JEE Mathematics Proportion MCQs Set 01

Answer: (b) 32

Explanation:
1. 7 : 8 = 28 : x, so 7x = 8 × 28 = 224.
2. x = 32.

Teacher's Note:
1. The fourth proportional d satisfies a : b = c : d.
2. d = bc/a.

Answer: (a) 6

Explanation:
1. If x is the mean proportional, 2 : x = x : 18, so x² = 36.
2. x = 6 (taking the positive value).

Teacher's Note:
1. The mean proportional is the geometric mean √(ac).
2. The arithmetic mean here would be 10.

Answer: (d) 9

Explanation:
1. 4 : 6 = 6 : x, so 4x = 36.
2. x = 9.

Teacher's Note:
1. The third proportional c to a and b satisfies b² = ac.
2. c = b²/a.

Answer: (c) 3

Explanation:
1. (3 + x) : (4 + x) = (9 + x) : (11 + x).
2. (3 + x)(11 + x) = (4 + x)(9 + x) gives 33 + 14x = 36 + 13x.
3. x = 3.

Teacher's Note:
1. Product of extremes equals product of means.
2. Check: 6 : 7 = 12 : 14.

Answer: (b) 6

Explanation:
1. With a : b = c : d, b + c = 7, a + d = 8 and bc = ad.
2. Squaring and adding: 65 + 2bc + 2ad = 49 + 64 = 113 gives bc = ad = 12.
3. Means are 3 and 4; extremes are 2 and 6. The proportion is 2 : 3 = 4 : 6 and the greatest term is 6.

Teacher's Note:
1. Use ad = bc to simplify the squares.
2. Check: 4 + 9 + 16 + 36 = 65.

Answer: (a) 5

Explanation:
1. (5 + x)/(9 + x) > 7/10 gives 50 + 10x > 63 + 7x, so x > 13/3.
2. The least integer greater than 4.33 is 5.
3. Check: 10/14 = 0.714 > 0.7, while 9/13 = 0.692 < 0.7.

Teacher's Note:
1. Cross-multiply since both denominators are positive.
2. Test the integers 4 and 5.

Answer: (d) 2 : 3 : 4

Explanation:
1. Equal ratios allow subtraction: from the equal fractions we get a + c = 2b.
2. Then (a + b + c)/(2a + b + 2c) = 3b/(5b) gives b/(a + b) = 3/5, so 2b = 3a.
3. So a : b = 2 : 3 and c = 2b − a = 4 when a = 2, b = 3: a : b : c = 2 : 3 : 4.

Teacher's Note:
1. Check with 2, 3, 4: b/(a + b) = 3/5, (a + c − b)/(b + c − a) = 3/5 and 9/15 = 3/5.
2. Use a + c = 2b to eliminate c.

Answer: (c) 30

Explanation:
1. Let the common ratio be r with b = ar, c = ar², d = ar³. Then 27/8 = r³, so r = 3/2.
2. b = 12 and c = 18, so b + c = 30.

Teacher's Note:
1. Continued proportion means a/b = b/c = c/d.
2. Check: 8/12 = 12/18 = 18/27 = 2/3.

Answer: (b) (c + d) : d

Explanation:
1. Adding 1 to both sides of a/b = c/d gives (a + b)/b = (c + d)/d.
2. This is componendo.

Teacher's Note:
1. Compare with dividendo, which uses subtraction.
2. Test with 2 : 3 = 4 : 6: 5 : 3 = 10 : 6.

Answer: (a) 5

Explanation:
1. By componendo and dividendo, (x + y)/(x − y) = (3 + 2)/(3 − 2) = 5.

Teacher's Note:
1. Put x = 3 and y = 2 to check: 5/1.
2. Use the property when ratios are given.

Answer: (d) 4

Explanation:
1. 3 × 12 = 9x, so x = 4.
2. Alternando: 3 : 9 = x : 12 gives the same result.

Teacher's Note:
1. Cross-multiply.
2. Check: 3/4 = 9/12.

Answer: (c) 12

Explanation:
1. Each ratio equals (a + c + e)/(b + d + f).
2. (a + c + e)/30 = 2/5, so a + c + e = 12.

Teacher's Note:
1. Equal ratios can be added numerator to numerator and denominator to denominator.
2. Check: 12/30 = 2/5.

Answer: (b) ab : cd

Explanation:
1. Put a = bk and c = dk.
2. (a² + b²)/(c² + d²) = b²(k² + 1)/(d²(k² + 1)) = b²/d².
3. ab/cd = b²k/(d²k) = b²/d², so the two are equal.

Teacher's Note:
1. Test with 2 : 3 = 4 : 6: 13/52 = 1/4 and 6/24 = 1/4.
2. The substitution a = bk, c = dk is the standard method.

Answer: (a) (c − d) : (c + d)

Explanation:
1. By componendo and dividendo (a + b)/(a − b) = (c + d)/(c − d).
2. Taking reciprocals (invertendo) gives (a − b)/(a + b) = (c − d)/(c + d).

Teacher's Note:
1. Check with 2 : 3 = 4 : 6: (2 − 3)/(5) = −1/5 and (4 − 6)/10 = −1/5.
2. The reciprocal of a proportion is again a proportion.

Answer: (d) x = 1

Explanation:
1. Componendo and dividendo give ((x + 3)/(x − 3))³ = (7 + 9)/(7 − 9) = −8.
2. So (x + 3)/(x − 3) = −2, hence x + 3 = −2x + 6 and x = 1.
3. Check: (1 + 27)/(9 + 27) = 28/36 = 7/9.

Teacher's Note:
1. Notice the numerator and denominator combine to (x + 3)³ and (x − 3)³.
2. The cube root of −8 is −2.

Answer: (c) \( \frac{a + c}{b + d} \)

Explanation:
1. Let a = bk and c = dk. Then (a + c)/(b + d) = k(b + d)/(b + d) = k = a/b.
2. The other combinations do not simplify to k in general.

Teacher's Note:
1. Use the constant ratio k.
2. Test with 2 : 3 = 4 : 6: 6/9 = 2/3.

Answer: (b) 10

Explanation:
1. 4 : x = x : 25 gives x² = 100.
2. x = 10.

Teacher's Note:
1. The middle term is the geometric mean √(4 × 25).
2. The arithmetic mean would be 14.5.

Answer: (a) 17 : 1

Explanation:
1. Take a = 3k and b = 4k.
2. 3a + 2b = 9k + 8k = 17k and 3a − 2b = 9k − 8k = k.
3. Ratio 17 : 1.

Teacher's Note:
1. Substitute the common multiple k.
2. The k cancels in the ratio.

Answer: (d) 29 : 20

Explanation:
1. Take x = 7 and y = 3.
2. x² + y² = 58 and x² − y² = 40.
3. 58/40 = 29/20.

Teacher's Note:
1. Always reduce the final ratio.
2. Substitute x = 7k, y = 3k; k² cancels.

Answer: (c) 7 : 5 : 9

Explanation:
1. Let a + b = 6k, b + c = 7k, c + a = 8k. Adding gives a + b + c = 10.5k.
2. a = 10.5k − 7k = 3.5k, b = 10.5k − 8k = 2.5k, c = 10.5k − 6k = 4.5k.
3. Ratio 3.5 : 2.5 : 4.5 = 7 : 5 : 9.

Teacher's Note:
1. Subtract each pair-sum from the total.
2. Check: a + b = 6k, b + c = 7k, c + a = 8k.