AMC 8 2019 Problem 22

~ using the answer choices (similar to ) Let our original cost be $\$ 100.$ We are looking for a result of $\$ 84,$ then. We try 16% and see it gets us higher than 84. We try 20% and see it gets us lower than 16 but still higher than 84. We know that the higher the percent, the less the value. We try 36, as we are not progressing much, and we are close! We try 40%, and we have the answer; it worked. (OR: try (C) first to eliminate 2 answer choices) Let our original cost be $\$100$, so we are looking for a whole number of $\$84$. Also, we can see that (A), (C), and (D) give us answers with decimals while we know that (B) and (E) give us whole numbers. Therefore, we only need to try these two: (B) $\$100$ increased by 20% = $\$120$, and $\$120$ decreased by 20% = $\$96$, a whole number, and (E) $\$100$ increased by 40% = $\$140$, and $\$140$ decreased by 40% = $\$84$, a whole number. Thus, 40 % or 40 is the answer. ~ SaxStreak First find the prime factors of 84. These are 2,2,3 and 7. Using this, try the answers. For example, $100×1.2$. This is equal to 120*0.8. If you find the factors of 12 and 8, you find that there is no seven and too many 2's. So 20 is out. Using the same method, you can find that 116*0.84 is out, as 84 already has all the factors of 84. 128*0.72 and 136*0.64 both don't have seven as a factor, so 16, 28, and 36 is out. This leaves 140*0.6, which has just the right amount of factors (2^2,3, and 7). Therefore, the answer is 40