JEE Mathematics Ratio MCQs Set 01

Answer: (b) 18 and 24

Explanation:
1. Let the ages be 3x and 4x. Then (3x + 12)/(4x + 12) = 5/6.
2. 6(3x + 12) = 5(4x + 12) gives 18x + 72 = 20x + 60, so x = 6.
3. Ages are 18 and 24.

Teacher's Note:
1. Always introduce one unknown multiplier x for a ratio.
2. Check: 30 : 36 = 5 : 6.

Answer: (a) 110°

Explanation:
1. Let the angles be 2x, 5x, 11x. Their sum is 180°.
2. 18x = 180, so x = 10.
3. The largest angle is 11x = 110°.

Teacher's Note:
1. The angle sum of a triangle is 180°.
2. The angles are 20°, 50°, 110°.

Answer: (d) 144°

Explanation:
1. Sum of angles = 360°, so x + 2x + 3x + 4x = 10x = 360°.
2. x = 36°, and the largest angle 4x = 144°.

Teacher's Note:
1. The angle sum of a quadrilateral is 360°.
2. The angles are 36°, 72°, 108°, 144°.

Answer: (c) \( \frac{10}{7} \)

Explanation:
1. Let the numbers be 7x and 3x. Then (7x + 10)/(3x + 10) = 7/6.
2. 6(7x + 10) = 7(3x + 10) gives 42x + 60 = 21x + 70, so x = 10/21.
3. The numbers are 10/3 and 10/7; the smaller is 10/7.

Teacher's Note:
1. Halving a ratio means halving the fraction a/b.
2. Check: (10/3 + 10)/(10/7 + 10) = (40/3)/(80/7) = 7/6.

Answer: (b) 10

Explanation:
1. (38 − x)/(31 − x) = 4/3.
2. 3(38 − x) = 4(31 − x) gives 114 − 3x = 124 − 4x, so x = 10.
3. Check: 28 : 21 = 4 : 3.

Teacher's Note:
1. Cross-multiply and collect like terms.
2. Verify by substituting the answer.

Answer: (a) 7 : 5

Explanation:
1. A ratio a : b is of greater inequality when a > b.
2. Only 7 : 5 has its first term larger than the second.
3. 6 : 6 is a ratio of equality; 5 : 7 and 3 : 8 are of lesser inequality.

Teacher's Note:
1. Compare the antecedent with the consequent.
2. Greater inequality means the ratio is more than 1.

Answer: (d) 7.5%

Explanation:
1. (3/40) × 100 = 7.5.

Teacher's Note:
1. A ratio a : b is (a/b) × 100 percent.
2. 3/40 = 0.075.

Answer: (c) \( \frac{1}{6} \)

Explanation:
1. 16 2/3 = 50/3, so the fraction is (50/3)/100 = 50/300.
2. 50/300 = 1/6.

Teacher's Note:
1. Convert the mixed number to an improper fraction first.
2. 1/6 = 0.1666… = 16.67%.

Answer: (b) 8%

Explanation:
1. Savings = 7650 − 7038 = Rs. 612.
2. (612/7650) × 100 = 8.

Teacher's Note:
1. Savings percentage uses the income as the base.
2. 612/7650 = 0.08.

Answer: (a) 2.5%

Explanation:
1. First mixture: (3/8) × 100 = 37.5%.
2. Second mixture: (2/5) × 100 = 40%.
3. Difference = 2.5 percentage points.

Teacher's Note:
1. Convert each fraction to a percentage.
2. The second is larger, so it is stronger.

Answer: (d) 200

Explanation:
1. Percentage of Chinese boys = 100 − (45 + 20 + 13) = 22%.
2. 22% of the total = 44, so the total = 44 × 100/22 = 200.

Teacher's Note:
1. Find the percentage of the remaining group first.
2. Check: 22% of 200 = 44.

Answer: (c) ab

Explanation:
1. (a² + x)/(b² + x) = a/b.
2. b(a² + x) = a(b² + x), so x(b − a) = ab² − a²b = −ab(a − b).
3. Hence x = ab.

Teacher's Note:
1. Cross-multiply and factor out (a − b).
2. Test with a = 2, b = 1: x = 2 gives 6 : 3 = 2 : 1.

Answer: (b) 1 : 2 or 1 : 1

Explanation:
1. Factorise: 2x² − 2xy − xy + y² = 2x(x − y) − y(x − y) = (2x − y)(x − y).
2. So y = 2x or x = y, giving x : y = 1 : 2 or 1 : 1.

Teacher's Note:
1. Divide by y² and put w = x/y if you prefer a quadratic in w.
2. Check: 2(1) − 3(2) + 4 = 0 for x = 1, y = 2.

Answer: (a) 4 : 7

Explanation:
1. (5x − 4y) = 4(3x − 2y) gives 5x − 4y = 12x − 8y.
2. So 4y = 7x, i.e. x/y = 4/7.

Teacher's Note:
1. Cross-multiply the ratio.
2. Collect x terms on one side and y terms on the other.

Answer: (d) −1 : 1 : 1

Explanation:
1. Multiply the first by 3 and the second by 2: 3x − 6y + 9z = 0, 4x + 6y − 2z = 0. Adding gives 7x + 7z = 0, so x = −z.
2. Multiply the first by 2 and subtract from the second: 7y − 7z = 0, so y = z.
3. Hence x : y : z = −1 : 1 : 1.

Teacher's Note:
1. Verify: −1 − 2 + 3 = 0 and −2 + 3 − 1 = 0.
2. Express x and y in terms of z.

Answer: (c) 1 : 27

Explanation:
1. Put x = 2k, y = 3k, z = 4k.
2. x³ + y³ + z³ − 3xyz = (8 + 27 + 64 − 72)k³ = 27k³.
3. (x + y + z)³ = (9k)³ = 729k³, so the ratio is 27/729 = 1/27.

Teacher's Note:
1. Use the multiplier k and cancel k³.
2. Identity check: (x + y + z)(x² + y² + z² − xy − yz − zx) = 9k · 3k² = 27k³.

Answer: (b) \( \frac{bc}{a} \)

Explanation:
1. (a + c)/(b + x) = a/b.
2. a(b + x) = b(a + c), so ax = bc.
3. x = bc/a.

Teacher's Note:
1. The same ratio means the second term is scaled by the same factor (a + c)/a.
2. Check: b(a + c)/a = b + bc/a.

Answer: (a) \( \frac{9}{16} \)

Explanation:
1. LCM of 16 and 24 is 48.
2. 9/16 = 27/48 and 13/24 = 26/48.
3. Since 27 > 26, 9/16 is larger.

Teacher's Note:
1. Use the common denominator method.
2. In decimals: 0.5625 and 0.5417.

Answer: (d) smaller, because a ratio of greater inequality decreases

Explanation:
1. 5/4 = 1.25 and 4/3 ≈ 1.333.
2. So the ratio decreases, as for any a > b > 0 with a positive quantity added to both terms.

Teacher's Note:
1. Adding the same positive number to both terms moves a ratio towards 1.
2. (a + c)/(b + c) < a/b when a > b.

Answer: (c) greater

Explanation:
1. (a + c)/(b + c) − a/b = c(b − a)/(b(b + c)).
2. This is positive because b − a > 0, so the new ratio is greater.

Teacher's Note:
1. Use the difference of the two fractions.
2. For example 2/3 becomes 3/4.