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Bunuel
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when you say Only 78 and 65 satisfy this condition how would you do the math quickly to solve which numbers work for the answer?
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Assuming Larry's rate is \(x\) apples per hour, Barry's rate will be \(3x\) apples per hour, and Garry's rate will be \(6x\) apples per hour. We need jointly consistent values for Garry's rate (apples per hour), \(6x\), and Total apples picked by Larry, which in 5 hours would be \(5x\) apples. Thus, the ratio of the value in the left column to the value in the right column must be 6 to 5. Only 78 and 65 satisfy this condition.


Correct answer:

Garry's rate (apples per hour) "78"

Total apples picked by Larry "65"
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silviao
when you say Only 78 and 65 satisfy this condition how would you do the math quickly to solve which numbers work for the answer?


Here the fastest approach is just trial and error. Since the ratio must be 6:5, test likely pairs from the list. Only 78 and 65 fit, because 78:65 = 6:5.
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Hi silviao,

Great question — here's the fast way to crack this.

First, set up the key relationship. Let Larry's rate = L.

- Barry's rate = 3L (three times Larry's)
- Barry's rate = G/2 (half of Garry's), so G = 2 × 3L = 6L

This gives you the golden ratio: Garry's rate is always [b]6 times Larry's rate.[/b]

Now, since Larry picks for 5 hours, Larry's total = 5L, which means Larry's rate = Larry's total ÷ 5.

Substitute that into the golden ratio:

Garry's rate = 6 × (Larry's total ÷ 5) = (6/5) × Larry's total

So the quick test is: Garry's rate must be exactly 6/5 of Larry's total. That's the only thing you need to check.

Now scan the answer choices6, 13, 52, 65, 78, 390 — and ask: which pair has a 6-to-5 ratio?

- 78 and 6578 ÷ 65 = 6/5

No other pair works. Try a few to confirm:
- 390 and 65390/65 = 6 (too big)
- 78 and 5278/52 = 3/2 (wrong ratio)
- 52 and 1352/13 = 4 (nope)

Only 78 and 65 have that 6-to-5 relationship.

So Garry's rate = 78 apples per hour, and Larry's total = 65 apples.

To verify: Larry's rate = 65 ÷ 5 = 13. Barry's rate = 3 × 13 = 39. Garry's rate = 2 × 39 = 78. Everything checks out.

Speed trick: derive the ratio between the two columns (here, 6/5), then scan for the pair of numbers with that ratio. That way you avoid testing every combination.

Answer: 5A, 4B
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