Question number 24. On a certain island, frogs are always either green or blue. The number of blue frogs increased by 60% while the number of green frogs decreased by 60%. Okay, so let's stop here and just write that information down before it gets too complicated. We have the number of blue frogs and I will denote that with capital B for the old amount and with N sub B for the new amount. Likewise, the green frogs here will be denoted by G for the old amount and N sub G for the new amount. From the first sentence, we know that these are all the possible frogs. They only come in two colors. From the second sentence, we know that the number of blue frogs increased by 60%. So N sub B for the blue frogs is 60% more than the initial amount, so that's 1.6 times the initial amount, and the number of green frogs is 60% less, so 0.4 of the original amount because they decreased in population by 60%. Now let's keep reading. From the next sentence, we know that the new ratio of blue frogs to green frogs is the same as the previous ratio but in the opposite order. So let's divide the number of blue frogs by the number of green frogs and that's the same as the old ratios but in the opposite order, so green to blue. Let's keep reading. Now we have the question, by what percentage did the overall number of frogs change? So we have to divide the new number of frogs, the total, so that would be the number of blue frogs, add it to the number of green frogs, and divide by the old amounts, B plus G. And from here, we can replace for N sub B and N sub G. We have 1.6B plus 0.4G divided by B plus G. And to keep going, to simplify this to a number, we need a relationship between B and G, so we obtain that from the ratios here. We can replace here likewise, N sub B is equal to 1.6B and 0.4G is the new amount of green frogs. Then we cross-multiply and we have 0.4G squared is equal to 1.6B squared. We divide by 0.4, so 4 times into 1.6B squared, and then we take the square root on both sides, so we have that the number of green frogs is twice the number of blue frogs. And then we can make that substitution into here, so this now is equal to 1.6B plus 0.4, G is twice B, so 2B over here, and we divide by B plus 2B. After simplifying, we have first in the denominator 3B, in the numerator 1.6B plus 0.8B, that gives us 2.4B in the numerator, 3B in the denominator. Now we see that the B's will cancel nicely, and so 2.4 divided by 3 is 0.8, and versus 1.00, that's a change of 20%. And so we know that the overall number of frogs changed by 20% from the old population to the new, and the answer is B.

frog population

ratio inversion

percentage change

green frogs

blue frogs