Amal, Geoff, and Julia each have some fireworks. If Amal gives half of his firework to Geoff, then Amal and Geoff will have firework in a 2:3 ratio. If Julia has one-fifth as half his fireworks as Geoff (before Amal gives half his fireworks to Geoff), then what is the ratio of the number of fireworks Amal has to the number of fireworks Julia has (before Amal gives half his fireworks to Geoff)?

Options

A. `20:1`
B. `15:4`
C. `3:2`
D. There is insufficient information to determine the answer.

Answer

A

Explanation

Difficult: Hard
Category: Problem Solving and Data Analysis/ Rates, Ratios, Proportions, and Percentages
Strategic Advice: When you see an answer choice indicating that there is insufficient information to answer the question, don't automatically assume that choice is correct.
Getting to the Answer: Let A and G equal the number of fireworks that Amal and Geoff originally had. If Amal gives half of his fireworks left while Geoff, then Amal will have `(A)/(2)` fireworks left while Greoff will have `G+(A)/(2)`.You're told that these two values will be in a 2:3 ratio, so set up a proportion and cross-multiply to simplify:
`((A)/(2))/(G+(A)/(2))=(2)/(3)`
`3(A)/(2)=2(G+(A)/(2))`
`(3A)/(2)=2G+A`
`3A=2(2G+A)`
`3A=4G+2A`
`A=4G`
While this doesn't give you the exact numbers for the Geoff or Amal, if does tell you that originally Amal had 4 times as many fireworks as Geoff. If Geoff, in turn, originally had 5 times as many fireworks as julia (because the question states that Julia had (1)/(5) as many as Geoff), then Amal must originally have `4times5=20` times as many fireworks as Julia, which makes the ratio 20:1. Choice (A) is correct.