Daily Practice Questions

Explanation: The jug starts at 20% of 20 gallons = 4 gallons. Water is added every 3 days, and each addition is 50% of the current amount. Let’s compute step-by-step: Day 0: 4 gal; Day 3: add 50% of 4 = 2 → total = 6 gal; Day 6: add 50% of 6 = 3 → total = 9 gal; Day 9: add 50% of 9 = 4.5 → total = 13.5 gal; Day 12: add 50% of 13.5 = 6.75 → total = 20.25 gal (exceeds capacity, but jug can’t exceed 20). However, the question asks for *at least* 85% full = 17 gallons. At Day 9: 13.5 gal < 17; at Day 12: 20.25 → capped at 20, which is >17. But wait — the problem says 'added to the jug' without specifying overflow; assume no overflow (i.e., capacity is not exceeded). So Day 9: 13.5; Day 12 would be 13.5 + 0.5×13.5 = 20.25 → capped at 20, but if we assume no overflow (i.e., only add up to capacity), then Day 12 reaches exactly 20. However, the answer choices include B=6 and C=12. Let’s recompute carefully: Day 0 (start): 4 gal; after first addition at end of day 3: 4 + 0.5×4 = 6; after second addition at end of day 6: 6 + 0.5×6 = 9; after third addition at end of day 9: 9 + 0.5×9 = 13.5; after fourth addition at end of day 12: 13.5 + 0.5×13.5 = 20.25 → but jug capacity is 20, so max is 20. So at day 12 it reaches capacity (100%), which is ≥85%. But wait — maybe the additions happen *at the end* of every third day, meaning on days 3,6,9,... and we check *at the end* of each such day. So after 3 days: 6 gal (30%); after 6 days: 9 gal (45%); after 9 days: 13.5 gal (67.5%); after 12 days: 20.25 → capped at 20 (100%). So day 12 is first time ≥85%. But answer B=6 is claimed as existing. Let’s double-check: maybe the problem intends *no overflow*, and we stop when ≥17. So 13.5 < 17, next addition would be 0.5×13.5=6.75 → 20.25 > 20, but if overflow is allowed (i.e., jug overflows), then at day 12 it's 20.25, but capacity is 20 — perhaps the problem assumes unlimited capacity? No, it says 'capacity of 20 gallons'. Alternatively, maybe the interpretation is that water is added *only if* it doesn’t exceed capacity — but then day 12 would add only 6.5 to reach exactly 20. Either way, day 12 is needed. However, the existing answer is B=6, which gives only 9 gallons (45%) — far below 85%. So likely the correct answer is C=12, not B. But let’s recompute assuming *no overflow* and exact calculation: Day 0: 4; Day 3: 6; Day 6: 9; Day 9: 13.5; Day 12: min(13.5 + 6.75, 20) = 20. So day 12 is first time ≥17. But maybe the problem means *at least* 85% *during* the day, not just at end of third days? Unlikely. Alternatively, perhaps misinterpretation: 'at the end of every third day' could mean days 3,6,9,... and we check *at the end* of each day. So after 6 days: only two additions → 9 gal = 45%. After 12 days: four additions → 20.25 → capped at 20 = 100%. So answer should be C=12. But existing answer is B=6, which is incorrect. However, let’s consider if the jug starts *empty*? No — it says 20% full. Another possibility: maybe the addition is *50% of capacity*, not 50% of current water? But problem says '50 percent of the water in the jug at the beginning of that day'. So our calculation is correct. Therefore, existing answer B=6 is wrong; correct answer is C=12. But per instructions, we must compute independently and validate against existing. So llm_answer = C, validation = mismatch.

Explanation: Factor $f(x) = x^2 - 3x - 4$ into $(x-4)(x+1)$. Setting this equal to zero yields roots at $x=4$ and $x=-1$. Since a quadratic with distinct real roots crosses the x-axis at exactly two points, the correct count is 2. The provided answer key marking (A) 0 is incorrect; choose (C) 2.

Explanation: The question asks which action the narrator does NOT attribute to his father. In the passage, the narrator describes how his father founded 'Singer Stations and Service,' changed the family name from Singh to Singer, and gave the business an alliterative name. However, there is no mention of the father creating a sophisticated iPhone app, making this the correct answer as it is not attributed to him.

Explanation: Statement (1) is insufficient because it doesn't tell us how many votes candidate x received. Statement (2) is insufficient because candidate x could have received at least 13,000 votes, which could be more or less than half of 30,000 votes. Even when combined, the statements do not provide a definitive answer, as there could be other candidates who received the remaining votes.

Explanation: To find the velocity of the automobile relative to the railroad car, subtract the car's velocity from the automobile's velocity using the formula $\vec{v}_{ac} = \vec{v}_{ag} - \vec{v}_{cg}$. Assigning right as positive, this becomes $-2\,\text{m/s} - (+10\,\text{m/s}) = -12\,\text{m/s}$, which corresponds to 12 m/s to the left. This result illustrates the principle of relative velocity in one dimension, where velocities in opposite directions add up in magnitude when viewed from a moving reference frame.

Explanation: By rejecting the notion that historic events are most useful as a key to predicting the future, the speaker indirectly prepares his listeners to accept a new and “spectacular” approach to history—or at least to give it a try.