Video: Solving Word Problems Involving Geometric Series

A gold mine produced 2,257 kg in the first year but decreased 14% annually. Find the amount of gold produced in the third year and the total across all 3 years. Give the answers to the nearest kg.

04:09

Video Transcript

A gold mine produced 2,257 kilograms in the first year but decreased 14 percent annually. Find the total amount of gold produced in the third year and the total across all three years. Give the answers to the nearest kilogram.

In this question, we’re given some information about a gold mine. We’re told that in the first year of production, the gold mine produces 2,257 kilograms. But we’re told that year on year, this amount is decreasing by 14 percent. The question wants us to find two things. It wants us to find the amount of gold which is produced in the third year of production, and it wants us to find the total amount produced in all three of the first years. And we need to give both of our answers to the nearest kilogram.

There’s actually two different ways we can answer this question. The first way is to directly find these values from the information given to us in the question. We’re told in the question, in the first year, the gold mine produces 2,257 kilograms of gold. We can find the amount of gold produced in the second year by remembering that this amount is going to decrease by 14 percent every year. There’s a few different ways of evaluating a decrease of 14 percent.

One way is to multiply by one minus 0.14. And it’s worth pointing out here we’re subtracting 0.14 because this is a decrease. So we need to subtract, and we get 0.14 because our rate, 𝑟, is 14 and we need to divide this by 100. What we’re really saying here is a decrease in 14 percent is the same as multiplying by 0.86. Therefore, the amount of gold produced in the mine in the second year is 2,257 multiplied by 0.86 kilograms. And we can evaluate this exactly; we get 1941.02 kilograms. And we shouldn’t round our answer until the very end of the question, so we’ll leave this in exact form.

We’re then going to want to do exactly the same for the third year. Once again, from the question, we know that the mine is going to produce 14 percent less gold in the third year than it did in the second year. So, one thing we could do is multiply the amount of gold we got in the second year by 0.86. However, it’s actually easy to just multiply our expression by 0.86. Multiplying this expression by 0.86 and simplifying, we get 2,257 multiplied by 0.86 squared kilograms. Calculating this expression exactly, we get 1669.2772 kilograms.

We can now use these three values to answer our question. First, we can find the amount of gold produced in the third year by rounding this number to the nearest kilogram. This would then give us 1,669 kilograms. Next, we can find the total amount of gold produced in three years by adding these three values together. This gives us 2,257 kilograms plus 1941.02 kilograms plus 1669.2772 kilograms. And if we evaluate this expression, we get 5867.2972 kilograms. And to the nearest kilogram, we can see our first decimal place is two, so we need to round down, giving us 5,867 kilograms.

However, what would have happened if we needed to find even more years of production? We can see that this method only really worked because we only had to calculate the first three years. If we were asked to find even more years in our example, we would need to notice something interesting. Each year we’re multiplying by a constant ratio of 0.86. And remember, in a sequence, if we’re multiplying by a constant ratio to get the next term in our sequence, we call this a geometric sequence.

So, the gold produced in our mine after 𝑛 years forms a geometric sequence with initial value 𝑎, 2,257 kilograms, and ratio 𝑟, 0.86. We can then use what we know about geometric sequences to find the amount of gold produced after 𝑛 years in our mine and the total amount of gold produced after 𝑛 years. We just substitute 𝑛 is equal to three and our values for 𝑎 and 𝑟 into the two formula to find these expressions. And after rounding, we get the same answers we had before. 𝑎 sub three will be 1,669 kilograms and 𝑆 sub three will be 5,867 kilograms.

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