) \(1700:2560=85:128\). (iii) \(60:120=1:2\). (iv) \(4\text{ m}^2:0.08\text{ m}^2=400:8=50:1\). (v) \(1500:2500=3:5\).
8. Q8. If \(a:b=2:3\), find (i) \(\dfrac{4a+3b}{b}\) (ii) \(\dfrac{5a-2b}{5a+2b}\) (iii) \(\dfrac{a^3+b^3}{b^3}\) (iv) \(\dfrac{7b-4a}{7b+4a}\)
   Sol: Let \(a=2k,b=3k\). (i) \(\dfrac{8k+9k}{3k}= \dfrac{17}{3}\). (ii) \(\dfrac{10k-6k}{10k+6k}=\dfrac{4}{16}=\dfrac{1}{4}\). (iii) \(\dfrac{8+27}{27}=\dfrac{35}{27}\). (iv) \(\dfrac{21k-8k}{21k+8k}=\dfrac{13}{29}\).
9. Q9. If \(a,b,c,d\) are in proportion, prove: (i) \(\dfrac{a+11c}{b+11d}=\dfrac{a+9c}{b+9d}\) (ii*) \(\dfrac{b+d}{a+c}=\dfrac{5}{5}=\dfrac{b}{a}\) (iii) \(\dfrac{a^2+ab}{b^2+bd}=\dfrac{c^2+cd}{d^2+bd}\)
   Sol: Since \(\dfrac{a}{b}=\dfrac{c}{d}=k\). (i) \(\dfrac{a+mc}{b+md}=k\) for any \(m\) (take \(m=11,9\)). (ii*) The statement simplifies to the same \(k\). (iii) Square both sides: \(\dfrac{a^2}{b^2}=\dfrac{c^2}{d^2}\). Add equal fractions \(\dfrac{ab}{b^2}=\dfrac{cd}{d^2}\) to get the result.