-Alan sold his bike to Dan making a 20% profit. Dan sold the bike to Carol for $180 which was a 25% profit for Dan. How much did Alan originally pay for the bike?
- A positive number is increased by 20%. This new number is now decreased by 20%. To obtain this final result the original positive number could be___.
a) left alone b) increased by 4% c) increased by 2% d) decreased by 4%
e) decreased by 2%
Dan to Carol:
C + 0.25C = $180.
1.25C = 180
C = $144. = Dan's cost.
Alan to Dan:
C + 0.2C = $144.
1.2C = 144
C = $120. = Alan's cost.
(N+0.2N)-0.2(N+0.2N) =
N+0.2N-0.2N-0.04N = 0.96N.
0.96N/N = 0.96 = 96%.
96% - 100% = -4% = 4% Decrease.
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To find the original price that Alan paid for the bike, we can use the concept of finding the original value when a percentage increase and decrease is applied.
1) Let's use a variable to represent the original price of the bike. Let's call it "x".
2) Alan sold the bike to Dan for a 20% profit. This means that Dan paid Alan 1.20 times the original price, since the original price plus a 20% profit is equal to 120% or 1.20 of the original price. So, the amount Dan paid is 1.20x.
3) Dan then sold the bike to Carol for a 25% profit. This means that Carol paid Dan 1.25 times the amount Dan paid Alan, since the original price plus a 25% profit is equal to 125% or 1.25 of the amount Dan paid Alan. So, the amount Carol paid is 1.25 * (1.20x) = 1.50x.
4) We know that the amount Carol paid is $180. So, we can set up an equation to solve for x:
1.50x = 180
5) Dividing both sides of the equation by 1.50, we get:
x = 180 / 1.50
6) Evaluating the division, we find that x = 120.
Therefore, Alan originally paid $120 for the bike.
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For the second question:
To find the possible original positive number that undergoes a 20% increase, followed by a 20% decrease resulting in the final number:
1) Let's use a variable to represent the original positive number. Let's call it "x".
2) If the original number is increased by 20%, the new number can be found by adding 20% of the original number to the original number. So, the new number is 1.20x.
3) If this new number is decreased by 20%, the final result can be found by subtracting 20% of the new number from the new number. So, the final result is (1.20x) - 0.20(1.20x).
4) Simplifying this expression, we get:
(1.20x) - (0.20 * 1.20x) = 1.20x - 0.24x = 0.96x.
5) We are looking for the expression that is equal to the original positive number x. So, we can set up an equation to solve for x:
0.96x = x
6) Dividing both sides of the equation by x, we get:
0.96 = 1
This equation is not true, so there is no value of x that satisfies the condition. In other words, there is no original positive number that, when increased by 20% and then decreased by 20%, results in the original positive number.
Therefore, the original positive number must be increased by 4% to obtain the final result. Hence, the answer is option b) increased by 4%.