Lisa saved 250 dollars and is trying to save an additional 25 cents per day. Mary saved two percent more money than Lisa did and is trying to save an additional five cents per day. After how many days will the amount of money that they will have saved be equal?
If we start with Lisa, the money she’s saved will equal 250 plus the additional 25 cents per day. We write that as 0.25 times 𝑑 where 𝑑 represents the number of days since she started saving. What about for Mary? To write an equation for the amount of money she saved, we know that the initial amount Mary saved is two percent more than the money Lisa had saved. And so we take the 250 Lisa saved. And we multiply that by 1.02. We do this because Mary saved 100 percent of the amount that Lisa saved plus two percent more. The amount of money Mary has is then 102 percent more than Lisa. And we write that as a decimal as 1.02. So we multiply 250 times 1.02 which equals 255.
We could consider a second method to solve this problem. If you multiply 250 by two percent, you will get five. And what that means is that Mary saved five dollars more than Lisa. So you would add the five dollars more that Mary saved to the 250 that Lisa saved to get 255. Either way, we see that Mary started with an initial savings of 255. And we know that she saved five cents per day. We’ll write that as 0.05 times 𝑑. If we want to know how many days it will take for the amount of money they have saved to be equal, we set these two equations equal to each other. If we look for the place for Lisa’s money will be equal to Mary’s, we have 250 plus 0.25𝑑 equals 255 plus 0.05𝑑.
To solve this, we need to get 𝑑 by itself. First, we’ll subtract 250 from both sides. 255 minus 250 equals five. Bring down the 0.05𝑑. To get the 𝑑’s on the same side of the equation, we’ll subtract 0.05𝑑 from both sides. 25 cents per day minus five cents per day is 20 cents per day. Bring down the five on the right side of the equation and then divide both sides by 0.20. On the left side, we’re left with 𝑑, five divided by 0.20. Five divided by 0.20 equals 25. After 25 days, Mary and Lisa will have the same amount of money. We can check that that’s true. For Lisa, we calculate 250 plus 0.25 times 25. Then Lisa would have 256 dollars and 25 cents. And, for Mary after 25 days, 255 plus 0.05 times 25 also equals 256 dollars and 25 cents.
We’ve just looked at a method that uses algebra completely to solve this problem; this is one method. Another method would be to take the information and write these equations in such a way that we could enter them into a calculator. Instead of using 𝑠 and 𝑑, we would use 𝑦 and 𝑥. Lisa’s function would be 𝑦 equals 250 plus 0.25𝑥. And Mary’s function would be 𝑦 equals 255 plus 0.05𝑥. You can enter these two functions into a calculator and then solve for the intersection.
The intersection of these two equations will be the place that they have saved the same amount of money. The intersection of these two graphs would be 25 and then 256.25. The 𝑥-coordinate 25 tells you the number of days when they would have equal amounts of money and the 𝑦-coordinate 256.25 tells you how much money they would have at that intersection. Either way we see that after 25 days Lisa and Mary would have saved the same amount of money.