Math MCQs

Factor n(n^2 - 1) as n(n - 1)(n + 1), the product of three consecutive integers. Any three consecutive integers contain a multiple of 3. Since n is odd, both n - 1 and n + 1 are even, and one of these two consecutive even numbers is a multiple of 4; therefore their product contributes a factor of 8. Hence the whole expression is always divisible by 3 x 8 = 24. It is not always divisible by 48, because n = 3 gives 3(9 - 1) = 24.

Let the quantities be x, y, and z. The weighted-average condition is 480x+576y+696z = 564(x+y+z), which simplifies to 7x = y+11z. Taking y=z=7 gives x=12, so 12:7:7 is a valid ratio. Its weighted cost is (12×480+7×576+7×696)/26 = ₹564. With three ingredients, several ratios can satisfy one mean-price equation, so the stem has been repaired to ask for a valid ratio.

Since xyz=1, we have 1/y=xz, 1/z=xy, and 1/x=yz. Rewriting the three denominators and multiplying suitable numerators and denominators converts all three fractions to a common denominator. Their numerators then add exactly to that denominator, so the total is 1. The key idea is to use xyz=1 to replace reciprocal variables by products of the other two variables.

Let the tens digit be t, so the units digit is t². Since a digit is at most 9, t can only be 1, 2, or 3. The absolute difference between a number and its reversal is 9|t - t²| = 54, so |t - t²| = 6. This gives t = 3 and units digit 9, so the number is 39. Forty percent of 39 is 0.4 × 39 = 15.6. The original source options did not include this correct value.

Let the original quantities be M and W. Since 0.70M+0.30W=0.55(M+W), we get W=5:3. The remaining milk and water are 0.30M and 0.70W, whose ratio is 0.30×5:0.70×3 = 5:7. Dividing the remaining 160 litres in the ratio 5:7 gives 66⅔ litres milk and 93⅓ litres water. The printed source answer incorrectly divided the remaining mixture in the original 5:3 ratio.

Let the individual amounts be 6x, 5x, and 4x. Since the total amounts paid to the three groups are in the ratio 12:15:16, the numbers of men, women, and children are proportional to 12/(6x):15/(5x):16/(4x)=2:3:4. With 45 people, their numbers are 10, 15, and 20. Total payment is 10(6x)+15(5x)+20(4x)=215x=₹430, so x=2. Hence each receives ₹12, ₹10, and ₹8.

The express needs 6 hours of running plus seven 3-minute intermediate stops before completing 600 km, for 381 minutes. The local's 25 km blocks take 30 minutes of running plus 1 minute stop. In 381 minutes it completes 12 blocks and begins the next, giving approximately 307.5 km. The printed source value near 307.25 km is an arithmetic/rounding defect.

A's capital-time product is 52000×12, while B's is 39000×8, giving a ratio of 2:1. Let total profit be P. B first receives commission P/4; the remaining 3P/4 is divided in the ratio 2:1, so B receives another P/4. Thus B's total is P/2=₹20000, giving total profit ₹40000. A receives two-thirds of the remaining ₹30000, or ₹20000.

Interpret 999 494/495 as the mixed number 999 + 494/495. Since 494/495 = 1 - 1/495, (999 + 494/495)×99 = (1000 - 1/495)×99 = 99000 - 1/5. Adding the leading 1/5 gives exactly 99000; dividing by 4 gives 24750. The scan visually resembles a multiplication sign, but the worked solution confirms that the source uses a mixed number.

Let the hundreds digit be x. Then the tens digit is 3x and the units digit is 4x. The original number is 100x + 30x + 4x = 134x. After interchanging tens and units, the number is 100x + 40x + 3x = 143x. Since the increase is 18, 143x - 134x = 18, so 9x = 18 and x = 2. The original number is 268, and 25% of 268 is 67.