Charlotte has 18 dollars in her piggy bank and Isabella has 11 dollars. Each week, Charlotte adds 50 cents to her piggy bank, while Isabella adds one dollar. How many weeks will it take for both girls to have the same amount of money in their respective piggy banks? What will that amount be?
Charlotte starts with 18 dollars and Isabella starts with 11 dollars. One way of solving this problem would be to work out how much money each girl has after week one, week two, week three, and so on.
As Charlotte is adding 50 cents each week, after week one, she will have 18 dollars and 50 cents. After week two, she will have 19 dollars. And after week three, she will have 19 dollars and 50 cents. Isabella adds one dollar to her piggy bank each week. So after week one, she will have 12 dollars; after week two, 13 dollars; and after week three, 14 dollars.
We could continue this process. And eventually, we would find the time where Charlotte and Isabella have the same amount of money. A much more efficient way of solving this problem would be to use algebra. If we let 𝑥 be the number of weeks, then the expression 18 plus 0.5 multiplied by 𝑥 tells us how much money Charlotte will have after 𝑥 weeks.
This is because she starts with 18 dollars and adds 50 cents or 0.5 dollars each week. Isabella started with 11 dollars and adds one dollar each week. This means that the amount of money that Isabella has is given by the expression 11 plus 𝑥.
We want to calculate how many weeks it will take for both girls to have the same amount of money. Therefore, we need to make the two expressions equal to each other. Subtracting 0.5𝑥 from both sides of this equation gives us 18 is equal to 11 plus 0.5𝑥. Subtracting 11 from both sides of the new equation gives us seven is equal to 0.5𝑥 or 0.5𝑥 equals seven. And finally, dividing both sides of this equation by 0.5 gives us 𝑥 is equal to 14.
Seven divided by 0.5 is equal to 14 as there are 14 0.5s in seven. As 𝑥 is equal to 14, we can say that it will take 14 weeks for both girls to have the same amount of money.
We were also asked to calculate the amount of money that that will be. In order to do this, we need to substitute 𝑥 equals 14 into either the expression for Isabella or Charlotte.
In this case, we will substitute 𝑥 equals 14 into the expression 11 plus 𝑥. 11 plus 14 is equal to 25. Therefore, both girls will have 25 dollars after 14 weeks.
We can check this by substituting 𝑥 equals 14 into the expression for Charlotte: 18 plus 0.5 multiplied by 14. 0.5 multiplied by 14 is equal to seven as a half of 14 equals seven. 18 plus seven is equal to 25. Therefore, we have proved that Charlotte also has 25 dollars.