(a) 15, 45, 40, 120
(b) 33, 121, 9,96
(c) 24, 28, 36, 48
(d) 32, 48, 70, 210
(e) 4, 6, 8, 12
(f) 33, 44, 75, 100
Solutions:
(a) 15, 45, 40, 120
\({15 \over 45} = {1 \over 3}\)
\({40 \over 120} ={ 1 \over 3}\)
Hence, 15: 45 = 40:120
∴ These are in proportion.
(b) 33, 121, 9, 96
\(\dfrac{33 }{ 121} = \dfrac{3 }{ 11}\)
\(\dfrac{9 }{ 96} = \dfrac{3 }{32}\)
Hence, 33:121 ≠ 9: 96
∴ These are not in proportion.
(c) 24, 28, 36, 48
\(\dfrac{24 }{28} = \dfrac{6 }{7}\)
\(\dfrac{ 36}{48 } = \dfrac{ 3}{7}\)
Hence, 24: 28 ≠ 36:48
∴ These are not in proportion.
(d) 32, 48, 70, 210
\(\dfrac{32 }{48} = \dfrac{2 }{3}\)
\(\dfrac{ 70}{210 } = \dfrac{ 1}{3}\)
Hence, 32: 48 ≠ 70: 210
∴ These are not in proportion.
(e) 4, 6, 8, 12
\(\dfrac{4 }{6} = \dfrac{2 }{3}\)
\(\dfrac{ 8}{12 } = \dfrac{2 }{3}\)
Hence, 4: 6 = 8: 12
∴ These are in proportion.
(f) 33, 44, 75, 100
\(\dfrac{33 }{44} = \dfrac{3 }{4}\)
\(\dfrac{ 75}{100 } = \dfrac{3 }{4}\)
Hence, 33:44 = 75: 100
∴ These are in proportion.
2. Write True ( T ) or False ( F ) against each of the following statements :
(a) 16 : 24 :: 20 : 30
(b) 21: 6 :: 35 : 10
(c) 12 : 18 :: 28 : 12
(d) 8 : 9 :: 24 : 27
(e) 5.2 : 3.9 :: 3 : 4
(f) 0.9 : 0.36 :: 10 : 4
Solutions:
(a) 16: 24 :: 20: 30
\({16 \over 24} = {2 \over 3}\)
\({ 20 \over 30} = {2 \over 3}\)
Hence, 16: 24 = 20: 30
Therefore, it is true.
(b) 21: 6:: 35: 10
\({21 \over 6} = {7 \over 2}\)
\({ 35 \over 10 } = {7 \over 2}\)
Hence, 21: 6 = 35: 10
Therefore, it is true.
(c) 12: 18 :: 28: 12
\({12 \over 18} = {2 \over 3}\)
\({ 28 \over 12} = {7 \over 3}\)
Hence, 12: 18 ≠ 28:12
Therefore, it is false.
(d) 8: 9:: 24: 27
\({24 \over 27} ={ (3 × 8)\over (3 × 9)} = {8 \over 9}\)
Hence, 8: 9 = 24: 27
Therefore, it is true.
(e) 5.2: 3.9:: 3: 4
\({5.2 \over 3.9} = {4\over 3}\)
Hence, 5.2: 3.9 ≠ 3: 4
Therefore, it is false.
(f) 0.9: 0.36:: 10: 4
\({0.9\over 0.36} = {90 \over 36} = {10\over 4}\)
Hence, 0.9: 0.36 = 10: 4
Therefore, it is true.
3. Are the following statements true?
(a) 40 persons : 200 persons = ₹ 15 : ₹ 75
(b) 7.5 litres : 15 litres = 5 kg : 10 kg
(c) 99 kg : 45 kg = ₹ 44 : ₹ 20
(d) 32 m : 64 m = 6 sec : 12 sec
(e) 45 km : 60 km = 12 hours : 15 hours
Solutions:
(a) 40 persons : 200 persons = ₹ 15 : ₹ 75
\({40 \over 200} = {1\over 5}\)
\({15 \over 75} = {1 \over 5 }\)
Hence, it is true.
(b) 7.5 litres : 15 litres = 5 kg : 10 kg
\({7.5 \over 15 } = { 1 \over 2 } \)
\({5 \over 10 } = { 1 \over 2 } \)
Hence, it is true.
(c) 99 kg : 45 kg = ₹ 44 : ₹ 20
\({ 99 \over 45 } = { 11 \over 5 } \)
\({ 44 \over 20 } = { 11 \over 5 } \)
Hence, it is true.
(d) 32 m : 64 m = 6 sec : 12 sec
\({32 \over 64 } = { 1 \over 2 } \)
\({6 \over 12 } = { 1 \over 2 } \)
Hence, it is true.
(e) 45 km : 60 km = 12 hours : 15 hours
\({ 45 \over 60} = {3 \over 4 } \)
\({ 12 \over15 } = { 4 \over 5} \)
Hence, it is false.
4. Determine if the following ratios form a proportion. Also, write the middle terms
and extreme terms where the ratios form a proportion.
(a) 25 cm : 1 m and ₹ 40 : ₹ 160
(b)39 litres : 65 litres and 6 bottles : 10 bottles
(c) 2 kg : 80 kg and 25 g : 625 g
(d) 200 mL : 2.5 litre and ₹ 4 : ₹ 50
Solutions:
(a) 25 cm : 1 m and ₹ 40 : ₹ 160
25 cm = \(25\over 100\) m
= 0.25 m
\({0.25 \over 1 }={ 1 \over 4 }\)
\({40 \over 160 }={ 1 \over 4 }\)
Yes, these are in proportion.
Middle terms are 1 m, ₹ 40, and Extreme terms are 25 cm, ₹ 160.
(b) 39 litres : 65 litres and 6 bottles : 10 bottles
\({39 \over 65 }={ 3 \over5 }\)
\({6 \over 10 }={ 3 \over 5 }\)
Yes, these are in proportion.
Middle terms are 65 litres, 6 bottles, and Extreme terms are 39 litres, 10 bottles.
(c) 2 kg : 80 kg and 25 g : 625 g
\({2 \over 80 }={ 1 \over 40 }\)
\({25 \over 625 }={ 1 \over 25}\)
No, these are not in proportion.
(d) 200 mL : 2.5 litre and ₹ 4 : ₹ 50
1 litre = 1000 ml
2.5 litre = 2500 ml
\({200 \over 2500 }={ 2 \over 25}\)
\({4 \over 50 }={ 2 \over 25 }\)
Yes, these are in proportion.
Middle terms are 2.5 litres, ₹ 4, and Extreme terms are 200 ml, ₹ 50.
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